BACHELIER FINANCE SOCIETY Third World Congress

Bachelier Finance Society 2004 Contributed Talks

A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | P | R | S | T | U |V | W | X | Y | Z Optimal Portfolio Control for Parabolic Type Infinite-dimensional Factor Model with Power Utility Shin Ichi Aihara and A. BAGCHI We consider the optimal portfolio construction problem for maximizing a power-utility at the final time.For managing the portfolio, we control the amounts of the bank account and several bonds with different maturities. The dynamics of bond price is given through the parabolic type infinite-dimensional factor model with boundary noises. By using the dynamic programming approach, we obtain the optimal portfolio in the incomplete market. Monte Carlo Static Replication of Barrier Options Emanuele Amerio, Antonio Vulcano, Gianluca Fusai The static hedge for barrier options, initially proposed by Derman.et.al (1995), is theoretically very appealing, since no adjustment is required once the replicating portfolio is in place. However, their procedure is practically flawed in assuming a perfect knowledge of the future implied volatility surface. As a possible solution, in this paper we propose a statistical static hedge, also called Monte Carlo static replication, consisting in generating risk-neutral dynamics for the volatility surface and in determining the portfolio minimizing the tracking error, i.e. the difference between the portfolio value and the barrier option price. This minimization is accomplished by a least square approach along the barrier level, where the dependent variable is the barrier option price, while the independent variable is the replicating portfolio value. Through our statistical approach, we can use the R² coefficient in order to have a preliminary measure of the goodness of the Monte Carlo static hedge; check the convergence of the statistical replicating portfolio to the barrier option price obtained by direct simulation of the underlying; construct confidence intervals around the estimated portfolio weights and hence around the barrier option price; evaluate the performance of our methodology when employing realistic dynamics of implied volatility surfaces. Mean-variance hedging for discontinuous asset price processes Takuji Arai Our goal in this paper is to give a representation of mean-variance hedging strategy for models whose asset price process is discontinuous as an extension of Gourieroux, Laurent and Pham(1998) and Rheinlander and Schweizer(1997). However, we have to impose some additional assumptions related to the variance-optimal martingale measure. No Arbitrage Conditions and Liquidity Fabian Astic and Nizar Touzi We extend the fundamental theorem of asset pricing to the case of markets with liquidity risk. Our results generalize those obtained by Kabanov, Rásonyi and Stricker (2002, 2003) and by Schachermayer (2004) for markets with proportional transaction costs. More precisely, we generalize the notion of robust no-arbitrage and prove that it is equivalent to the existence of a so-called strictly consistent price system. Moreover, we give another generalization of the usual no-arbitrage condition that has the existence of a consistent price system as a dual characterization, which is a question that remained open even in the case of transaction costs. This concept, called weak no-arbitrage condition, roughly states that one does not systematically create an arbitrage by enlarging the "positive orthant". Accurate Yield Curve Scenarios Generation using Functional Gradient Descent Francesco Audrino, Fabio Trojani We propose a multivariate nonparametric technique for generating reliable historical yield curve scenarios and confidence intervals. The approach is based on a Functional Gradient Descent(FGD) estimation of the conditional mean vector and volatility matrix of a multivariate interest rate series. It is computationally feasible in large dimensions and it can account for non-linearities in the dependence of interest rates at all available maturities. Based on FGD we apply filtered historical simulation to compute reliable out-of-sample yield curve scenarios and confidence intervals. We back-test our methodology on daily USD bond data for forecasting horizons from 1 to 10 days. Based on several statistical performance measures we find significant evidence of a higher predictive power of our method when compared to scenarios generating techniques based on (i) factor analysis,(ii) a multivariate CCC-GARCH model, or (iii) an exponential smoothing volatility estimators as in the RiskMetrics approach. Optimal impulse control for a multidimensional cash management system with nonlinear cost functions Stefano Baccarin We consider the optimal impulse control of a multidimensional cash man- agement system which fluctuates randomly in accordance with a homogeneous diffusion process in : Holding/penalty costs are continuously incurred at a rate which is a positive function of the state of the system. The controller can at any time increase (or decrease) the cash funds by transferring money among them and by selling (or buying) short-term securities. However the presence of a fixed component in the transaction costs structure makes a continuous control unprofitable. Under general assumptions we show that the value function of the problem is the minimum solution of a quasi-variational inequality in a suitable Sobolev space. From this solution it is always possible to deduce the existence of an optimal impulse policy. Furthermore we show that the value function of the corresponding problem on a bounded domain, enlarging the domain converges to the value function in : This result allows us to prove the convergence of a numerical scheme in by using a numerical procedure already existing for problems on bounded domains. Universal Exercise Signals for American Options: A New Approach to Optimal Stopping Peter Bank We present a new approach to optimal stopping problems where the usually considered Snell envelope is replaced by the solution to a stochastic representation problem. The main advantage of this approach is that it provides what we call a universal stopping signal, i.e., a single stochastic process which simultaneously yields optimal stopping rules for a whole family of stopping problems. This result is illustrated by considering American put options on a common underlying. We also discuss numerical aspects of our approach and present an ecient algorithm to solve the central stochastic representation problem. Maximum Likelihood Estimation of Latent Affine Processes David Bates This article develops a direct filtration-based maximum likelihood methodology for estimating the parameters and realizations of latent affine processes. Filtration is conducted in the transform space of characteristic functions, with a version of Bayes' rule used for recursively updating the joint characteristic function of latent variables and the data conditional upon past data. An application to daily stock returns over 1953-96 reveals substantial divergences from EMM-based estimates; in particular, more substantial and time-varying jump risk. The implications for option prices are discussed. Optimal stopping and American options with discrete dividends and exogenous risk Anna Battauz and Maurizio Pratelli We analyze the effects of discrete dividends on option pricing. First of all, we allow for the presence of a discrepancy between the stock jump and the dividend amount, in order to be consistent with the empirical evidence of excess returns at the dividend payment date. Following Heath and Jarrow (Journal of Business, 1988) we introduce a stochastic discrepancy factor to avoid arbitrage opportunities in the market model, and characterize the set of equivalent martingale measures as well as the minimal martingale/variance optimal measure. Then, we fit the model to Italian market data and analyze the impact of the discrepancy on the prices of American call options written on such discrete-dividend paying stocks. In a continuous-time model (like Black-Scholes) the optimal stopping time of an American call option does not exist, since formally it is not possible to exercise the option at the end of the cum-dividend date. To this aim, we study an optimal stopping problem with restrictions on exercise dates, in order to circumvent the failure of the continuous-time model. Monte carlo method using malliavin calculus on poisson space for the computation of greeks. Marie-pierre Bavouzet-morel, Vlad Bally, Marouen Messaoud We use the Malliavin calculus for Poisson processes in order to compute sensibilities for European options with underlying following a jump type diffusion. The key point of the calculus is to state an integration by parts formula for general random variables: we have to define the differential operators which are involved in the weight following from this formula. Then, we compute them to perform Monte Carlo simulations of this weight. Quickest Detection of the Poisson Disorder with Exponential Delay Cost Erhan Bayraktar and Savas Dayanik We solve the Poisson disorder problem when the delay is penalized exponentially. The relevance of the quickest detection problems to the financial data analysis are discussed by Shiryaev [Mathematical finance---Bachelier Congress, 2000 (Paris), Springer Finance, Springer, Berlin, pp.~487--521]. In the Poisson disorder problem, the intensity of a Poisson process changes at some unobservable random time, and the objecive is to detect the change point as accurately as possible. The change point delimits two different regimes in which one employs distinct strategies (e.g., investment, advertising, manufacturing). We seek a stopping rule that minimizes the frequency of false alarms and an exponential (unlike previous formulations which use a linear) cost function of the detection delay. Especially in the financial applications, the exponential penalty is a more apt measure for the cost of delay because of the compounding of the investment growth. The Poisson disorder problem with a linear delay cost is studied by Peskir and Shiryaev [Advances in Finance and Stochastics, Springer, Berlin, 295-312], which is a special limiting case of ours. Arbitrage in a Discrete Version of the Wick Fractional Black-Schole Market Christian Bender and Robert J. Elliott We consider binary market models based on the discrete Wick product instead of the pathwise product and provide a sufficient criterion for the existence of an arbitrage. This arbitrage is explicitly constructed in the class of self-financing one-step buy-and-hold strategies, (i.e., the investor holds shares of the stock only at one time step). Using coefficients obtained from an approximation of a fractional Brownian motion with Hurst parameter bigger than a half, the result is applied to a discrete version of the Wick-fractional Black-Scholes market. Modeling of spot and forward/futures contracts in markets for electricity and weather Fred Benth, Steen Koekebakker, Jurate Saltyte-Benth The problem of pricing forward/futures products and options written on these in electricity and weather markets is analyzed. Empirical studies motivate the introduction of Levy processes and seasonal volatility for temperature and spot electricity. Based on such models, we derive the forward/futures prices which are based on averages of the underlying product to be delivered. A dynamics for the option premium can be derived, at least numerically. Taking the alternative Heath-Jarrow-Morton approach, we discuss no-arbitrage models when the forward/futures can be overlapping in their delivery period, and present models which we study empirically on data collected from the Norwegian power exchange NordPool. Measuring Default Risk Premia from Default Swap Rates and EDFs Antje Berndt, Rohan Douglas, Darrell Duffie, Mark Ferguson This work estimates recent default risk premia for U.S. corporate debt, based on a close relationship between default probabilities, as estimated by Moody's KMV EDFs, and default swap (CDS) market rates. The default-swap data, obtained through CIBC from 22 banks and specialty dealers, allow us to establish a strong link between actual and risk-neutral default probabilities for the 69 firms in the three sectors that we analyze: broadcasting and entertainment, healthcare, and oil and gas. We find dramatic variation over time in risk premia, from peaks in the third quarter of 2002, dropping by roughly 50\% to late 2003. This is joint work with R. Douglas, D. Duffie, M. Ferguson, and D. Schranz. Incomplete Information, Heterogeneous Beliefs and Bounded Rationality Tony Berrada We consider a continuous time pure exchange economy with incomplete information populated by agents with heterogeneous beliefs and learning rules. There is a bayesian learner who constructs an optimal filter based on the available observations and a number of other agents displaying different learning bias, namely under / over reaction and over confidence. By simulation we study the distribution of terminal and average consumption share in 2 agent (bias + unbiased) and 3 agent (bias 1, bias 2, unbiased) economies. We find that for relatively long horizons (50 years), an unbiased agent does not necessarily dominate when facing 2 other biased agents. In this model, agents, biased or not, never assign 0 probability to observable events. We show that, as a result, there are no agent with consumption equal to 0 at anytime. We also consider the effect of irrational agent on equilibrium prices and trading volume, and we find that depending on other agents learning rules the average stock ownership of the rational agent varies considerably. Finally we consider the impact of irrational agent on the equilibrium stock volatility. Replication and Mean-Variance Approaches to Pricing and Hedging of Credit Risk Tomasz Bielecki, Monique Jeanblanc, Marek Rutkowski The paper presents some methods and results related to the valuation and hedging of defaultable claims (credit-risk sensitive derivative instruments). Both the exact replication of attainable defaultable claims and the mean-variance hedging of non-attainable defaultable claims are examined. For the sake of simplicity, the general methods are then applied to simple cases of defaultable equity derivatives, rather than to the more complicated examples of real-life credit derivatives. Fractional Heath-Jarrow-Morton Models Jaya Bishwal We introduce a class of one factor Heath-Jarrow-Morton term structure models driven by fractional Brownian motion with Hurst parameter H > 1/2. This class of models is important since it captures the long memory behavior of the forward rate and it captures, as a special case, all term structure models where the short term represents a time-homogeneous univariate fractional diffusion in the equivalent fractional risk neutral economy. We introduce several fractional short rate models. We obtain the fractional bond option pricing formula. Where as in the classical case of short memory the option price depends only on the length of time to exercise, here it depends both on the the exercise time and the maturity time. We then study the asymptotic behavior of estimator of the term structure's volatility. Finally we study estimation in fractional stochastic volatility model as a filtering problem. Towards a General Theory of Good Deal Bounds Tomas Björk and Irina Slinko We consider a Markovian factor model consisting of a vector price process for traded assets as well as a multidimensional random process for non traded factors. All processes are allowed to be driven by a general marked point process (representing discrete jump events) as well as by a standard multidimensional standard Wiener process. Within this framework we provide the following results. 1. We extend the Hansen-Jagannathan bounds for the Sharpe Ratio to the point process setting. 2. We study arbitrage free good deal pricing bounds for derivative assets along the lines of Cochrane and Saa-Requejo (2000). Using martingale techniques we derive the relevant Hamilton-Jacobi-Bellman equation for the upper and lower good deal bound functions, thus extending the results from Cochrane and Saa-Requejo to the point process case. 3. In particular we study the case of a single price process driven by a scalar Wiener process as well as by a marked point process. For this case we provide a detailed analysis of the dynamic programming equation and the optimal market prices of risk. As a concrete application we present numerical results for the classic Merton jump-diffusion model. Modelling forward curves for seasonal commodities Svetlana Borovkova and Heylette Geman We introduce a new way to model futures prices for seasonal commodities such as natural gas, electricity and agricultural commodities. We extend the well-known cost-of-carry relationship between spot and futures prices, by introducing a deterministic seasonal premium and a stochastic convenience yield. This leads to a better understanding of the futures prices; separating the deterministic seasonal component of the forward curve allows us to study features of futures prices, that are normally obscured by dominant seasonal effects. Our model is a two-factor model with the factors given by the average level of the forward curve and the stochastic convenience yield. We describe a method for estimating the seasonal cost-of-carry model and apply it to natural gas and electricity futures. We outline some properties of the stochastic convenience yield and illustrate them on the examples of energy futures. Applications of the model to derivatives pricing will be discussed as well. Stochastic Volatility Models: a Large Deviation Approach Phelim Boyle, Shui Feng, Weidong Tian When volatility is stochastic the market is incomplete and there is an infinite number of equivalent martingale measures. This paper analyzes the situation where we have a sequence of stochastic volatility models which converge in the limit to a complete market model with deterministic volatility. We examine the convergence of derivative prices as the stochastic volatility model converges to its deterministic limit. Using some recent results from large deviation theory we are able to demonstrate convergence for a wide class of diffusion processes. We also examine the speed of convergence and establish a theoretical result which will be used to analyze the hedging risk in stochastic volatility models in a subsequent version of this Keywords: Stochastic volatility, incomplete market, hedging, large deviations. Tractable Hedging - An Implementation of Robust Hedging Strategies Nicole Branger and Antje Mahayni This paper provides a theoretical and numerical analysis of robust hedging strategies in a diffusion-type setup including stochastic volatility models. A robust hedging strategy avoids any losses as long as the realised volatility stays within a given interval. We focus on the effects of restricting the set of admissible strategies to tractable strategies which are defined as the sum over Gaussian strategies. While the cheapest robust hedge of a mixed payoff-profile is given by a numerical solution of a stochastic control problem, firstly analyzed in Avellaneda, Levy, and Parás (1995), a tractable hedge still allows for a closed form solution. The cheapest tractable hedge can be represented by one long and one short position in convex claims where each claim is hedged separately. We show that although a trivial Gaussian hedge may be prohibitively expensive compared to the cheapest overall hedge, this is not the case for the cheapest tractable hedge. Furthermore, we use a Monte Carlo simulation to illustrate the hedging performance and the distribution of terminal losses in a stochastic volatility model. The results show that after taking the different initial capital into account, the optimal tractable strategies behave quite similar to the cheapest robust hedge. A Complete Market Model for Implied Volatility Oliver Brockhaus This paper presents a new framework for modelling equities in discrete time. Within this framework di erent smile dynamics such as "sticky delta" and "sticky strike" can be represented. The approach builds upon ideas by Schönbucher [3], Hobson and Rogers [2] and an earlier paper by the author [1] in that • implied volatility is modelled directly • the model is assumed to be complete but not Markovian • the asset distribution is only generated for a small number of deal relevant dates using implied sampling. The general framework provides a parameterisation of all arbitrage free distributions of an asset process observed at discrete times. Specific parameterisation examples allowing to match both spot and forward implied volatilities demonstrate the practical relevance of the approach. Entropic Calibration Revisited Ian Buckley, Dorje C Brody, Bernhard K Meister Whilst the entropic calibration of the risk-neutral density function is effective in recovering the strike dependence of options, it encounters difficulties in determining the relevant greeks. By use of put-call reversal we apply the entropic method to the dual, time-reversed economy, which allows us to obtain the spot price dependence of options and the relevant greeks. The failure of the calibration to satisfy global constraints can be used as a litmus test for the price consistencies in the market data, and thus can be used as a powerful tool for risk management. Numerical examples will include calibration to real market call prices at single and multiple strike prices. Wherever possible, ideas will be presented graphically and using animations. Optimal Dividend Policy with Mean-Reverting Cash Reservoir Abel  Cadenillas, Sudipto Sarkar, Fernando Zapatero Motivated by empirical evidence and economic arguments, we assume that the cash reservoir of a financial corporation follows a mean reverting process. The firm must decide the optimal dividend strategy, which consists of the optimal times and the optimal amounts to pay as dividends. We model this as an stochastic impulse control problem, and succeed in finding an analytical solution. We also find a formula for the expected time between dividend payments. A crucial and surprising result of our paper is that, as the dividend tax rate decreases, it is optimal for the shareholders to receive smaller dividend payments. [Joint work with S. Sarkar and F. Zapatero]. A Family Of Term-structure Models with Stochastic Volatility Andrew Cairns and Samuel A. Garcia Rosas In this paper we extend the class of multifactor term-structure models proposed by Cairns (2004) to incorporate a more explicit form of stochastic volatility. The models are built up within the framework proposed by Flesaker \& Hughston (1996). Our general aim is to work with models in which zero-coupon bond prices can be expressed in the form \[ P(t,T)=\frac{\int_{T-t}^\infty e^{A(u) + B(u)^T X(t)} du}{ \int_{0}^\infty e^{A(u) + B(u)^T X(t)} du} \] for some $n$-dimensional, stationary diffusion $X(t)$ and for suitable deterministic functions $A(u)$ and $B(u)$. We prove that the models require a multivariate affine state-variable $X(t)$ as developed previously by Duffie \& Kan (1996). The remainder of the paper describes some numerical experiments for specific two and three-factor models which incorporate one stochastic volatility component. The models have a close relationship with recently developed market models incorporating stochastic volatility. The new models can therefore be used to provide practitioners with a parsimonious benchmark against which more elaborate market models can be compared. Some results on quadratic hedging with insider trading Luciano Campi We consider the hedging problem in an arbitrage-free incomplete financial market, where there are two kinds of investors with different levels of information about the future price evolution of a stock, described by two filtrations $\mathbf F$ and $\mathbf G =\mathbf F \vee \sigma (G)$ where $G$ is a given r.v. representing the additional information. We focus on two types of quadratic approaches to hedge a given square-integrable contingent claim: local risk minimization (LRM) and mean-variance hedging (MVH). By using initial enlargement of filtrations techniques, we solve the hedging problem for both investors and compare their optimal strategies under both approaches. Generalised Fractional-Black-Scholes Equation: pricing and hedging Alvaro Cartea In this paper we show that when the underlying risk-neutral dynamics of securities follow either a regular Lévy process of the exponential type (RLPE) or a maximally skewed Lévy-Stable process (also known as the FMLS) it is possible to express the corresponding Black-Scholes operator in terms of fractional derivatives. Based on this Fractional-Black-Scholes operator we show how portfolios can be hedged. Finally, we show an interesting connection between factional derivatives, also known as the Riemann-Liouville operators, and the Lévy measure of the corresponding processes we use. Reservation Prices on Order Driven Markets Boyer Cécile On order driven financial markets, a limit order investor faces two types of risk: one concerns the execution of his order and the other concerns the profit when the order is executed, through the future price evolution. Investors trade for liquidity and/or speculative purpose. In any cases, any executed order is more likely to generate a negative profit ex post. This problem induces a winner's curse risk. In this paper, we define a transfer premium in order to describe the behavior of an investor facing such risks. This premium corresponds to the amount of money an investor is willing to pay for sure execution. This premium allows us to characterize the order driven market reservation prices and, as a result, to compare the prevailing spreads on quote and order driven markets. The Risk of Optimal, Continuously Rebalanced Hedging Strategies and Its Efficient Evaluation via Fourier Transform Ales Cerny This paper derives a closed-form formula for the hedging error of optimal and continuously rebalanced hedging strategies in a model with leptokurtic IID returns and, in contrast to the standard Black-Scholes result, shows that continuous hedging is far from riskless even in the absence of transaction costs. The paper provides an efficient implementation of the hedging error formula via FFT and demonstrates its speed and accuracy. We compute the size of hedging errors for individual options based on the historical distribution of returns on FT100 equity index as a function of moneyness and time to maturity. The resulting option price bounds are non-trivial, and largely insensitive to model parameters. Our result is an extension of the Capital Asset Pricing Model and the Arbitrage Pricing Theory, allowing for intertemporal risk diversification, and at the same time it represents an important generalization of the Black-Scholes pricing formula. Optimal financing policies via a stochastic control problem with exit time Roy Cerqueti The life of a firm can be influenced by several events, whose impact can change drastically the evolution of the dynamic associated to the firm's value. We propose in these pages a particular case of an equilibrium model, in which such event is represented by the external financing. The complexity of the financing modelling is due to the differences between the financiers. In the work that we present, we want to construct a model that can take in account two fundamental cases of external financiers: a bank and an illegal financier. We start from a work due to Peccati et al., but the aim of our research is different. We want to search for the best interest rate that the financier has to apply to a firm in order to catch up a certain objective. The mathematical tools adopted in order to solve the problem are associated to the stochastic control theory via dynamic programming approach. Good-deal equilibrium pricing bounds on option prices Marie Chazal and Elyés Jouini We consider the problem of finding bounds for European option prices when the underlying asset dynamics is unknown. More precisely, given a consensus on the actual distribution of the underlying asset at maturity, we derive an upper bound for the call option price which only depends on the second moment of the stock terminal value under the risk-neutral probability measure. We further restrict to equilibrium prices, i.e. that are obtained under some probability measure which has a Radon-Nikodym density with respect to the true probability measure which is a nonincreasing function of the stock price at maturity. We formulate an associated dual problem and obtain some sufficient condition for strong duality and existence to hold, in a fairly general context. Explicit bounds are provided for the call option. Finally, we provide a numerical example. DrawDown Measure in Portfolio Optimization Alexei Chekhlov, Stanislav Uryasev, and Michael Zabarankin A new one-parameter family of risk measures called Conditional Drawdown (CDD) has been proposed. These measures of risk are functionals of the portfolio drawdown (underwater) curve considered in active portfolio management. For some value of the tolerance parameter Alpha, in the case of a single sample path, drawdown functional is defined as the mean of the worst (1 - Alpha)* 100% drawdowns. The CDD measure generalizes the notion of the drawdown functional to a multi-scenario case and can be considered as a generalization of deviation measure to a dynamic case. The CDD measure includes the Maximal Drawdown and Average Drawdown as its limiting cases. Mathematical properties of the CDD measure have been studied and efficient optimization techniques for CDD computation and solving asset-allocation problems with a CDD measure have been developed. The CDD family of risk functionals is similar to Conditional Value-at-Risk (CVaR), which is also called Mean Shortfall, Mean Excess Loss, or Tail Value-at-Risk. Some recommendations on how to select the optimal risk functionals for getting practically stable portfolios have been provided. A real-life asset-allocation problem has been solved using the proposed measures. For this particular example, the optimal portfolios for cases of Maximal Drawdown, Average Drawdown, and several intermediate cases between these two have been found. Time Series Properties of Cross-Sectional Equity Returns Gib Bassett, Chen Chen, Rong Chen We investigate time series properties of the cross sectional distributions of equity returns. Changes in cross sectional dispersion affect the ability of portfolio managers to exploit skill. It also changes portfolio risk. The analysis is based on monthly returns of the 1000 largest stocks on the NYSE from 1991 to 2003. Dispersion is measured by the variance, and also by differences in the extreme quantiles. We show that the quantile analysis picks up properties of the changing distribution that are missed by focusing on just the variance. We find that: (1) dispersion has varied widely over the decade; (2) there is short-term predictability in dispersion; and (3) dispersion is associated with market volatility. We suggest theoretical reasons for the connection between dispersion and volatility and indicate how it is related to the cross sectional dispersion of market ?s and ?s. On Modeling Firm-Specific Correlations between Bonds and Stocks Li Chen, Damir Filipovic, H. Vincent Poor In this paper, a model is presented to establish the correlation between the returns on individual stocks and the yield changes of individual bonds issued by the same firm. The dynamics of a firm's credit migration, earnings process and default event are jointly characterized in a general affine setting together with consideration of market risk. This new modeling strategy provides a unifying framework of pricing corporate bonds and stocks subject to default risk. Closed-form price formulas for both securities are derived. Furthermore, the model is implemented using integrated data from both bond and stock markets. Its empirical fitting ability and the pricing performance are investigated. It turns out that by incorporating the information from the bond market, the model exhibits a strong predictive power on stock prices. General arbitrage pricing model: probability and possibility approaches Alexander Cherny The paper has 5 main goals: 1) We propose the general arbitrage pricing model. (It is similar to the general model introduced by Harrison and Kreps, but there is a number of important differences.) This model includes various models used in the theory of pricing by arbitrage as particular cases. Within the framework of the general model, we obtain the Fundamental Theorem of Asset Pricing; the form of fair prices of a contingent claim; the form of fair prices of a controlled contingent claim (this notion is introduced in the paper). 2) The obtained general results are applied to several particular models (one-period model, multiperiod model, continuous-time model, etc.) The "projection" of general results on these models leads us, in particular, to the revision of the Fundamental Theorem of Asset Pricing in the continuous-time setting. 3) The general approach mentioned above allows us to narrow the class of risk-neutral measures (and thus to make the intervals of fair prices shrink) by taking into consideration the current prices of traded securities (options, bonds, etc.). 4) Furthermore, the obtained results are extended to models with friction, i.e. models with proportional transaction costs; restrictions on short selling; costs of short selling. 5) Finally, we introduce the possibility approach to pricing by arbitrage. When using this approach, one does not need to know the original probability measure. It is shown that all the results described above can be transferred to the possibility framework. Pricing Swap Credit Risk with Copulas Umberto Cherubini We apply copula functions to evaluate counterpart risk in swap transactions. Using copulas allows to generalise the approach proposed by Sorensen and Bollier (1994), allowing for dependence between swap rates and counterparty default. Counterpart risk is represented by a sequence of vulnerable swaptions, which are priced using Cherubini and Luciano (2002) approach. Using copulas grants maximum flexibility with the choice of term structure and default risk models, as well as with the specification of the dependence structure between interest rate and credit risk. Closed form hedging and pricing formulas are derived for extreme dependence cases and for copula functions of the Fréchet family. An empirical application based on actual market data has shown that dependence affects both the level and the slope credit spreads, particularly for the case in which a credit institution is paying fixed. The effect is reversed in the case in which the financial institution pays floating. Free boundary near the maturity for an American option on several assets Etienne Chevalier We consider an American put option on a linear function of d dividend-paying assets. The value function of this option can be viewed as the solution of a free boundary problem, which can be formulated as a variational inequality. When d=1, the behavior of the free boundary near the maturity of the option is well-known: when the payoff function is regular near the limit of the boundary at maturity, the behavior is parabolic. In less regular situations however, an extra logarithmic factor appears. We will extend the study of the free boundary near maturity to the case d>1. A parametrisation of the stopping region at time t is given. That enables us to define and give a convergence rate for this region when t goes to the maturity. A General Benchmark Model for Stochastic Jump Sizes Morten Christensen and Eckhard Platen This paper extends the benchmark framework of Platen (2004). It introduces a sequence of incomplete markets, having uncertainty driven by an m-dimensional Wiener Process and a marked point process. By introducing an idealized market, in which all fundamental economical variables exist, but may not all be traded, a generalized growth optimal portfolio (GOP) is obtained and calculated explicitly. The problem of determining the GOP is solved in a general setting, which extends existing treatments. It also provides a clear link to a fundamental variable of the economy, the market price of risk. The connection between traded securities, arbitrage and market incompleteness is analyzed. This paper aims to provide a framework for analyzing the degree of incompleteness associated with jump processes, a problem well known from insurance and credit risk modelling. Furthermore, by staying under the empirical measure, the resulting benchmark model has potential advantages for various applications in finance and insurance. Reset and Withdrawal Rights in Dynamic Fund Protection Chi Chiu Chu and Yue-Kuen KWOK We analyze the nature of the dynamic fund protection which provides an investment fund with a floor level of protection against a reference stock index (or stock price). The dynamic protection feature entitles the investor the right to reset the value of his investment fund to that of the reference stock index. The reset may occur automatically whenever the investment fund value falls below that of the reference stock index, or only allowed at pre-determined time instants. The protected funds may allow a finite number of resets throughout the life of the fund, where the reset times are chosen optimally by the investor. We examine the relation between the finite-reset funds and automatic-reset funds. We also analyze the premium and the associated exercise policy of the embedded withdrawal right in protected funds, where the investor has the right to withdraw the fund prematurely. The impact of proportional fees on the optimal withdrawal policies is also analyzed. The holder should optimally withdraw at a lower critical fund value when the rate of proportional fees increases. Under the assumption that the fund value and index value follow the Geometric Brownian processes, we compute the grant date and mid-contract valuation of these protected funds. Pricing properties of the protected fund value and the cost to the sponsor are also discussed. Mispricing of S&P 500 Index Options Jens Carsten Jackwerth, George M. Constantinides, Stylianos Perrakis We address the implied volatility smile of the S&P 500 index options before and after the October 1987 crash. In a single-period model, the cross sections of one-month calls and puts violate stochastic dominance even with realistic bid-ask spreads and transaction costs on the options and the index (SPDRs). The violations are frequent even prior to the crash, in contrast to the extant literature that considers the post-crash pronounced smile to be the primary challenge to economic theory. In a multiperiod model, we address potential stochastic dominance violations via the bounds on bid and ask option prices developed by Constantinides and Perrakis (2002) because the linear programming methodology applied in the one-period model provides unusably weak restrictions. We find that the violations of stochastic dominance persist both before and after the crash. By Force of Habit: An Exploration of Asset Pricing Models using Analytic Methods Thomas  Cosimano, Yu Chen, Alex Himonas Classical analysis is used to explore solution methods for asset pricing models. Campbell and Cochrane's (1999) habit persistence model provides a prototypical example to illustrate these methods. We show that the integral equation for the price-dividend function yields a unique, bounded, continuous and infinitely differentiable solution. Using real analysis we are able to demonstrate that the price-dividend solution is analytic within a small interval. Switching to complex analysis we are able to find the maximum radius of convergence so that the price-dividend function may be portrayed accurately as a Taylor series for any consumption growth within [-16.25%, 16.25%] per month. I have a preference for presenting this paper on or after Thursday afternoon, since I have to teach in South Bend on thursday morning. Thanks. Tom Cosimano A Comparison between the SSRD Model and a Market Model for CDS Options Pricing Laurent Cousot and Damiano Brigo We investigate implied volatility patterns in the Shifted Square Root Diffusion (SSRD) model as functions of the model parameters. We introduce a candidate market model for Credit Default Swap (CDS) options that is consistent with a market Black-like formula. We introduce an analytical approximation for the SSRD implied volatility that follows the same patterns in the model parameters and that can be used to have a first rough estimate of the implied volatility following a calibration. We also find an increasing CDS-rate volatility smile for the adopted SSRD model, with one exception in one negative interest-rate / intensity correlation. We find a decreasing pattern of SSRD implied volatilities in the correlation. We finally find one testing set out of four where correlation has a possibly relevant impact on CDS option prices and on the implied CDS volatility smile shape. This work is joint with Dr. Damiano Brigo. Optimal Contracts and Principal-Agent Problems in Continuous Time Jaksa Cvitanic, Abel Cadenillas, Fernando Zapatero Principal-agent problems involve an interaction between two parties to a contract: an agent and a principal. Through the contract, the principal tries to induce the agent to act in line with the principal's interests. In our framework, the agent can control both the drift (the "mean") and the volatility (the "variance") of the underlying stochastic process. The question is: What is the optimal contract from the principal's point of view, among all the contracts that the agent is willing to consider? Main applications include optimal reward of portfolio managers and optimal compensation of company executives. There are two main modeling frameworks for these problems: 1) the case of full information; 2) the case of hidden information in which the principal cannot observe the agent's actions. In the first case we develop a methodology based on martingale/duality methods for stochastic optimization, that can be applied to general semimartingale models. In the second case we assume that the randomness is driven by Brownian Motion, and we derive necessary and sufficient conditions (the "stochastic maximum principle") for the problem, in the form of Forward-Backward Stochastic Differential Equations. The latter methodology covers a number of less general frameworks and examples considered in the existing literature. The Swing Option on the Stock Market Martin Dahlgren and Ralf Korn The valuation of a Swing option for stocks under the additional constraint of a minimum time distance between two different exercise times is considered. We give an explicit characterization of its pricing function as the value function of a multiple optimal stopping problem. The solution of this problem is related to a system of variational inequalities. We prove existence of a solution to this system and discuss the numerical implementation of a valuation algorithm. Finally we present numerical examples, which highlight the typical form of the optimal exercise strategy. Heterogenous Beliefs, Trading Risk, and the Equity Premium Alexander David Agents who have heterogeneous beliefs about the states of fundamental growth agree to not hedge each other perfectly, leading to heterogeneity in consumption growth. In addition to fundamental risks, they face trading risks, the risks of incurring losses due to equilibrium prices responding to beliefs of other agents. These trading risks are priced, causing larger premiums than in a similar benchmark economy with homogeneous beliefs. Agents with a coefficient of relative risk aversion of less than one have a `speculative' demand for risky assets in periods of high disagreement, but choose to remain in safe assets in periods of low disagreement, resulting in a low riskless rate in such times. Calibrated to fundamentals as well as survey data, the model resolves most features of the equity premium puzzle. A self exciting threshold term structure model Marc Decamps, Marc Goovaerts, Wim Schoutens One-factor models assume that all the information about the term structure of interest rates can be summarized by a single state variable which is usually the short-term rate. Among many others, the Vasicek, the CIR and the CEV models define the short rate process as a linear diffusion with continuous scale and speed densities. In this contribution, we permit the scale and speed densities to be discontinuous at some level. We derive stochastic representations for the transition probability as well as for the state-price density using the skew brownian motion and the skew three-dimensional Bessel process recently introduced by the same authors (2003). Similarly to Linetsky (2002) and Gorovoi and Linetsky (2003), we obtain eigenfunction expansions for the price of general contingent claims when analytical expressions exist for the continuous case. We interpret the resulting term structure as a continuous-time version of the Self Exciting Threshold AutoRegressive models (SETAR) popular in time series analysis. Following Goldstein and Keirstead (1997), we adapt the Heath-Jarrow-Morton procedure to forward rates with discontinuous scale density and we discuss possible generalization to Libor market models. Finally, we calibrate a SET model with two Vasicek regimes to the U.S. yield curve. Reward-Risk Portfolio Selection and Stochastic Dominance Enrico De Giorgi The portfolio selection problem is traditionally modelled by two different approaches. The first one is based on an axiomatic model of risk-averse preferences, where decision makers are assumed to possess a utility function and the portfolio choice consists in maximizing the expected utility over the set of feasible portfolios. The second approach, first proposed by Markowitz (1952), is very intuitive and reduces the portfolio choice to a set of two criteria, reward and risk, with possible tradeoff analysis. Usually the reward-risk model is not consistent with the first approach, even when the decision is independent from the specific form of the risk-averse expected utility function, i.e. when one investment dominates another one by second order stochastic dominance. In this paper we generalize the reward-risk model for portfolio selection. We define reward measures and risk measures by giving a set of properties these measures should satisfy. One of these properties will be the consistency with second order stochastic dominance, to obtain a link with the expected utility portfolio selection. We characterize reward and risk measures and we discuss the implication for portfolio selection. Unconditional Return Disturbances: a non Parametric Approach Rita DEcclesia and Robert G. Tompkins In this paper, we propose an alternative historical simulation approach. Given a historical set of data, we define a set of standardized disturbances and we generate alternative price paths by perturbing the first two moments of the original path or by reshuffling the disturbances. This approach is either totally non-parametric when constant volatility is assumed; or semi-parametric in presence of GARCH (1,1) volatility. Without a loss in accuracy, it is shown to be much more powerful in terms of computer efficiency than the Monte Carlo approach. It is also extremely simple to implement and can be an effective tool for the valuation of financial assets. We apply this approach to simulate pay off values of options on the S&P 500 stock index for the period 1982-2003. To verify that this technique works, the common back-testing approach was used. The estimated values are insignificantly different from the actual S&P 500 options payoff values for the observed period. Option valuation in a non-affine stochastic volatility jump diffusion model Griselda Deelstra and Ahmed Ezzine This paper proposes an alternative option pricing model in which the stock prices follow a diffusion process with non-affine stochastic volatility and random jumps. Our class contains and generalises the usual square root stochastic volatility model of Heston (1993) and the affine jump diffusion models. Approximative European option pricing formulae are derived by transforming a non-linear PDE in an approximate linear PDE which is explicitly solved by using Fourier transformations. We check that these approximative prices are close to the (very time-consuming) Monte Carlo estimates. We also state the original affine model corresponding to the approximations. Model parameters are estimated from joint time series of the S&P 500 index and option prices and by using the simulated method of moments. We evaluate the impact of the different submodels on option prices and on implied volatility. In particular, the relative performance based on mean-squared error (MSE) is measured for each model. We further study the sensibility of the implied volatility curves on the model parameters. Pension funds with a minimum guarantee under short selling and borrowing constraints Marina Di Giacinto and Fausto Gozzi We propose a continuous time stochastic model of optimal allocation for a defined contribution pension fund with a minimum guarantee. Traditionally, portfolio selection models are interested in maximizing the total expected discounted utility from consumption and from final wealth, whereas our target is to maximize the total expected discounted utility from current wealth. In our model the dynamics of wealth takes directly into account the flows of contributions and benefits, so that in general the portfolio is not self-financing and the level of wealth is constrained to stay above a "solvability level". The fund manager can invest in a riskless asset and in a risky asset but borrowing and short selling are prohibited. Applying dynamic programming techniques, we discuss the existence, uniqueness and regularity of the value function which solves the related Hamilton-Jacobi-Bellman equation. Lastly we obtain the existence and uniqueness of the allocation strategy in a feedback form. An algorithm for early detection of volatility change Vladimir Dobric Consider the standard geometric Brownian motion stock price model with a constant volatility and drift. The goal is to detect as soon as possible, with probability at least p, that one of the two parameters has changed within a fixed time interval L. The problem is solved using a stopping time algorithm. For the given p and L explicit upper and lower envelopes are drawn starting at the current price and into the future. If the stock price crosses either of the envelopes within the time L, with probability at least p, the crossing is due to parameters changes. The algorithm is based on new estimates of the distribution of exit times of Brownian motion through "almost" modulus of continuity boundaries. Those estimations are obtained using the natural wavelet expansions of Brownian motion. The algorithm can be modified to hold for any stock price model based on an increasing function of a Gaussian-Markov process. On the Market Price of Volatility James Doran and Ehud Ronn In this paper we examine the extent of the bias between Black-Scholes (1973)/Black (1976) implied volatility and realized term volatility. To examine this bias we institute a stochastic volatility data generating process, and demonstrate the bias through Monte Carlo simulation of the underlying parameters. This provides us with the numerical justification for testing the importance of a risk premia for volatility. We implement empirical tests for the market price of volatility risk by analyzing at-the-money options on the S&P 500 and S&P 100. Further, we extend the study by considering options on natural gas contracts. Our findings suggest a negative market price of volatility risk, and this risk contributes to the bias between Black-Scholes/Black implied volatility and realized term volatility. Asymptotic Analysis of Portfolio Trading with Transaction Costs Petr Dostal We consider an agent who invests in a stock and a money market, but he/she does not consume. We seek for a strategy of investment such that the asymptotic behavior of expected utility is as good as possible. We restrict ourselves to the utility functions with hyperbolic absolute risk aversion. Instead of considering the singular case corresponding to logarithmic utility, we maximize the asymptotic behavior of total wealth form view of ergodic theory. We restrict ourselves to such strategies that do not trade when the position of the investor is inside a given interval (a,b) and on the other hand they buy or sell the stock in order to keep this position within the interval [a,b]. This approach together with the above mentioned restrictions enable us to obtain almost explicit results. The stock market price is supposed to be a geometric Brownian motion while the transaction costs are assumed to be proportional to the amount of sales and purchases of shares. The lognormal approximation in financial and other computations Daniel Dufresne Sums of lognormals frequently appear in a variety of situations, including engineering and financial mathematics. In particular, the pricing of Asian or basket options is directly related to finding the distributions of such sums. There is no general explicit formula for the distribution of sums of lognormal random variables. This paper looks at the limit distributions of sums of lognormal variables when the second parameter of the lognormals tends to zero or to infinity; in financial terms, this is equivalent to letting the volatility, or maturity, tends either to zero or to infinity. The limits obtained are either normal or lognormal, depending on the normalization chosen; the same applies to the reciprocal of the sums of lognormals. This justifies the lognormal approximation, much used in practice, and also gives an aymptotically exact distribution for averages of lognormals with a relatively small volatility; it has been noted that all the analytical pricing formulas for Asian options perform poorly for small volatilities. Asymptotic formulas are also found for the moments of the sums of lognormals. Results are given for both discrete and continuous averages. More explicit results are obtained in the case of the integral of geometric Brownian motion. The Levy Libor Model Ernst Eberlein and Fehmi Oezkan Models driven by Lévy processes are attractive in finance because of their greater flexibility compared to classical diffusion models. First we derive conditions for arbitrage-freeness of the Libor rate process in a Lévy Heath--Jarrow--Morton setting. Then we introduce a Lévy Libor market model. In order to guarantee positive rates, the Libor rate process is constructed as an ordinary exponential. Via backward induction we get that the rates are martingales under the corresponding forward measures. An explicit formula to price caps and floors which uses bilateral Laplace transforms is derived. This is joint work with Fehmi Oezkan. Optimal Stopping Problems for Asset Management Masahiko Egami and Savas Dayanik We study two optimal stopping problems of an institutional asset manager hired by ordinary investors. The investors entrust initial funds to the asset manager and receive coupons from the asset manager; in return, the asset manager collects dividend or a management fee. The asset manager has the right to terminate the contract and to walk away with the net terminal value of the portfolio after the payment of the investors' initial funds without any responsibilities for any amount of shortfall. The asset manager's first problem is to find a nonanticipative stopping rule which maximizes her expected discounted total income. To address the possibility of the investors losing initial funds, the asset manager could offer limited protection in the form that the contract will terminate as soon as the market value of the portfolio goes below a predetermined threshold. Her second problem is to find the fair price for this protection and the best stopping rule under this additional clause. We assume that the market value of the asset manager's portfolio follows a geometric Brownian motion subject to instantaneous downward jumps with jump times arriving according to an independent Poisson process. The problems and the setting are motivated by those faced by the managers for the portfolios of defaultable bonds as in collateralized debt obligations (CDOs). Pricing Claims on Non Tradable Assets Robert Elliott and John van der Hoek A discrete two time period model is considered where there are both tradable and non tradable assets. The indifference bid and ask prices for the non tradable asset are determined for general utilities. The results generalize those of Musiela and Zariphopoulou. Properties of European and American barrier options Jonatan Eriksson We investigate monotonicity in the volatility and convexity in the underlying asset for barrier option prices when using a time and level-dependent volatility. It is well known that for non-barrier contracts, convexity of the contract function implies that the price is increasing in volatility and convex in the underlying asset. We show that under some additional conditions depending on the interest rate this result holds for certain barrier options. We give an example to illustrate the necessity of the additional conditions. Results are obtained for barrier options of both European and American type. A Series Solution for Bermudan Options Ingmar Evers This paper presents closed-form expressions for pricing Bermudan options in terms of an infinite series of standard solutions of the Black-Scholes equation. These standard solutions are combined for successive exercise dates using backward induction. At each exercise date, the optimal exercise price of the underlying asset is the root of a one-dimensional nonlinear algebraic equation. Numerical examples demonstrate the convergence of the series to the solution obtained using alternative methods. Duality and Derivative Pricing with Lévy processes José Fajardo and Ernesto Mordecki The aim of this work is to use a duality approach to study the pricing of derivatives depending on two stocks driven by a bidimensional Lévy process. The main idea is to apply Girsanov's Theorem for Lévy processes, in order to reduce the posed problem to the pricing of a one Lévy driven stock in an auxiliary market, baptized as "dual market". In this way, we extend the results obtained by Gerber and Shiu (1996) for two dimensional Brownian motion. Also we examine an existing relation between prices of put and call options, of both the European and the American type. This relation, based on a change of numeraire corresponding to a change of the probability measure through Girsanov's Theorem, is called put{call duality. It includes as a particular case, the relation known as put{call symmetry. Necessary and sufficient conditions for put-call symmetry to hold are obtained, in terms of the triplet of predictable characteristic of the Lévy process. On the Valuation of Options in Jump-Diffusion Models by Variational Methods Vadim Linetsky, Liming Feng, Michael Marcozzi We consider the valuation of European and American-style options under jump-diffusion processes by variational methods. In particular, the value function is seen to satisfy a parabolic partial (spatial) integro-differential variational inequality. A theoretical framework is developed and an analysis of a finite element implementation presented. A key feature is the introduction of separate approximation domains for both the state space and jump process variables. When coupled with any semi-implicit time integrator, this procedure presents a full discretization which is of optimal efficiency; the additional computational cost of evaluating the value function associated with a jump-diffusion as compared to pure diffusion process is, as a practical matter, negligible. Multi-dimensional computations are presented that validate the applicability and efficiency of the method. Which input in the calibration of a stochastic volatility model? Gianna Figa' Talamanca In this paper we investigate the informational content, for pricing purposes, of historical data of the underlying stock and of past implied data from European option prices in the context of a specific stochastic volatility model suggested by Heston (1993). First of all, some calibration techniques, based respectively on the partial information contained in the stock historical data or in the implied volatilities data, are analysed in terms of robustness and efficiency in the estimates provided. Then the techniques are applied for fitting Heston model to market data obtained for the MIB30 and the SP500 stock indexes. Preliminary results show a more powerful pricing forecast for the technique based on implied data of option prices when suitable restrictions are imposed on input data to be considered in the calibration. Finally, a mixed calibration technique which takes as input both historical and implied data is suggested, which seems capable to exploit all the information available, "taking the best" from the partial techniques analysed. Nonparametric estimation of Exponential Levy Models for Asset Prices Jose Figueroa-Lopez and Christian Houdre Accurate asset price models is of crucial importance in modern mathematical finance. Recent works have introduced asset price models driven by Levy processes with superior performance compared to the classical Black-Scholes model. However, the high computational intensity involved in the calibration of such models has prevented them from being more widely used in practice. In this talk, novel methods of model selection and nonparametric estimation for Levy processes are presented. The estimation relies on properties of Levy processes for small time spans, on the nature of the jumps of the process, and on recent methods of estimation for spatial Poisson processes. The procedures are illustrated in the case of Gamma Levy processes as well as variance Gamma processes, models of key relevance in asset price modeling. Credit Derivatives in an Affine Framework Damir Filipovic and Li Chen We develop a general and efficient method for valuating credit derivatives based on multiple entities in an affine framework. This includes interdependence of market and credit risk, joint credit migration and counterparty default risk of multiple firms. As an application we provide closed form expressions for the joint distribution of default times, default correlations, and credit default spreads in the presence of counterparty default risk. High Dimensional Radial Barrier Options Neil Firth and J. N. Dewynne Pricing high dimensional American options is a difficult problem in mathematical finance. Many simulation methods have been proposed, but Monte Carlo is numerically intensive, and therefore slow. We derive an analytic expression for a new type of multi-asset barrier option using Laplace transform methods. The solution is assumed to be radially symmetric in the normalized non dimensional variables, hence the name "Radial Barrier Options". In the single-asset case our results reduce to published results for American binary barrier options. Asset Substitution and Debt Renegotiation Christian Riis Flor In a structural modeling framework this paper analyzes how the asset substitution problem is affected when debt renegotiation is possible. The wellknown effect of asset substitution is that it destroys ex ante firm value. In this paper asset substitution can be used as a threat in a debt renegotiation, instead of simply being enforced. Thus, if the firm's earnings decrease, the equity holders will threaten the debt holders with either an asset substitution or a default in order to obtain lower coupon payments and, hence, reestablish the firm's optimal capital structure. Including asset substitution in the renegotiation game mitigates the negative ex ante effect but does not necessarily eliminate the problem. Furthermore, the paper analyzes the optimal threat and ex ante costs depending on the riskiness and expected growth rate of the substituted assets. Default and Volatility Time Scales Jean-Pierre Fouque, Ronnie Sircar, Knut Sølna In the first passage structural approach, default occurs when the underlying reaches a default barrier. In the classical Black-Cox model the underlying follows a geometric Brownian motion with constant volatility. Probability distributions of first passage times are used to compute default probabilities. One of the undesirable features of this model is that the yield spreads at short maturities is almost zero which is in contradiction with observed yield spreads. We propose here to look at the effect of stochastic volatility on the yield spreads. We show that volatility time scale is an essential concept in understanding this effect. Perturbation methods are used to approximate defaultable bond prices for instance. From the probabilistic point of view we propose approximations to the probability distributions of hitting times of "Brownian motion with stochastic diffusion". Markov Models for Interacting Defaults and Counterparty Risk Rüdiger  Frey and Jochen Backhaus In this talk we will be concerned with dynamic models for portfolios of dependent defaults. We concentrate on models for credit contagion, where the default of one company has a direct impact on the default intensity of other firms. We introduce a Markovian model and discuss the various types of interaction. We present limit results for large portfolios in a homogeneous model with mean-field interaction and analyze the impact of credit contagion on the portfolio loss distribution. Finally we discuss the pricing of basket credit derivatives in our model. A Financial Approach to Machine Learning with Application to Credit Risk Craig Friedman and Sven Sandow We review a coherent, financially based approach for measuring model performance and building probabilistic models that learn from data. We give information theoretic interpretations of our model performance measures and provide new generalizations of entropy and Kullback-Leibler relative entropy. For investors with utility functions in a three-parameter logarithmic family, our model building method leads to a regularized relative entropy minimization. We review applications of this methodology to two credit problems: estimating the conditional probability of default, given side information and estimating the conditional density of recovery rates of defaulted debt, given side information. Liquidity Discovery and Asset Pricing Michael  Gallmeyer, Burton Hollifield, Duane Seppi Long-dated securities are are risky, in part, because of uncertainty about the preferences of potential counter-parties and the terms-of-trade at which they will be willing to provide liquidity in the future. We call such randomness liquidity risk. We argue that liquidity risk is a important source of asymmetric information in addition to private information about future cash flows. We model the endogenous dynamics of liquidity risk, the risk premium for bearing liquidity risk, and the role of market trading in the liquidity discovery process through which investors learn about their counter-parties' preferences and future demands for securities. We show that market liquidity is a forward-looking predictor of future risk and, as such, is priced. Our model also provides rational explanations for price support levels and flights to quality." Theory and Calibration of Swap Market Models Stefano Galluccio We introduces a general framework for Swap Market Models which includes the co-terminal, the co-initial and the co-sliding model specification. The standard LIBOR Market Model appears as a special case of the co-sliding class. We aim at classifying Market Models according to their practical use in pricing and risk managing complex interest-rate derivatives. We show that swap market models can be easily handled both in theory and practice; for the co-terminal class, we introduce and numerically compare several approximating analytical formulas for caplets, while swaptions can be priced by a simple Black-type formula. A novel calibration technique is introduced to allow simultaneous consistency with caplet and swaption markets. In particular, we show that the calibration of the co-terminal swap market model is faster, more robust and more efficient than the same procedure applied to the LIBOR market model. We numerically study efficient pricing algorithms for exotic derivatives. Finally, suitable smile-consistent extensions of the diffusion-based dynamics are introduced and meaningful classes are selected according to trading and risk-management practice. General Quadratic Term Structures of Bond, Futures and Forward Prices Raquel Gaspar For finite dimensional factor models, the paper studies general quadratic term structures. These term structures include as special cases the affine term structures and the Gaussian quadratic term structures, previously studied in the literature. We show, however, that there are other, non-Gaussian, quadratic term structures and derive sufficient conditions for the existence of these general quadratic term structures for bond, futures and forward prices. As forward prices are martingales under the T-forward measure, their term structure equation depends on properties of bond prices' term structure. We exploit the connection with the bond prices term structure and show that even in quadratic short rate settings we can have affine term structures for forward prices. Finally, we show how the study of futures prices is naturally embedded in a study of forward prices and show that the difference between the two prices have to do with the correlation between bond prices and the price process of the underlying to the forward contract and this difference may be deterministic in some (non-trivial) stochastic interest rate settings. A Hidden Markov Model of Default Interaction Giacomo Giampieri, Mark Davis, Martin Crowder The occurrence of defaults within a bond portfolio is modeled as a simple hidden Markov process. The hidden variable represents the risk state, which is assumed to be common to all bonds within one particular sector and region. After describing the model and recalling the basic properties of hidden Markov chains, we show how to apply the model to a simulated sequence of default events. Then, we consider a real scenario, with default events taken from a large database provided by Standard & Poor's. We are able to obtain estimates for the model parameters, and also to reconstruct the most likely sequence of the risk state. Finally, we address the issue of global vs. industry-specific risk factors. By extending our model to include independent hidden risk sequences, we can disentangle the risk associated with the business cycle from that specific to the individual sector. Beyond Single Factor Affine Term Structure Models Javier Gil-Bazo This paper proposes a new approach to testing for the hypothesis of a single priced risk factor driving the term structure of interest rates. The method does not rely on any parametric specification of the state variable dynamics or the market price of risk, and simply exploits the constraint imposed by the no arbitrage condition on instantaneous expected bond returns. In order to achieve our goal, we develop a Kolmogorov-Smirnov test and apply it to data on Treasury bills and bonds for both the U.S. and Spain. We find that the single factor cannot be rejected for either dataset. The Market Price of Credit Risk Kay Giesecke and Lisa Goldberg We describe the relationship between actual probability of default and defaultable security prices. Our starting point is I-squared, a first passage model of default based on incomplete information. This model incorporates the unpredictable nature of default and thereby accounts for positive short spreads and the abrupt drops in defaultable security prices that occur at default. To connect prices with actual default probabilities, we analyze post-default recovery and the credit risk premium. Our recovery model is a generalization of the fractional market value convention introduced by Duffie and Singleton for intensity-based credit models. We derive I-squared pricing formulae for defaultable securities subject to fractional recovery. The I-squared credit risk premium has two components. One accounts for investors' aversion towards diffusive price volatility. The other reflects aversion toward the price jumps that occur at default. An Intensity-Based Approach to Valuation of Mortgage Contracts Subject to Prepayment Risk Yevgeny Goncharov This paper gives a rigorous treatment of modeling of mortgage securities subject to sub-optimal prepayment risk. A general model is developed and a new alternative representation of a mortgage price obtained. The proposed classification of approaches to the option-based and mortgage-rate-based (MRB) is validated by the fact that these two approaches require different analytical and numerical tools. With the option-based specification of the prepayment intensity, our model is the first continuous-time model; this gives advantage in numerical treatment of the model. We propose to generalize MRB approach by considering the endogenous mortgage rate. The mortgage rate process is defined and existence of a solution is proven for the option-based specification of the prepayment process. Wiener chaos and the Cox-Ingersoll-Ross model Matheus Grasselli and T. R. Hurd In this paper we recast the Cox--Ingersoll--Ross model of interest rates into the chaotic representation recently introduced by Hughston and Rafailidis. Beginning with the ``squared Gaussian representation'' of the CIR model, we find a simple expression for the fundamental random variable X. By use of techniques from the theory of infinite dimensional Gaussian integration, we derive an explicit formula for the n-th term of the Wiener chaos expansion of the CIR model, for n=0,1,2,.... We then derive a new expression for the price of a zero coupon bond which reveals a connection between Gaussian measures and Ricatti differential equations. Impulse Response Analysis and Immunization in Affine Term Structure Models Martino Grasselli and Claudio Tebaldi Affine Term Structure Models (ATSM) are traditionally used in finance as a statistical model for bond portfolio immunization and, more recently, to modelize the stochastic response of the term structure to unanticipated macroeconomic shocks and monetary policy. A natural stochastic approach to both problems requires some new tools from stochastic analysis. Their computation in ultimate analysis involves the Jacobian for the ATSM‡ow in the forward measure, which appears to be the natural multifactor generalization of the duration measure. We reduce its computation to the solution of the same set of Riccati ODE required for pricing. Necessary Conditions for the Existence of Utility Maximizing Strategies under Transaction Costs Paolo Guasoni and Walter Schachermayer For any utility function with asymptotic elasticity equal to one, we construct a market model in countable discrete time, such that the utility maximization problem with proportional transaction costs admits no solution. This proves that the necessity of the reasonable asymptotic elasticity condition, established by Kramkov and Schachermayer [KS99] in the frictionless case, remains valid also in the presence of transaction costs. Robust Utility Maximization for Complete and Incomplete Market Models Anne Gundel We investigate the problem of maximizing the robust utility functional infQ2Q EQu(X) for some set of subjective measures Q. We give the dual characterization for its solution for both a complete and an incomplete market model. To this end, we introduce the new notion of reverse f-projections and use techniques developed for f-divergences. This is a suitable tool to reduce the robust problem to the classical problem of utility maximization under a certain measure: the reverse f-projection. Furthermore, we give the dual characterization for a closely related problem, the minimization of expenditures given a minimum level of expected utility in a robust setting and for an incomplete market. Explicit solution of a stochastic irreversible investment problem and its moving threshold Ulrich Haussmann, Maria B. Chiarolla We consider a firm producing a single consumption good, that makes irreversible investments to expand its production capacity. The firm aims to maximize its expected total discounted real profit net of investment on a finite horizon. The capacity is modeled as a controlled Ito process where the control is the investment, an increasing process. The resulting singular stochastic control problem and the associated optimal stopping problem are solve explicitly in the case of CRRA production functions. The moving free boundary is the threshold at which the shadow value of invested capital exceeds the capital's replacement cost. Then we use the equation of the free boundary to evaluate the optimal investment policy and the corresponding optimal profits. On Covariance Estimation for High-Frequency Financial Data Takaki Hayashi and Nakahiro Yoshida We consider the problem of estimating the covariance/correlation of two diffusion prices that are observed at discrete times in a nonsynchronous manner. A de facto standard approach in the literature, "realized" estimator, which is based on the sum of cross-products of intraday log-price changes measured on regularly-spaced intervals over a day, is problematic because choice of regular interval size and data interpolation scheme may lead to unreliable estimation. We present a new estimation procedure recently proposed by Hayashi and Yoshida(03)(04), which is free of such "synchronization" of data, hence, free of biases or other problems caused by it. In particular, the estimators are shown to have consistency as the observation frequency (or the market liquidity) tends to infinity, which is not possessed by realized estimators. Valuing Real Options without a Perfect Spanning Asset Vicky Henderson The real options approach to corporate investment decision making recognizes a firm can delay an investment decision and wait for more information concerning project cashflows. The classic model models of McDonald and Siegel (1986) and Dixit and Pindyck (1994) value the investment decision as a perpetual American option and in doing so, essentially assumes the real asset underlying the option is traded, or that there is a perfect spanning asset available. Most real projects however can only be partially hedged by traded securities. Our model relaxes this assumption and assumes only a partial spanning asset can be found. In this model, we obtain in closed form the value of the option to invest and the optimal investment trigger level, above which investment takes place. These both depend on the correlation between project cashflows and the spanning asset, risk aversion of the firm's shareholders, and volatilities of project cashflows and the partial spanning asset. We observe that the value of the option to invest and the trigger level are both lowered when the spanning asset is less than perfect. This implies the firm should invest earlier than the classic models suggest. Although the partial spanning model contains the classic model as a special case, it is much richer. In particular, there are situations where the classic model recommends the firm always postpones investment, whereas if a highly (but not perfectly) correlated spanning asset were assumed, the firm should invest at a certain trigger level. On the tradeoff between consumption and investment in incomplete financial markets Daniel Hernandez-Hernandez and Wendell H. Fleming In this paper we are concerned with the tradeoff between long term growth of the expected utility of wealth and consumption. The goal is to find a consumption policy for which the optimal rate of capital growth is zero, i.e. a policy for which balance between consumption and investment is reached. The asymptotic limit of this investment problem when the HARA parameter $\gamma\to-\infty$ is also studied. Pricing electricity risk by interest rate methods Juri Hinz, Lutz von Grafensein, Michel Verschuere, Martina Wilhelm We address a method for pricing electricity contracts based on valuation of ability to produce power, which is considered as the true underlying for electricity derivatives. This approach shows that an evaluation of free production capacity provides a framework where a change--of--numeraire transformation converts electricity forward market into the common settings of money market modeling. Using the toolkit of interest rate theory, we derive explicit option pricing formulas. Arbitrage-free bounds for basket options David Hobson, Peter Laurence, Tai-Ho Wang In this paper we investigate the possible values of basket options. Instead of postulating a model and pricing the basket option using that model, we consider the set of all models which are consistent with the observed prices of vanilla options, and, within this class find the model for which the price of the basket option is largest. This price is an upper bound on the prices of the basket option which are consistent with no-arbitrage. In the absence of additional assumptions on the market it is the least upper bound on the price of the basket option. Associated with the bound is a simple super-replicating strategy involving trading in calls. The integral of a geometric Brownian motion is indeterminate by its moments Per Hoerfelt This talk proves that the integral of a geometric Brownian motion is indeterminate by its moments. The proof is based on geometric inequalities in Gauss space and the Pedersen criterion for the moment problem. The question if the integral of a geometric Brownian motion is indeterminate by its moments was raised by Yor. The presented paper corrects previous false proofs showing that the integral of a geometric Brownian motion is indeterminate by its moments. Stochastic Cascades, Credit Contagion, and Large Portfolio Ulrich Horst We analyze an interactive model of credit ratings where external shocks, initially affecting only a small number of firms, spread by a contagious chain reaction to the entire economy. Counterparty relationships along with discrete adjustments of credit ratings generate a transition mechanism that allows the financial distress of one firm to spill over to its business partners. Such a contagious infectious of financial distress constitutes a source of intrinsic risk for large portfolios of credit sensitive securities that cannot be ``diversified away.'' We provide a complete characterization of the fluctuations of credit ratings in large economies when adjustments follow a threshold rule. We also analyze the effects of downgrading cascades on aggregate losses of credit portfolios. We show that the loss distribution has a power-law tail if the interaction between different companies is strong enough. Matched asymptotic expansions for discretely sampled barrier options Sam Howison and Mario Steinberg The problem of calculating the approximate `continuity correction' for discretely sampled barrier options was solved for the Black-Scholes model by Broadie, Glasserman and Kou using probabilistic techniques. (An explicit solution to the full problem has recently been presented by Abrahams et al. using the Wiener-Hopf method.) The problem can also be viewed in a PDE framework and the method of matched asymptotic expansions used. This has the advantage that it is more flexible and can, for example, be generalised to the case of local volatility surfaces. I will describe the approach in general terms and in relation to the specific example of discretely sampled barrier contracts. Indifference pricing and hedging in stochastic volatility models Tom Hurd and Matheus Grasselli We introduce "reciprocal affine" stochastic volatility models whose elegant analytic properties lead to tractable formulas for the indifference pricing and hedging of pure volatility claims. These ``unhedgable'' claims are not well understood theoretically, and are difficult to price, but in our model we have the opportunity of testing the usefulness and practicality of utility based methods for incomplete markets. Optimal Static-Dynamic Hedges for Barrier Options Aytac Ilhan and Ronnie Sircar In a general semimartingale market model, we study optimal hedging of barrier options using a combination of a static position in vanilla options and dynamic trading of the underlying asset. Only a finite number of types of vanilla options are available to the investor, who chooses the optimal combination by maximizing her expected terminal utility. The problem reduces to computing the Fenchel-Legendre transform of the utility-indifference price as a function of the number of vanilla options used to hedge, evaluated at the market price of these options. Using the well-known duality between exponential utility and relative entropy, we provide a new characterization of the indifference price in terms of the minimal entropy martingale measure, and give conditions guaranteeing differentiability and strict convexity in the hedging quantity, and hence a unique solution to the hedging problem. We discuss computational approaches within the context of Markovian stochastic volatility models. Evaluating the Switching Options by Simulation Junichi Imai This paper develops the valuation model of a switching option with the simulation methods. Since the switching option contains a wide range of contingent claims the development of the valuation model of the switching option is important. Especially, the idea of a switching option is useful in evaluating the project value with real options because many types of real options are regarded as switching options. This paper extends two existing simulation methods that can evaluate an American style option to value the switching option, which are called the low-discrepancy mesh method and the least square method. The efficiency and the accuracy of these methods are examined theoretically and in the numerical experiences. Futures Trading Model with Transaction Costs Karel Janecek and Steven E. Shreve We consider an agent who invests in a futures contract and a money market and consumes in order to maximize the utility of consumption over an infinite planning horizon in the presence of a proportional transaction cost $\lambda>0$. The utility function is of the form $U(c)=c^{1-p}/(1-p)$ for $p>0$, $p\neq 1$. The asymptotic analysis for a standard stock price process was rigorously done by Janecek and Shreve in \emph{'Asymptotic Analysis for Optimal Investment and Consumption with Transaction Costs'} (to appear in \textbf{Finance and Stochastics}). The authors use the technique of super and subsolutions to the corresponding HJB equation. Unfortunately, similar technique is not readily available for more general market models. A fundamentally different approach is based on purely probabilistic arguments. The advantage of this approach is that, besides being intuitively appealing, it can be generalized to more general market setup, e.g. stochastic volatility, which is necessary for a successful practical implementation. Measuring default premium using the Cox process with shot noise intensity Jiwook Jang We employ the Cox process with shot noise intensity to model the default time. The survival probability is derived based on the Cox process with shot noise intensity that has doubly stochastic property. As an interest rate process for non-defaultable bond, i.e. a government bond, we use a generalised Cox-Ingersoll- Ross (CIR) model (1985). As Lando (1998) has shown that we can combine the effects of default and of discounting for interest rates, it is used to obtain the default premium between non-defaultable bond and defaultable bond (i.e. a corporate bond). Using an equivalent martingale probability measure obtained via the Esscher transform, risk-neutral default premium formula is derived. The asymptotic distribution of the shot noise intensity is used not to have its initial value. We also assume that the jump size of shot noise intensity follows an exponential distribution to illustrate the calculation of default premium. For simplicity, we ignore the recovery rate. On valuation before and after tax in no arbitrage models: Tax neutrality in the continuous time model Bjarne  Jensen We establish necessary and sufficient conditions for a linear taxation system to be neutral - within the continuous-time ``no arbitrage'' model - in the sense that asset valuation is invariant to the process for tax rates and choice of realization dates as well as immune to timing options attempting to twist the time profile of taxable income through wash sale transactions. We also demonstrate that despite neutrality the portfolio choice can be quite different across investors subject to different tax rates. Invariance Tests of Forward Rate Models Malene Jensen and Bent Jesper Christensen We introduce a statistical invariance test for the consistency between the shape of the yield curve and the stochastic process driving interest rates. The analysis is cast in the Heath-Jarrow-Morton framework, and utilizes factor analysis of forward rates estimated from coupon-bearing bond data. The theoretical properties of the invariance tests are investigated, and an application to U.S. government bonds is offered. Keywords: factor analysis; interest rate models; invariant manifolds; JEL Codes: C51; C52; Continuous-Time Mean--Variance Portfolio Selection with Bankruptcy Hanqing Jin, Tomasz R. Bielecki, Stanley R. Pliska, Xun Yu Zhou A continuous-time mean--variance portfolio selection problem is studied where all the market coefficients are random and the wealth process under any admissible trading strategy is not allowed to be below zero at any time. The trading strategy under consideration is defined in terms of the dollar amounts, rather than the proportions of wealth, allocated in individual stocks. The problem is completely solved using a decomposition approach. Specifically, a (constrained) variance minimizing problem is formulated and its feasibility is characterized. Then, after having solved a system of equations for two Lagrange multipliers, variance minimizing portfolios are derived as the replicating portfolios of some contingent claims, and the variance minimizing frontier is obtained. Finally, the efficient frontier is identified as an appropriate portion of the variance minimizing frontier after the monotonicity of the minimum variance on the expected terminal wealth over this portion is proved, and all the efficient portfolios are found. In the special case where the market coefficients are deterministic, efficient portfolios are explicitly expressed as feedback of the current wealth, and the efficient frontier is represented by parameterized equations. Our results indicate that the efficient policy for a mean--variance investor is simply to purchase a European put option that is chosen, according to his or her risk preferences, from a particular class of options. Malliavin Monte Carlo Greeks for jump diffusions Martin Johansson and Mark Davis In recent years efficient methods have been developed for calculating derivative price sensitivities using Monte Carlo simulation. Malliavin calculus has been used to transform the simulation problem in the case where the underlying follows a Markov diffusion process. In this work, recent developments in the area of Malliavin calculus for Levy processes are applied and slightly extended. This permits derivation of stochastic weights, as in the continuous case, for a certain class of jump diffusion processes. Bayesian Analysis of Stochastic Betas Gergana Jostova and Alexander Philipov This paper proposes a mean-reverting stochastic process for the market beta. In a simulation study, the proposed model generates significantly more precise beta estimates relative to competing GARCH betas, betas scaled by aggregate or firm-level variables, and betas based on rolling regressions, even when the true betas are generated based on these competing specifications. Applying our model to US industry portfolios, we document significant improvement in out-of-sample hedging effectiveness relative to the traditional OLS beta estimate. In asset-pricing tests, our model provides substantially stronger support for the conditional CAPM relative to competing beta models. It also helps resolve asset-pricing anomalies such as the size, book-to-market, and idiosyncratic volatility effects in the cross-section of stock returns. On asymptotic pricing of securities in a multivariate extension of Scotts stochastic volatility model Joerg Kampen, Joerg Kampen, Nicolae Surulescu We consider a natural multivariate extension of Scotts stochastic volatility model and analyse the pricing and its asymptotics in the sense of \cite{papanic1}. The pricing is done via the mean-variance hedging approach, where the market risk function of volatility is determined by a fully nonlinear PDE, and where we point out that explicit stationary solutions can be constructed in case of non-leverage. We provide explicit formulas for the effective volatility matrix and the correction constants by an explicit solution of a multivariate Poisson equation. From the computational point of view the high dimensional asymptotic option pricing to lower order reduces to solving Gaussian (explictly given) integrals which can be done very effectively by using sparse grids. We show that explicit asymptotic formulas (in terms of fast converging univariate power series) can be obtained for standard bivariate options such as the call on the maximum of two assets. Computational Solution of the American Put using the Moving Free Boundary Method Michael Kelly Computational Solution of the American Put using the Moving Free Boundary Method: The solution of the American Put option has been shown by Jamshidian, Carr, Buchen and Kelly to be theoretically expressible in terms of an integro-differential Fredholm equation. Unfortunately there is no known method for the general solution of such equations. However it is shown in this paper that such equations can be described symbolically and solved numerically using recent mathematical software such as Mathematica. These solutions are also compared with other approximate results. Optimal bank capital with costly recapitalization Jussi Keppo and Samu Peura We study optimal bank capital holding behavior as a dynamic tradeoff between the opportunity cost of equity, the loss of franchise value following a regulatory minimum capital violation, and the cost of recapitalization. Our model indicates that a recapitalization option may be valuable despite substantial delays and fixed costs in capital issuance, and that a significant fraction of the value of low capitalized banks may be attributable to the option to recapitalize. Due to the downward bias in banks' accounting return volatility, we operate the model with implied bank return volatilities that correctly replicate observed bank capital ratios. We argue that the option to recapitalize (at a lower cost) can explain the lower capital ratios of large banks relative to small banks. A Parallel time stepping approach using meshfree approximations for pricing options with non-smooth payoffs Greg Fasshauer, A. Q. M. Khaliq, D. A. Voss We consider a meshfree radial basis function approach for the valuation of pricing options with non-smooth payoffs. By taking advantage of parallel architecture, a strongly stable and highly accurate time stepping method is developed with computational complexity comparable to the implicit Euler method implemented concurrently on each processor. This, in collusion with radial valuation of exotic options, such as American digital options. Valuation and Hedging of Power-Sensitive Contingent Claims for Power with Spikes: a Non-Markovian Approach Valery Kholodnyi We present a new approach to modeling spikes in power prices proposed earlier by the author. In contrast to the standard approaches, we model power prices with spikes as a non-Markovian stochastic process. This allows for modeling spikes directly as self-reversing jumps. This also allows for the analytical valuation of European contingent claims on power with spikes as well as for the analytical valuation and dynamic hedging of European contingent claims on forwards on power for power with spikes. Moreover, the proposed non-Markovian process provides a natural mechanism to explain the absence of spikes in the values of European contingent claims on power far from their expiration time while power prices exhibit spikes. This mechanism also explains why power forward prices, far from the maturity time of the forward contracts on power, do not exhibit spikes while power prices do. Finally, we obtain a linear evolution equation for European contingent claims on power with spikes. We also obtain a semilinear evolution equation for universal contingent claims, including Bermudan and American options, on power with spikes. Universal contingent claims and the semilinear evolution equation for universal contingent claims were introduced earlier by the author. Market Price of Risk Specifications for Affine Models: Theory and Evidence Robert Kimmel, Patrick Cheridito, Damir Filipovic We extend the standard specification of the market price of risk for affine yield models of the term structure of interest rates, and estimate several models using the extended specification. For most models, the extended specification fits US data better than standard specifications, often with extremely high statistical significance. Our specification yields models that are affine under both objective and risk-neutral probability measures, but is never used in financial applications, probably because of the difficulty of applying traditional methods for proving the absence of arbitrage. Using an alternate method, we show that the extended specification does not permit arbitrage opportunities, provided that under both measures the state variables cannot achieve their boundary values. Likelihood ratio tests show our extension is statistically significant for four of the models considered at the conventional 95% confidence level, and at far higher levels for three of the models. The results are particularly strong for affine diffusions with multiple square-root type variables. Although we focus on affine yield models, our extended market price of risk specification also applies to any model in which Feller's square-root process or a multivariate extension is used to model asset prices. Pricing Options in Electricity Markets Tino Kluge In this working paper we examine a particular stochastic process describing the spot price development in electricity markets. The process under investigation is mean-reverting with a jump component and two mean-reverting rates: one for the diffusion part and one for the jump part. Setting the mean-reverting rate for the jump component to a high value allows for a realistic modelling of spikes, a phenomenon peculiar to electricity markets. This model is highly incomplete, not only because it allows for jumps but also because of the inability to store electricity efficiently, which rules the underlying out to be used as a hedging instrument. We describe a (subset) of risk neutral measures Q and show how Q can be determined based on a continuous forward curve observed in the market. Furthermore we give a semi-analytic formula for call options. Optimal Portfolios with Fixed Consumption or Income Streams Ralf Korn and Martin Krekel We consider some portfolio optimisation problems where either the investor has a desire for an a priori specified consumption stream or/and follows a deterministic pay in scheme while also trying to maximize expected utility from final wealth. We derive explicit closed form solutions for continuous and discrete monetary streams. The mathematical method used is classical stochastic control theory. Neutral Derivative Pricing in Incomplete Markets Christoph Kühn and Jan Kallsen This talk is about some recent developments in neutral derivative pricing. Neutral prices occur if investors are utility maximizers and if derivative supply and demand are balanced. Provided that the dual pricing measure exists unique price processes can be derived for derivatives of European, American, and game type. More delicate is the case that the dual pricing measure does not exist. As stated in Hugonnier, Kramkov, and Schachermayer (2002) there can be a whole interval of neutral prices. In addition it turns out that in this case some initial prices which are neutral in a model allowing only for buy-and-hold strategies in the derivative cannot be extended to neutral price processes in the same model allowing also for intermediate trades in the derivative. We prove an existence theorem for neutral price processes and characterize them as martingales under a special set of finitely additive set functions (joint work with Jan Kallsen). Linkage between lookback and reset features Yue Kuen  Kwok and Hoi Ying Wong The lookback feature in an option contract refers to the payoff structure where the terminal payoff depends on the realized extreme value of the underlying state variables. The reset feature is the privilege given to the holder to reset certain terms in the contract according to specified rules at the moment of reset. The reload provision in an employee stock option entitles its holder to receive one new (reload) option from the employer for each share tendered as payment of strike upon the exercise of the stock option. The dynamic protection feature in equity-indexed annuities entitles the investor the right to reset the fund value to that of the reference stock index. When the number of reset or reload rights becomes infinite, the non-deterministic free boundary associated with the optimal reset or reload decision becomes deterministic. The optimal stopping problems associated with multiple reset or reload rights become derivative models with an additional lookback variable. The non-linear free boundary value problem of optimal reset becomes linear, though the model formulation involves an additional path dependent lookback variable. Optimal Portfolio Delegation when Parties have different Coefficients of Risk Aversion Kasper Larsen We consider the problem of delegated portfolio management when the involved parties are risk-averse. The agent invests the principal's money in the financial market, and in return he receives a compensation, depending on the value that he generates over some period of time. We use a dual approach to explicitly solve the agent's problem and subsequently use this solution to solve the principal's problem numerically. The interaction between the risk coefficients and the optimal compensation scheme is studied for different coefficients. E.g.\ in the case of a more risk averse agent the principal should according to common folklore optimally choose a fee schedule such that the agent's derived risk aversion decreased. We illustrate that this is not always the case. Generating Functions for Stochastic Integrals Stephan Lawi and C. Albanese Generating functions for stochastic integrals have been known in analytically closed form for just a handful of stochastic processes: namely, the Ornstein-Uhlenbeck, the Cox-Ingerssol-Ross (CIR) process and the exponential of Brownian motion. In virtue of their analytical tractability, these processes are extensively used in modelling applications. In this paper, we construct broad extensions of these process classes. We show how the known models fit into a classification scheme for diffusion processes for which generating functions for stochastic integrals and transition probability densities can be evaluated as integrals of hypergeometric functions against the spectral measure for certain self-adjoint operators. We also extend this scheme to a class of finite-state Markov processes related to hypergeometric polynomials in the discrete series of the Askey classification tree. Sharpe Ratio as a Performance Measure in a Multi-Period Setting Ali Lazrak, Jaksa Cvitanic, Tan Wang We study Sharpe ratio as a performance measure in a multi-period setting. We show that the typical mean-variance efficiency justification for using Sharpe ratio, valid in a static setting, typically fails in a multi-period setting. To focus on the contrast between static vs multi-period settings, we maintain the mean-variance utility assumption of the static model. We show that the Sharpe ratio maximization criterion is subject to the horizon problem noted by Jensen (1969). That is, the trading strategy that leads to the most desirable portfolio for, say, each quarter and for four consecutive quarters is not same as the strategy that gives the highest Sharpe ratio for a year. As a consequence, unless the investor's investment horizon matches exactly the performance measurement period of the portfolio manager, the portfolio with the highest Sharpe ratio is not necessarily the most desirable from the investor's point of view. We also show that quantitatively the difference is significant even when the investor's investment horizon is only six months. The difference becomes larger has the investor's horizon becomes longer. Robust Replication of Volatility Derivatives Roger Lee, Peter Carr Suppose the positive asset price S follows a diffusion process. Define realized variance to be the quadratic variation of log(St) on [0; T]. (In practice, one typically uses the sample variance of the daily or weekly logarithmic returns of S.) Without knowing the dynamics of S, one can replicate a contract which pays realized variance, as follows: hold a particular static position in European-style options expiring at T, and trade S according to a particular dynamic strategy. Does this known result extend to general functions of realized variance? For example, it is of practical interest to replicate contracts which pay realized volatility, the square root of realized variance. Previous replication efforts sacrificed robustness, by imposing specific models on the S-dynamics. We show that, with trading in options, we can replicate contracts which pay general functions of realized variance. Instead of imposing a volatility model, we make only a correlation assumption. We also show how to correct for deviations from the correlation condition. The American put and European options near expiry, under Levy processes Sergey Levendorskiy We derive explicit formulas for time decay for the European call and put options at expiry, and use them to calculate analytical approximations to the price of the American put and early exercise boundary near expiry. We show that for many families of non-Gaussian processes used in empirical studies of financial markets, the early exercise boundary for the American put without dividends is separated from the strike price by a non-vanishing margin. As the riskless rate vanishes and the drift decreases accordingly so that the stock remains a martingale, the boundary goes to zero uniformly over the interval. Mean-Reverting and Co-Integrated Energy Futures Curve Models for Pricing and Risk Management Alex Levin We present a broad class of multifactor no-arbitrage mean-reverting and co-integrated HJM-type energy futures curve models. They are equally suited for the long-term simulation in Credit Risk systems and short-term simulation in Market Risk systems in the real-world measure or pricing in the risk-neutral measure of commodity derivatives. This approach generalizes a PCA methodology for a class of no-arbitrage entire futures curve models represented as a static non-linear transformation of linear combinations of the time-homogeneous volatility functions driven by Gaussian mean-reverting or co-integrated stochastic systems with constant coefficients. A full characterization of such family is obtained. It is proven that only four following types of models with the corresponding types of static transformations and volatility functions are consistent with the no-arbitrage dynamics: Generalized Vasicek and Generalized Black-Karasinski model corresponding to linear and exponential transformations and exponential-polynomial volatility functions; a new no-arbitrage model with capped (bounded) futures and spot prices corresponding to Normal CDF transformation and volatility functions derived in closed analytical form from the non-linear dynamic system; and a new "explosive" model corresponding to some integral transformation. Presented results are generalized for the state variables driven by jump-diffusion processes and the real world futures curve dynamics. A new "historical" multifactor parameter calibration procedure based on Lyapunov equations and introduced Modified Principal Components is developed for mean-reverting and multi-curve cross-mean-reverting models. Practical examples of effective long-term simulation for the NYMEX Crude Oil and Natural Gas futures curves are considered. Mean-variance hedging when there are jumps Andrew Lim and Thaisiri Watewai This paper concerns the problem of hedging a random liability in a market when the uncertainty is modelled by Brownian motion as well as a jump diffusion. Explicit representations of the optimal hedging portfolio are obtained using the theory of backward stochastic differential equations. The Spectral Decomposition of the Option Value Vadim Linetsky This paper develops a spectral expansion approach to the valuation of contingent claims when the underlying state variable follows a one-dimensional diffusion with the infinitesimal variance $a^2(x)$, drift $b(x)$ and instantaneous discount (killing) rate $r(x)$. The Spectral Theorem for self-adjoint operators in Hilbert space yields the spectral decomposition of the contingent claim value function. Based on the Sturm-Liouville (SL) theory, we classify Feller's natural boundaries into two further subcategories: non-oscillatory and oscillatory/non-oscillatory with cutoff $\Lambda\geq 0$ (this classification is based on the oscillation of solutions of the associated SL equation) and establish additional assumptions (satisfied in nearly all financial applications) that allow us to completely characterize the qualitative nature of the spectrum from the behavior of $a$, $b$ and $r$ near the boundaries, classify all diffusions satisfying these assumptions into the three spectral categories, and present simplified forms of the spectral expansion for each category. To obtain explicit expressions, we observe that the Liouville transformation reduces the SL equation to the one-dimensional Schr\"{o}dinger equation with a potential function constructed from $a$, $b$ and $r$. If analytical solutions are available for the Schr\"{o}dinger equation, inverting the Liouville transformation yields analytical solutions for the original SL equation, and the spectral representation for the diffusion process can be constructed explicitly. This produces an explicit spectral decomposition of the contingent claim value function. Pricing Vulnerable European Options with Stochastic Default Barriers Chi-Fai Lo, C.H. Hui, K.C. Ku This paper develops a valuation model of European options incorporating a stochastic default barrier, which extends a constant default barrier proposed in the Hull and White model. The default barrier is considered as an option writer's liability. Closed-form solutions of vulnerable European option values based on the model are derived to study the impact of the stochastic default barriers on option values. The numerical results show that negative correlation between the firm values and stochastic default barriers of option writers gives material reductions in option values where the options are written by firms with leverage ratios corresponding to BBB or BB ratings. Geometric Brownian Motion of Skorohod Type as a Canonical Model for Assets with Correlated Returns and Heavy Tails Andrew Lyasoff Essentially all statistical studies of market data indicate that returns from stock are slightly negatively autocorrelated. Usually, this finding is interpreted as an evidence that returns from stocks are very close to random walks. I will demonstrate with a concrete example that a very small — in fact much smaller than the one observed in market data — autocorrelation may reveal a strong self-regulatory pattern which is inconsistent with the random walk hypothesis. I will then consider a model for stock prices that differs from the standard geometric Brownian motion only in that its starting point depends on the driving Brownian motion. Somewhat surprisingly, this model leads to probability distributions with much heavier tails than the tails of the standard geometric Brownian motion. Risk Control of Dynamic Investment Models William Ziemba, Leonard MacLean, Y. Zhao The risk inherent in the accumulation of investment capital depends on the true return distributions of risky assets, the accuracy of estimated returns, and the investment strategy. This paper considers risk control with Value-at-Risk and Conditional Value-at-Risk constraints, using control limits to determine times for rebalancing the portfolio. Optimal strategies and control limits are determined for a geometric Brownian motion asset pricing model, with random parameters. The approaches to risk control are applied to the fundamental problem of investment in stocks, bonds, and cash over time. The advantages of portfolio rebalancing at random times determined from control limits are illustrated. A Multinomial Approximation of American Option Prices in a Levy Process Model Ross Maller, David Solomon, Alex Szimayer Models in which the the logarithm of the stock price process follows a Levy process (a continuous time process with independent stationary increments which generalises Brownian motion) are presently under vigorous investigation. Of particular interest and difficulty is the pricing of American options written on stocks which follow these models. A couple of existing schemes are able to deal with some but not all aspects of this. This paper proposes a multinomial tree setup which can be viewed as generalising the binomial model of Cox et al.(1979) for Brownian motion. Under very mild conditions, we show that the American option prices obtained under the multinomial model converge to the corresponding prices under the continuous time Levy process model. Our procedure is very general and can handle infinite activity processes such as the variance gamma (VG) model straightforwardly. It also overcomes some practical difficulties that have previously been encountered. Explicit illustrations are given for jump diffusion models and for the VG model. Detecting the presence of a diffusion in asset prices Cecilia Mancini We consider a process evolving by $$dY_t = a_t dt+sigma_t dW_{t}+dJ_t, t \leq T,$$ where $a$, $\sigma$ are progressively measurable stochastic processes, $W$ is a standard Brownian motion, and $J_t$ is a pure jump Lévy kind process. Such a process can be used to model the evolution of the logarithm of the price of a stock, of an index or of a commodity. Given a discrete record of equally spaced observations $\{Y_{0},Y_{t_1},...,Y_{t_{n-1}},Y_{t_n}\}$, with $t_i=ih$, $hn=T$, we are going to manipulate them to deteremine whether the diffusion component is zero or not. This is to make a comparison between the widely used jump-diffusion processes and the pure jump infinite activity processes used for instance y Madan (1999), Geman&al, for financial stock prices or indexes. At this aim we construct quadratic variation based estimators of the integrated volatility and of the jump part of $Y$, which separate the two parts of $Y$. We check the performance of our estimators on a variety of simulated models, both in the finite jump activity case and in the infinite activity (both finite and infinite variation) for any type of dependence of sigma on W (cfr Barndorff-Nielsen and Shephard, Ait-Sahalia, Woerner). We then apply our estimators to financial high frequency data.\\ Harmonic analysis methods for volatility computation Emilio Barucci, Paul Malliavin, Maria Elvira Mancino We provide two nonparametric methods based on harmonic analysis to compute instantaneous and realized volatility of a diffusion process. The methods are well suited to use high frequency data. We reconstruct the volatility of a diffusion process as a function of time by establishing a connection between the Fourier transform of the price process and that of the volatility function. Then the volatility is obtained as a series of trigonometric polynomials. A relationship is also established between the Laplace transform of the price process and of the volatility function. Continuous time option valuation with discrete hedging subject to transaction costs and trading delays Michael Marcozzi We consider a model for the writing price of a European option based on the utility maximization formulation of Hodges and Neuberger (1989) and Davis et al. (1993). Transaction costs, block trading, and transaction delays within the model are implemented through the velocity of the wealth process, representing holding a portfolio of securities. In particular, the value function is seen to satisfy a temporally weak ultraparabolic Hamilton-Jacobi-Bellman equation. Hedging under the Minimal Potential Measures Massimo Masetti In this paper we show a new characterization of the Minimal Mar- tingale Measure bP, used for the local risk minimization hedging ap- proach. First we derive the Föllmer-Schweizer decomposition under an absolutely continuous change of measure from bP to another equiv- alent martingale measure Q 2 Me(S). Second we derive the optional decomposition of a Q-submartingale under an absolutely continuous change of measure from the optimal measure Q to another Q 2 Me(S). We show that bP is the unique Minimal Potential Measure such that, the potential generated by the Riesz optional decomposition of the value process bV under Q, is reduced to zero bP-a.s. and P-a.s. Finally, we extend the interpretation of E bP [H] in terms of the arbitrage-free prices range. A simple calibration procedure of stochastic volatility models with jumps by short term asymptotics Alexey Medvedev and Olivier Scaillet In this paper we propose a simple calibration procedure to infer key model relationships from short term option prices. We develop and make use of an approximating formula for European options prices based on short maturity asymptotics, i.e. when time-to-maturity tends to zero. The analysis is performed in a general setting where stochastic volatility and jumps drive stock price dynamics. In a numerical study we show that the calibration procedure is accurate for a reasonable set of model parameters. An empirical application illustrates its application to S&P 500 index options. Efficient Hedging and Equity-Linked Life Insurance Alexander Melnikov The talk is devoted to how hedging methodologies developed in the modern financial mathematics can be exploited to price equity-linked life insurance contracts. We study pure endowment life insurance contracts with fixed and flexible guarantees. In our setting, these insurance instruments are based on two risky assets of the market controlled by Black-Scholes model during a contract period. The first asset is responsible for the maximal size of a future profit while the second, more reliable, asset provides a flexible guarantee for the insured. The insurance company is considered as a hedger of a maximum of these assets conditioned by remaining life time of a client in the framework of this market. The main attention is paid to new types of hedging (quantile hedging and efficient hedging with power loss function), which, together with Black-Scholes (fixed guarantee) and Margrabe (flexible guarantee) formulae, creates effective actuarial analysis of such contracts. We show also how this approach is extended to a jump-diffusion scheme and discuss some connections with the pricing of credit risks and defaultable derivative securities. Finally, we give numerical examples based on financial indices the Dow Johns Industrial Average and the Russell 2000 to demonstrate how our results can be applied to actuarial practice. A Note on Esscher Transformed Martingale Measures for Geometric Levy Processes Yoshio Miyahara The Esscher transform is one of the very useful methods to obtain a reasonable equivalent martingale measure, and it is defined with relation to the corresponding risk process. In this article we consider two kinds of risk processes (compound return process and simple return process). Then we obtain two kinds of Esscher transformed martingale measures. The first one is the one which was introduced by Gerber and Shiu, and the second one is identified with the MEMM (minimal entropy martingale measure). We set up the economical characterization of these two kinds of Esscher transforms, and then we study the properties of the two kinds of Esscher transformed martingale measures, comparing each others. Key words: incomplete market, geometric Levy process, equivalent martingale measure, Esscher transform, minimal entropy martingale measure. JEL Classification: G12, G13 On the Starting and Stopping Problem: Application in reversible investment Jeanblanc Monique and Hamademe Said In this work we solve completely the starting and stopping problem when the dynamics of the system are a general adapted stochastic process. We use backward stochastic differential equations and Snell envelopes. A power station produces electricity whose selling price fluctuates. We suppose that electricity is produced only when its profitability is satisfactory. Otherwise the power station is closed up to time when the profitability is coming back. So for the power station there are two modes: operating and closed. At the initial time, we assume it is in its operating mode. On the other hand, like every economic unit, there are expenditures when the station is in its operating mode as well as in the closed one. In addition, switching from a mode to another is not free and generates costs. The problem we are interested in is to find the sequence of times where one should make decisions to stop the production and to start it again successively in order to maximize the profitability of the station and then to determine the maximum profit. Optimal Asset Allocation and Ruin-Minimization Annuitization Strategies Kristen Moore, Virginia R. Young, Moshe A. Milevsky In this paper, we derive the optimal investment and annuitization strategy for a retiree whose objective is to minimize the probability of lifetime ruin, namely the probability that a fixed consumption strategy will lead to zero wealth while the individual is still alive. Recent papers in the insurance economics literature have examined utility-maximizing annuitization strategies. Others in the probability, finance, and risk management literature have derived shortfall-minimizing investment and hedging strategies given a limited amount of initial capital. This paper brings the two strands of research together. Our model pre-supposes a retiree who does not currently have sufficient wealth to purchase a life annuity that will yield her exogenously desired fixed consumption level. Therefore, she decides to self-annuitize, while dynamically managing her investment portfolio and possibly purchasing some annuities to minimize the probability of lifetime ruin. However, it turns out that she will not annuitize any of her wealth until she can fully cover her desired consumption by annuities. We derive a variational inequality that governs the ruin probability and optimal strategies and demonstrate that the problem can be recast as a related optimal stopping problem that yields a free-boundary problem that is more mathematically tractable. We numerically approximate the ruin probability and optimal strategies and examine how they change as we vary the mortality assumption and parameters of the financial model. Moreover, we solve the problem implicitly for exponential future lifetime. As a byproduct, we are able to quantify the reduction in lifetime ruin probability that comes from being able to dynamically manage the investment portfolio. Pricing American Options: a Variance Reduction Technique for the Longstaff-Schwartz Algorithm Nicola Moreni We propose a drift-based variance reduction technique that applies to the computation of American option prices via the Longstaff-Schwartz algorithm. It is well known that pricing American options is not an easy task and that difficulties arise as we increase the dimension of the underlying (stock) asset. In the unidimensional case, we dispose of efficient numerical techniques that approximate prices and that unfortunately cannot be extended to high-dimensional cases. A possible alternative consists in using Monte Carlo simulation and in the last decade, many authors proposed algorithms to price American options on multidimensional assets. We analyse the one introduced by Longstaff and Schwartz and which is based on a least squares approach, obtaining an efficient importance sampling variance reduction technique by means of Girsanov's theorem. The main idea is to jointly change the drift of the driving Brownian motion and the form of the payoff function, in order to obtain a set of equivalent option pricing problems. These pricing problems return the same price but have different convergence rates and among them, we search for the one which guarantees the fastest convergence. It is possible to prove the convergence of the modified algorithm as well as the existence of an optimal estimator. As it is not straightforward to locate the optimal estimator, we suggest some euristic approximations which make the algorithm more time performing. We analyse in details the numerical implementation and the results obtained by chosing different payoff functions and evolution models for the underlying. An empirically efficient cascade calibration of the LIBOR Market Model based only on directly quoted swaption data Damiano Brigo and Massimo Morini This work focuses on the swaptions automatic cascade calibration algorithm (CCA) for the LIBOR Market Model (LMM) first appeared in Brigo and Mercurio (2001). This method induces a direct analytical correspondence between market swaption volatilities and LMM parameters, and allows for a perfect recovery of market quoted swaption volatilities if a common industry swaptions approximation is used. We present explicitly an extension of the CCA to calibrate the entire swaption matrix rather than its upper diagonal part. Then, while previous tests on earlier data showed the appearance of numerical problems, we present here different calibration cases leading to acceptable results. We analyze the characteristics of the configurations used and concentrate on the effects of different exogenous instantaneous historical or parametric correlation matrices. We also investigate the influence of manipulations in input swaptions data for missing quotes, and devise a new algorithm maintaining all the positive characteristics of the CCA while relying only on directly quoted market data. Empirical results on a larger range of market situation and instantaneous covariance assumptions show this algorithm to be more robust and efficient than the previous version. Calibrated sigma parameters are in general regular and financially satisfactory, as confirmed by the analysis of various diagnostics implied structures. Finally we Monte Carlo investigate the reliability of the underlying LMM swaption analytical approximation in the new context, and present some possibilities to include information coming from the semi-annual tenor cap market. Square-root process and Asian options Jayalaxshmi Nagaradjasarma, Angelos Dassios Although the square-root process has long been used as an alternative to the Black-Scholes geometric Brownian motion model for option valuation, the pricing of Asian options on this diffusion model has never been studied analytically. However, the additivity property of the square-root process makes it a very suitable model for the analysis of Asian options. We here develop explicit prices for digital and regular Asian options. We also obtain a number of distributional results concerning the square-root process and its average over time, including analytic formulae for their joint density and moments. We also show that the distribution is actually determined by those moments. Valuation of Mortgage-Backed Securities Based on Unobservable Prepayment Cost Processes Hidetoshi Nakagawa and Tomoaki Shouda We propose a new prepayment model of mortgage in order to discuss the analysis of mortgage-backed securities (MBS) market. In our model, it is assumed that each mortgager in the underlying loan pool shall prepay rationally at the first time when her or his prepayment cost process falls below zero. Prepayment cost is defined as a stochastic process that consists of common cost and idiosyncratic cost. Common cost, called fundamental cost, depends on refinance rate and is observable at each time, while idiosyncratic cost is impossible to directly observe in the MBS market (hence so is the whole prepayment cost). Besides, there are several classes for mortgagers and the dynamics of idiosyncratic cost is specified according to the class. MBS investor can only the probability of which class each mortgager belongs to, but the probability depends upon an unobservable parameter that is recognized as state variable peculiar to MBS market. We calculate the conditional distributions of prepayment cost and the state variable given discrete-time observation to estimate parameters in the model and to deduce expected prepayment probability and hazard rate. Moreover we argue risk-neutrality in MBS market and a fair value of MBS. Fractional Volatility Models and Malliavin Calculus Chi Tim Ng and Ngai Hang Chan The purpose of this thesis is to develop European option pricing formulae for fractional market models. Although there have been previous papers discussing the option pricing problem for a fractional Black Scholes model using Wick calculus and Malliavin calculus, the formula obtained is similar to the classical Black Scholes formula, which cannot explain the volatility smile pattern that is observed in the market. In this thesis, a fractional version of the Constant Elasticity of Volatility (CEV) model is developed. An European option pricing formula similar to that of the classical CEV model is also obtained and a volatility skew pattern is revealed. The Futures Market Model and No-Arbitrage Conditions on the Volatility Jorgen Aase Nielsen, Kristian R. Miltersen, Klaus Sandmann Interest rate futures are basic securities and at the same time highly liquid traded objects. Despite this observation, most models of the term structure of interest rate assume forward rates as primary elements. The processes of futures prices are therefore endogenously determined in these models. In addition, in these models hedging strategies are based on forward and/or spot contracts and only to a limited extent on futures contracts. Inspired by the market model approach of forward rates by Miltersen, Sandmann, and Sondermann (1997), the starting point of this paper is a model of futures prices. Using the prices of futures on interest related assets as the input to the model, new no-arbitrage restricions on the volatility structure are derived. Moreover, these restrictions turn out to prevent an application of a market model based on futures prices. Higher order numerical algorithms for the solution of some path dependent options pricing problems Maria R Nogueiras, Alfredo Bermudez, Carlos Vazquez In this paper we deal with the numerical solution of some degenerate partial differential equations (PDE) or inequalities (PDI). These problems arise, for instance, when pricing path dependent options, as fixed-strike Amerasian options. We propose a general methodology that can be applied to one or several factors models. When there are restrictions on the solution, as early exercise features, the problem is formulated as a mixed problem (including a Lagrange multiplier) and then solved by a numerical iterative algorithm.This algorithm also provides the optimal exercise boundary. For numerical solution of PDE, we propose Lagrange-Galerkin methods of order two both in time and space. These are combinations of the characteristics method with the finite element method. A rigourous analysis of the stability and convergence properties of these methods has been developed. Finally, we will show an application to pricing (two-factors) Asian options, and compare our results with others existing in the bibliography. An agent market model using evolutionary game theory Craig Nolder and Benoit Montin We develope a simple agent based economic model. Stock prices evolve by trades between agents. Agents learn from the past by replicator dynamics of evolutionary game theory. By casting the model as a dynamical game we show from previous results the existence of a unique limiting distribution which serves as a stochastic equilibrium. We present a numerical study of this limiting distribution. It displays features similar to those of actual financial data. On utility based super replication prices Mark Owen We consider a financial market in which an agent can trade with only utility-induced restrictions on negative wealth. For a sufficiently integrable (but possibly unbounded) contingent claim, we give a representation of the utility-based super-replication price of the claim as the supremum of its discounted expectations under pricing measures with finite generalised entropy. Central to our proof is a bipolar relation between the cone of super replicable contingent claims with zero initial endowment, and the cone generated by pricing measures with finite loss-entropy. The cone of super replicable claims is shown to be the closure, under a relevant weak topology, of the cone of claims which are super replicable using only admissible strategies. If the agent has a utility function which is unbounded above, the set of pricing measures with finite loss-entropy can be slightly larger than the set of pricing measures with finite entropy. Indeed, the former set is also the closure of the latter under a weak topology. Symmetries and Pricing of exotic options in Levy models Antonis Papapantoleon and Ernst Eberlein This talk surveys symmetries and pricing methods for vanilla and exotic options in models driven by Lévy processes. Properties of time-inhomogeneous Lévy processes, more specifically processes with indepedent increments and absolutely continuous characteristics (PIIAC), are presented. Empirical research, using data from equity and bond markets, supports their use for financial modelling alongside Lévy processes. A method to explore symmetries between different payoffs –which involves a choice of numéraire, a subsequent change of measure and the characteristic triplet of a dual PIIAC– is described. This allows one to reduce the complexity of problems in option pricing. Firstly, symmetries between European (plain vanilla) call and put options are derived and a valuation method is outlined. Secondly, symmetries between floating and fixed strike Asian and Lookback options are described and some valuation methods are discussed. Finally, symmetries for payoffs involving two assets, such as Margrabe options, are explored. Satisfying convex risk limits by trading Traian Pirvu, Kasper Larsen, Steven E Shreve, Reha Tutuncu A random variable, representing the final position of a strategy, is deemed acceptable if under each of a variety of probability measures its expectation dominates a floor associated with the measure. The set of random variables representing pre-final positions from which it is possible to trade to final acceptability is characterized. In particular, the set of initial capitals from which one can trade to final acceptability is shown to be a closed half-line. Methods for computing it , and the application of these ideas to derivative security pricing is developed. Modeling the volatility and expected value of a diversified world index Eckhard Platen This paper considers a diversified world stock index in a continuous financial market with the growth optimal portfolio (GOP) as reference unit or benchmark. Diversified broadly based indices and portfolios, which include major world stock market indices, are shown to approximate the GOP. It is demonstrated that a key financial quantity is the trend of a world index. It turns out that it can be directly observed since the expected increments of the index equal four times those of the quadratic variation of its square root. Using a world stock index as approximation of the discounted GOP it is shown that, in reality, the trend of the discounted GOP does not vary greatly in the long term. This leads for a diversified world index to a natural model, where the index is a transformed square root process of dimension four. The squared index volatility appears then as the inverse of the square root process. This feature explains most of the properties of a stock index and its volatility. Optimal Mortgage Refinancing with Endogenous Mortgage Rates Stanley Pliska This paper is based upon earlier work with and by Goncharov, where risk neutral martingale methods together with the intensity-based approach borrowed from credit risk modeling were used to derive new, continuous time results for the value of a mortgage contract that is subject to prepayment risk. In particular, with the mortgagor's prepayment behavior suitably modeled by an intensity process, in the absence of arbitrage opportunities the interest rate for a fixed rate mortgage was shown to be an endogenous variable that can be computed. This paper extends these results by presenting three new developments. First, the earlier results are derived for a discrete time environment. Second, a complementary problem is solved, where the mortgage market is fixed, refinancing incurs transaction costs, and the mortgagor seeks to choose the refinancing schedule in an optimal manner. The solution is obtained via an infinite horizon, but nonstandard, Markov decision chain. Third, this paper addresses and solves an economic equilibrium problem where the mortgagor is a representative agent, he/she refinances in an optimal fashion, and the value of the mortgage rate is such that the mortgage market is free of arbitrage opportunities. In other words, the resulting solution provides both the optimal refinancing schedule as well as the endogenous mortgage rate. The Critical Kurtosis Value and Skewness Correction Vassilis Polimenis In the empirical option pricing literature, it is generally agreed that the pronounced volatility smirks are signs of a strongly negative risk neutral skewness. The paper provides exact conditions under which skewness in a Lévy process is corrected. The initial observation is that, unable to dynamically hedge pure jump gamma risk, agents scale risk neutral volatility. Such scaling violates arguments in the literature that leptokurtosis always leads to skewness correction. Actually, it is shown that such arguments are correct only for symmetric Lévy processes. The general formula for skewness correction for homogeneous Lévy processes is developed. Jump induced skewness is corrected only when kurtosis is greater than a critical value. Efficient trading strategies with transactions costs Vincent Porte and Elyès Jouini In this article, we characterize efficient contingent claims in a context of transaction costs and multidimensional utility functions. The dual formulation of utility maximization helps us outline the key notion of cyclic anticomonotonicity. Moreover, after defining a utility price in this multidimensional setting, we provide a measure of strategies inefficiency and a tool allowing to effectively compute this measure with the help of cyclic anticomonotonicity. Exotic Options: Proofs Without Formulas Rolf Poulsen We review how reflection results can be used to give simple proofs of price formulas and derivations of static hedge portfolios for barrier and lookback options in the Black-Scholes model. A Theory of stochastic integration for bond markets. Maurizio Pratelli and Marzia DE DONNO (Based on a joint paper with Marzia De Donno). In the Bond Market model there is a continuum of securities, and this gives rise to the problem of what exactly should be meant by the word "portfolio" in this setting: for this reason, Bjork-DiMasi- Kabanov and Runggaldier (1997) considered the bond market as a stochastic process with values in the space of continuous functions C([0,T]), and introduced a construction of a stochastic integral where the integrand process (i.e. the mathematical model for a self-financing strategy) is a process with values in the dual on C([0,T]), i.e. the set of Radon measures on [0,T]. But this approach, which seems to be the natural one, has some drawbacks: for instance the "uniqueness" of martingale probability is equivalent to the "approximate completeness". We introduce a theory of stochastic integration, starting from "elementary integrands" (the mathematical representation of portfolios based on a finite number of bonds) and going to the limit with "generalized integrands": with our theory, the stochastic integral is "isometric" and we prove an extension of a result due to Memin (limit of stochastic integrals, for the semimartingale topology, is still a stochastic integral). This gives a good answer to the problem of completeness, and with an appropriate definition of "admissible" strategy it is possible to extend to this infinite dimensional setting the method of convex duality for the problem of "utility maximization". But the "generalized integrands" cannot be characterized in an explicit way: with further regularity assumptions we can state explicitly which kind of processes are integrable. Finally, we give some applications of these results to the Bond Market models. Weak Convergence of Option Quantile Hedging Strategies Jean-Luc Prigent Option hedging of contingent claims is a prominent problem in finance. This problem is straightforward when dealing with complete markets. However, most of financial markets are incomplete. In that case, the risk-neutral probability is no longer unique and contingent claims are not all attainable. The problem of hedging can be seen by another point of view: what should an investor do if he is unwilling to invest all the amount needed for a perfect hedging or a superhedging strategy ? Alternatively, we can ask: which initial amount the investor can save by accepting a certain fixed ``shortfall'' probability? Two seminal papers of Follmer and Leukert (1999), (2000) deal with quantile hedging. In Prigent (1999) and Prigent and Scaillet (2002), it is proved that both prices and hedging strategies associated to the locally risk-minimizing criteria are stable under weak convergence. The goal of this paper is to examine the same problem for the quantile approach. It is proved that the stability under convergence is generally satisfied in the complete case. Nevertheless, for the incomplete case, usually there is no longer stability under weak convergence. This property has potential applications since it indicates that we have to take care when using quantile approach from the convergence point of view. AMS 2000 classification : 60 F05, 91 B 28. A Chaotic Approach to Interest Rate Modelling Avraam Rafailidis and Lane Hughston This paper presents a new approach to interest rate dynamics. We consider the general family of arbitrage-free positive interest rate models, valid on all time horizons, in the case of a discount bond system driven by a Brownian motion of one or more dimensions. We show that the space of such models admits a canonical mapping to the space of square-integrable Wiener functionals. This is achieved by means of a conditional variance representation for the state price density. The Wiener chaos expansion technique is then used to formulate a systematic analysis of the structure and classification of interest rate models. We show that the specification of a first-chaos model is equivalent to the specification of an admissible initial yield curve. A comprehensive development of the second-chaos interest rate theory is presented in the case of a single Brownian factor, and we show that there is a natural methodology for calibrating the model to at-the-money-forward caplet prices. The factorisable second-chaos models are particularly tractable, and lead to closed-form expressions for options on bonds and for swaptions. In conclusion we outline a general "international" model for interest rates and foreign exchange, for which each currency admits an associated family of discount bonds, and show that the entire system can be generated by a vector of Wiener functionals. Nonparametric estimation of the diffusion coefficient via Fourier analysis, with an application to short interest rates Roberto Reno' We introduce an original fully nonparametric estimator of the diffusion coefficient of an univariate diffusion, which makes use of discrete observations. Infill asymptotic properties of the estimator are fully derived: the estimator is proven to be consistent and asymptotically normally distributed. Monte Carlo simulations are conducted to explore the small-sample properties of the estimator and compare it to already known estimators. Finally, the estimator is implemented on short rate time series and the diffusion coefficient is estimated. A Two-Factor Model for Commodity Prices and Futures Valuation Diana Ribeiro and Stewart Hodges This paper develops a reduced form two-factor model for commodity spot prices and futures valuation. This model extends Schwartz's (1997) two-factor model by adding two new features. First the Ornstein-Uhlenbeck process for the convenience yield is replaced by a Cox-Ingersoll-Ross (CIR) process. This ensures that our model is arbitrage-free. Second, spot price's volatility is proportional to the square root of the convenience yield level. We empirically test both models using weekly crude oil futures data from 17th of March 1999 to the 24th of December 2003. In both cases, we estimate the model parameters using the Kalman filter. Correcting for Simulation Bias in Monte Carlo methods to Value Exotic Options in Models Driven by Lévy Processes Claudia Ribeiro and Nick Webber Lévy processes can be used to model asset return's distributions. Monte Carlo methods must frequently be used to value path dependent options in these models, but Monte Carlo methods can be prone to considerable simulation bias when valuing options with continuous reset conditions. In this paper we show how to correct for this bias for a range of options by generating a sample from the extremes distribution of the Lévy process on subintervals. We work with the variance-gamma and normal inverse Gaussian processes. We find the method gives considerable reductions in bias, so that it becomes feasible to apply variance reduction methods. The method seems to be a very fruitful approach in a framework in which many options do not have analytical solutions. A synthetic measure of multivariate risk and its empirical implications for portfolio risk management Andrea Roncoroni and Stefano Galluccio We define a new measure of multivariate risk based on the typical shapes displayed by term structure movements. The resulting cross-shape covariance is decomposed into uncorrelated shape factors. Empirical tests conducted on a U.S. database over a wide range of hedging scenarios suggest that these factors represent more accurately the yield curve risk than those stemming from the classical principal components analysis of cross-yield covariance. Estimating the Commodity Market Price of Risk for Energy Prices Ehud Ronn and Sergey P. Kolos The purpose of this paper is to determine the magnitude and sign of the commodity "market price of risk" (MPR) for electricity and natural-gas prices. This MPR variable determines whether forward prices in energy are upward- or downward-biased predictors of future expected spot prices. We evaluate that risk premium by estimating the drift term in spot and forward prices. In futures prices, we explicitly account for the Samuelson hypothesis "term structure of volatility." In spot prices of electricity, we examine the relationship between Day-Ahead Prices and Real-Time Prices. The results have implications for understanding the relationship between energy markets and other physical and financial markets, for incorporating the risk premium in making informed hedging decisions in industry, and for relating futures prices to the forecast prices produced by industry. Pathwise Optimality for Benchmark Tracking Wolfgang Runggaldier, Paolo Dai Pra, Marco Tolotti We consider the problem of investing in a portfolio in order to track a given benchmark. We study this problem from the point of view of almost sure/pathwise optimality. We first obtain a control that is optimal in the mean and this control is then shown to be also pathwise optimal. The standard Merton model leads to lognormality of the value process so that it does not possess the ergodic properties required for pathwise optimality. We obtain ergodicity by transforming the process so that it remains bounded thereby using a method that can be related to random time change. We furthermore describe a general approach to solve the Hamilton-Jacobi-Bellman equation corresponding to the given problem setup. A numerical study of the smile effect in implied volatilities induced by a nonlinear feedback model Simona Sanfelici, Maria Elvira Mancino, Shigeyoshi Ogawa The Black-Scholes model requires that the volatility is constant. Nevertheless this hypothesis is recognized to cause option price distortions. In order to give explanation to this behavior the literature has proposed both of modelling directly volatility as a stochastic process either of relaxing the hypothesis on which Black-Scholes model is based, such as the facts that the market is complete, frictionless and that the agents act as price takers. We propose a model where the price is defined in equilibrium with some traders employing a trading strategy to hedge a contingent claim. As a consequence we get a fully nonlinear partial differential equation which implicitly incorporates the demand induced by hedging. In this paper we present numerical results which show the impact of the hedging strategies for derivative asset analysis. In particular we obtain the smile pattern and term structure of implied volatility. The numerical results show that in our model the implied volatility smile can be reproduced as a consequence of dynamical hedging. The simulations are performed using the Infinite Elements Method. Utility maximization for unbounded processes Sara Biagini and Frittelli Marco When the price processes of the financial assets are described by possibly unbounded semimartingales, the classical concept of admissible trading strategies may lead to a trivial utility maximization problem because the set of bounded from below stochastic integrals may be reduced to the zero process. However, it could happen that the investor is willing to trade in such a risky market, where potential losses are unlimited, in order to increase his/her expected utility. We translate this attitude into mathematical terms by employing a class $\mathcal{H}^{W}$ of $W-$admissible trading strategies which depend on a loss random variable $W$. These strategies enjoy good mathematical properties and the losses they could give rise to in the trading are compatible with the preferences of the agent. We formulate and analyze by duality methods the utility maximization problem on the new domain ${H}^{W}$. We show that, for all loss variables W contained in a properly identified set $\mathcal{W}$, the optimal value on the class $\mathcal{H}^{W}$ is constant and coincides with the optimal value of the maximization problem over a larger domain ${K}_{\Phi }.$ The class ${K}_{\Phi }$ doesn't depend on the single $W\in\mathcal{W},$ but it depends on the utility function $u$ through its conjugate function $\Phi $. By duality methods we show that the optimal solution exists in ${K}% _{\Phi }$ and it can be represented as a stochastic integral that is a uniformly integrable martingale under the minimax measure. We provide the economic interpretation of the larger class ${K}%_{\Phi }$ and we analyze some examples that show that this enlargement of the class of trading strategies is indeed necessary. Asymptotic Analysis for American Options on Alternative Stochastic Processes David Saunders and John Chadam We study the free-boundary problem arising from the pricing of the finite horizon American put option. We extend known results on the asymptotic behaviour of the optimal exercise boundary (critical stock price) to the case where the underlying follows a process other than geometric Brownian motion. In particular, we derive higher order asymptotic expansions of the boundary near expiry, as well as approximations for larger times. Our results cover both the case of a diffusion process with a volatility surface (e.g. the constant elasticity of variance, or CEV model) and a class of jump diffusion processes. The Valuation of Callable Contingent Claims with Applications Katsushige Sawaki and Susumu Seko The purpose of this paper is to introduce and analyze callable contingent claims with an application to callable American options. These contingent claims are bilateral contracts which entitle both a seller(issuer) and a buyer(holder) to exercise the rights respectively. The seller can cancel(call) the contract by paying some penalty to the buyer and the buyer can exercise(put) the right at any time before the maturity of the contract. In this paper we establish a pricing formula for the callable contingent claims that can be decomposed into the difference of the value of an American option and the callable discount. Moreover, following our decomposition of the callable contingent claim, the value of the claim can be solved only with the numerical technique based upon the discrete binomial model, which provides a visual understanding of the analytical properties of the value. Analytic American Option Pricing: The Flat-Barrier Lower Bound Alessandro Sbuelz I study the pricing performance of a closed-form lower bound to American option values based on an exercise strategy corresponding to a flat exercise boundary. The lower bound has a simple two-step implementation akin to the Barone-Adesi and Whaley (1987) formula and shows superior pricing performance in the Out-of-The-Money region and for long maturities. JEL-Classification: G12, G13. Keywords: American Options, Flat Barrier Options. Risk measures and capital requirements for processes Giacomo Scandolo In this paper we propose a generalization of the concepts of convex and coherent risk measures to a multi-period setting, in which payoffs are spread over different dates. To this end, a careful examination of the axiom of translation invariance and the related concept of capital requirement in the one-period date is performed. These two issues are then suitably extended to the multi-period case, in a way that makes their operative financial meaning clear. A characterization in terms of expected values is derived for this class of risk measures and some examples are presented. Evolutionary Stable Stock Markets Klaus Reiner  Schenk-Hoppé, Igor V. Evstigneev, Thorsten Hens This paper shows that a stock market is evolutionary stable if and only if stocks are evaluated by expected relative dividends. Any other market can be invaded by portfolio rules that will gain market wealth and hence change the valuation. In the model the valuation of assets is given by the wealth average of the portfolio rules in the market. The wealth dynamics is modelled as a random dynamical system. Necessary and sufficient conditions are derived for the evolutionary stability of portfolio rules when (relative) dividend payoffs form a stationary Markov process. These local stability conditions lead to a unique evolutionary stable strategy according to which assets are evaluated by expected relative dividends. Optimal investments for robust utility functionals Alexander Schied We introduce a systematic approach to the problem of maximizing the robust utility of the terminal wealth in a general market model, where the robust utility functional is defined via a lower expectation of a set Q of probability measures. In a complete model, this problem can be reduced to determining a least favorable measure in Q. This least favorable measure often is universal in the sense that it does not depend on the particular utility function and can be constructed by using the Huber-Strassen theory from robust statistics. Applications include the problem of robust utility maximization with uncertain drift and optimal investment under weak information. Building on results by Kramkov and Schachermayer, we also discuss the duality theory for general incomplete market models. Infinite Factor Models for Credit Risk Thorsten Schmidt The defaultable term structure is modeled in two ways, both in an infinite dimensional framework. First, we use stochastic differential equations in Hilbert spaces, which allows for incorporating different recovery scenarios as well as ratings. Second, a special case is described via transformations of Gaussian random fields. This approach concentrates on deriving explicit pricing formulas and calibration of the model. In the calibration procedure we account for the fact, that credit derivatives data is scarce, so the model can be calibrated using only a small set of credit derivatives. Hedging issues are also discussed. Information-Driven Default Contagion Philipp Schoenbucher Much of the existing literature on default contagion assumes a direct causal relationships between two obligors' defaults. In this paper we present a model in which default contagion arises without causal links solely from information effects if investors are imperfectly informed about some common factors affecting the true riskiness of the obligors. We model this effect in a simple extension of the intensity-based modelling framework using unobserved frailty variables. This allows the modelling of much higher (and more realistic) levels of default dependence between the obligors than what purely diffusion-based intensity models were able to capture previously, without adding too much additional complexity. The parameters of the dependence can be implied directly from spread jumps observed in the market, thus enabling a full specification of the model under pricing probabilities without recourse to historical default correlations. We furthermore present an extension of the model in which the size of the contagion effect can depend on the reason for the default and not just the identity of the defaulted obligor, thus introducing stochastic dependency. Pricing Swaptions in Affine Term Structure Models David Schrager and A. A. J. Pelsser We propose an approach to find an approximate price of a swaption in Affine Term Structure Models. Our approach is based on the derivation of approximate dynamics in which the volatility of the Forward Swap Rate is itself an affine function of the factors. Hence we remain in the Affine framework and well known results on transforms and transform inversion can be used to obtain swaption prices in ways similar to bond options (i.e. caplets). We demonstrate that we can even obtain a closed form formula for the approximate price which is based on square-root dynamics for the swap rate. The latter approximation is extremely fast to compute while remaining accurate. Computational time compares favorably with other known approximation methods. The results on the quality of the approximation are excellent. Our results show that, analogously to the LIBOR Market Model, LIBOR and Swap rates are driven by approximately the same type of (in this case affine) dynamics. On the Martingale Measures in Exponential Levy Models Andrey Selivanov We study the existence and the uniqueness of martingale measures in 1) exponential Levy models and 2) time-changed exponential Levy models. We consider models with finite and infinite time horizon. By the First Fundamental Theorem of Asset Pricing our problem is equivalent to the problem of absence of arbitrage (and completeness) of the corresponding model. We discuss two arbitrage concepts: traditional Free Lunch with Vanishing Risk and so-called Generalized Arbitrage. We obtain criteria for the absence of arbitrage for the first model both with finite and infinite time horizon and for the second model with finite time horizon. Bermudan Guaranteed Return Contracts: Analysis and Valuation Steven Simon A guaranteed or minimum return can be found in different financial products, e.g. 'guaranteed investment contracts' (GIC's) issued by investment banks and life-insurance contracts. We consider the so-called multi-period or compounding guaranteed return contracts. With such a contract a minimum return is guaranteed over a series of sub-periods, making the pay-off at maturity path-dependent. We first derive some important features of the European style contract in a general Heath-Jarrow-Morton framework. Secondly, we analyze the effect of adding a Bermudan put feature to this type contract and we derive some interesting properties about the optimal exercise behavior. More precisely, we derive the sufficient and necessary conditions on the dynamics of the underlying asset for the optimal exercise decision to be a stopping time with respect to the term structure of interest rates. The Merton Problem in an Illiquid Financial Market Surbjeet Singh and L.C.G. Rogers Liquidity is an important effect in the markets, yet it is hard to come up with a definition which not only has some economic explanation but also retains a reasonable degree of tractability. In this paper, we propose a simple microeconomic model in discrete time which carries over to the continuous-time setting; this results in a modification of the usual dynamics of portfolio wealth, which appears to be impossible to analyse exactly. We investigate the Merton problem numerically, and through some asymptotic analysis. A Two-Person Game for Pricing Convertible Bonds Mihai Sirbu and Steven E. Shreve A firm issues a convertible bond. At each subsequent time, the bondholder must decide whether to keep the bond, thereby collecting coupons, or to convert it to stock. The bondholder wishes to choose a conversion strategy to maximize the bond value. Subject to some restrictions, a convertible bond can be called by the issuing firm, which presumably acts to maximize equity value and thus to minimize the bond value. This creates a two-person game, and we model the bond price as the value of this game. We show, however, that under our standing assumption (dividends are paid at a lower rate than the money market rate) this game reduces to one of two optimal stopping problems, and the relevant stopping problem can be determined a priori, i.e., without first solving the convertible bond pricing problem. Because of dividend payments, the partial differential equation describing the pricing function becomes nonlinear. This means that our analysis involves a fixed point problem. We also prove that for large time to maturity the value of the convertible bond approaches the value of the perpetual convertible bond. The presentation is based on joint work with Steven E. Shreve. Asymptotic Option Pricing under a Pure Jump Process Seongjoo Song This paper studies the problem of option pricing in an incomplete market. Under the market incompleteness from the discontinuity of the asset price process, we try to find a reasonable price for a European contingent claim by adopting an asymptotic approach. First, we find the unique minimal martingale measure and get a price by taking an expectation of the payoff under this measure. We also show that it converges weakly to the equivalent martingale measure in the limit. To get a closed-form price, we use an asymptotic expansion. In case where the minimal martingale measure is a signed measure, we use a sequence of martingale measures that converges to the equivalent martingale measure in the limit to compute the price of an option. Good Deal Bounds for Valuation of Real and Financial Options Jeremy Staum We explore an approach to constructing good deal bounds for pricing contingent claims in incomplete markets. The approach applies to trades in financial options and to real investment decisions, which may involve real options. It is grounded in equilibrium theories (such as the capital asset pricing model) which support the decision tools (such as net present value and real options analyses) currently used in practice to make real investment decisions. An equilibrium model yields a specification of the acceptance set used to construct good deal bounds. Good deal bounds enable trading and investment decisions to incorporate robustness or aversion to ambiguity about stochastic models, for instance, regarding market risk premia. Optimal Investments in Markets with Stochastic Opportunity Sets Sasha Stoikov and Thaleia Zariphopoulou A class of optimal investment and consumption models in incomplete market environments will be analyzed. The focus will be on a universal characterization of the optimal portfolios (myopic and excess risky demand) in terms of hedging strategies of supporting pseudoclaims. These claims are written on the market price of risk and are priced by indifference. Recent results on indifference prices will be used for the sensitivity and robustness analysis of the optimal investments. Issues related to model specification and to the interplay between market incompleteness and risk preferences will also be discussed. Optimal Statistical Decisions About Some Alternative Financial Models Wolfgang Stummer and Igor Vajda We deal with Bayesian decisions between the following two models for the price dynamics X(t) of a financial asset X: (i) the prominent geometric Brownian motion, which is a special diffusion process with constant growth rate and constant volatility, and (ii) a (generally non-lognormally-distributed) diffusion process with non-constant growth rate but with the same volatility. We derive some bounds on the corresponding decision-theoretic quantities such as e.g. the Bayes factor, the Bayes loss (minimal mean loss) and the Bayes probability of (discrimination) error. This is achieved with the help of obtained bounds on generalized relative entropies between (i) and (ii). Furthermore, we also investigate the total variation distance between the two models. Endogenous Risk Aversion and Ockham's Razor Michael Stutzer Influential early articles by Paul Samuelson advocated use of expected concave utility of wealth criteria in T-repeated betting and investment problems. He and other founders of modern decision theory viewed their work as normative prescriptions for choice under uncertainty; not just as predictive theories of pre-existing behavior. Results of Rabin (Econometrica, 2000), exposited and applied in Rabin and Thaler (Journal of Economic Perspectives, 2001, pp. 219-232), directly challenged both the prescriptive and predictive usefulness of ANY expected concave utility criterion in these settings. As a predictive alternative, they advocated the use of loss averse preferences as a substitute for expected concave utility. While different systems of preference axioms have been found that imply the use of specific alternatives to concave utility, they are not normatively convincing. Moreover, none of them do anything to solve a problem that plagues both the prescriptive and predictive use of expected utility: results critically depend on adjustable preference parameters that are difficult to directly measure, and hence must be "fit" to the observable data that they attempt to explain. As a simpler alternative criterion, this paper proposes the probability of outperforming an observable benchmark the agent wants to beat. This criterion does not suffer from the possible ills of some other probabilistic criteria that were (influentially) critiqued by Samuelson. Large deviations theory is used to show that for suitably large T, this criterion is equivalent to maximizing an expected CRRA (power) habit-formation utility, but with a coefficient of risk aversion that varies endogenously with the alternative evaluated. This eliminates the adjustable curvature parameter used in other expected and non-expected utility (e.g. loss aversion) preference theories, in accord with the scientific principle of parsimonious parameterization called Ockham's Razor. A Bayesian formulation of Ockham's Razor is used to illustrate the advantages of this parsimony. A Bayesian Learning Model of Risk Taking by Fund Managers Ajay Subramanian, Ping Hu, Jayant Kale  We propose a multi-period bayesian learning model to examine the impact of explicit and implicit incentives on the risk taking behavior of fund managers. We show that implicit incentives arising from career concerns and, especially, employment risk lead to a non-monotonic U-shaped relation between a fund manager's relative risk choices and her prior performance relative to her peers. Moreover, the incentive to increase relative risk, ceteris paribus, declines with the manager's experience. Our empirical analysis of fund managers' risk taking behavior documents significant support for our theory. Our study therefore demonstrates the importance of both explicit and implicit incentives in driving the risk-taking behavior of fund managers. On an Alternative Approach to Pricing General Barrier Options Michael Suchanecki In this paper, an alternative approach to pricing barrier options is presented that relies on the use of the first hitting time density to the barrier. Laplace transforms with respect to time are used in order to determine option prices. It turns out that this approach allows for pricing more general barrier options (at least numerically). As an example, a simple down-and-in call option is considered. In this case, all Laplace transforms (even the finite Laplace transforms) can be inverted analytically and the well-known closed form pricing formula is obtained. Environment &Financial Markets Wojciech Szatzschneider and Monique Jeanblanc We argue that practical solutions for the environmental degradation are in a short supply. Most of the increasingly complex models set off different opinions about their applicability. Models should be well specified. This requirement is hard to meet in environmental studies. Thus, the efficient global environmental decision--making becomes very difficult. Moreover politicians often tend to justify their decisions by inappropriate theories. This situation leads to proliferation of ineffective studies and waste of resources. We shall propose to apply the market approach in the solutions of several environmental problems. It could result in more transparent transfer of funds and the involvement of everybody concerned. Also we can expect that the transparency could stem in an increment of these funds. We shall focus on the issue of deforestation due to its importance for the global well--being, and the possibility to assess the number of trees. Our approach is based on a positive involvement of holders of "good" options bought or, in the first stage, obtained for free. In the case of the forest "good" means a kind of Asian call option. We will show that, in a natural way, three kinds of optimization problems crop up: 1)Individual agent problem. 2)Local optimization problem. 3)Global optimization problem. The first one is how the holder of a good option could eventually contribute to reforestation. The second one is how to choose prices of options, to maximize the space mean of the temporal mean of the "asset" in given place. The last one is how to distribute funds into particular projects. In the final part we analyze the dynamical control for bounded processes and awards partially based on the mean of the underlying value. A new fast and accurate method to calculate Value-at-Risk and other tail risk measures Tanya  Tamarchenko and Rabi De We present a highly efficient methodology for calculating Value-at-Risk (VAR) and other tail-risk measures for a portfolio of derivative securities. This new methodology, named "Reliability-VAR" provides an answer to the question that is important to risk managers: what are the most likely values of underlying risk factors that cause a loss of a certain size? We show that the ability to answer this question enables us to construct highly efficient hybrid numerical procedures to calculate the tail of distribution for portfolio losses. We present numerical examples that demonstrate the superior performance of Reliability VAR in comparison to commonly used VAR methods. Pricing CEV moving barrier options with time-dependent parameters — Lie algebraic approach Hoi-man Tang, C.F. Lo, C.H. Hui In this paper we apply the Lie-algebraic technique for the valuation of moving barrier options with time-dependent parameters. The value of the underlying asset is assumed to follow the constant elasticity of variance (CEV) process. By exploiting the dynamical symmetry of the pricing partial differential equations, the new approach enables us to derive the analytical kernels of the pricing formulae straightforwardly, and thus provides an efficient way for computing the prices of the moving barrier options. The method is also able to provide tight upper and lower bounds for the exact prices of CEV barrier options with fixed barriers. In view of the CEV model being empirically considered to be a better candidate in equity option pricing than the traditional Black-Scholes model, our new approach could facilitate more efficient comparative pricing and precise risk management in equity derivatives with barriers by incorporating term-structures of interest rates, volatility and dividend into the CEV option valuation model. Solvable affine term structure models Claudio Tebaldi and Martino Grasselli Pricing of contingent claims in the Affine Term Structure Models (ATSM) can be reduced to the solution of a set of Riccati-type Ordinary Differential Equations (ODE), as shown in Duffie, Pan and Singleton (2000) and in Duffie, Filipovic and Schachermayer (2003). We discuss the solvability of these equations. While admissibility is a necessary and sufficient condition in order to express their general solution as an analytic series expansion, we prove that, when the factors are restricted to have continuous paths, these ODE admit a fundamental system of solutions if and only if all the positive factors are independent. Finally, we classify and solve all the consistent polynomial term structure models admitting a fundamental system of solutions. Optimal Portfolio Strategies with Different Constraints: A Unified Treatment Phelim Boyle and Weidong Tian The traditional portfolio selection problem concerns an agent who wishes to maximize expected utility of consumption and or terminal wealth during some planning horizon. This basic problem can be modified by adding constraints and portfolio insurance provides one example. In this paper we investigate the portfolio selection problem under different constraints within a unified framework. These constraints reflect differing objectives and in this paper we analyze three types of objectives. First we consider an investment strategy which aims to maximize expected utility subject to a guaranteed (or minimal performance) wealth level. Under this strategy a benchmark index is chosen to reflect the particular objective. The benchmark index can be either deterministic or stochastic. The second strategy is designed to maximize expected utility with the added objective of taking on investment risk with the aim of realizing higher returns. In this case the constraint is expressed in probabilistic terms as for example in a conventional Value at Risk framework. Once again the benchmark index can be either deterministic or stochastic. Under the third strategy a portfolio manager follows a mixed strategy, by combining the previous two investment objectives. In this case the investor maximizes expected utility while achieving a guaranteed return and also maintaining features of the second strategy as well. We analyze these three investment problems and provide explicit solutions and discuss several applications. Efficient Computation of Hedging Parameters for Discretely Exercisable Options Stathis Tompaidis, Ron Kaniel, Alexander Zemlianov We describe a method to obtain bounds on the hedging parameters of discretely exercisable options using Monte-Carlo simulation. The method is based on a combination of the duality formulation of the optimal stoping problem for pricing discretely exercisable options and Monte-Carlo estimation of hedging parameters for European options. For a given computer budget and exercise strategy we provide an algorithm that achieves the tightest bounds. The method can handle arbitrary payoff functions, general diffusion processes, and a large number of random factors. We also present a fast, heuristic, alternative method and use our method to evaluate its accuracy. Flexible Complete Models with Stochastic Volatility: Generalising Hobson & Rogers (1998) Robert Tompkins, Friedrich Hubalek, Josef Teichmann Hobson and Rogers (1998) propose an option pricing model where the volatility is a deterministic function of the moving average of past (logarithm of) underlying prices. They show that such a model can also generate implied volatilities that vary across striking price and term to expiration. In this research, this model is tested on actual option markets. While the Hobson and Rogers (1998) model produces divergences from Black-Scholes (1973) prices on a microscopic scale, we have not been able to replicate actual option prices with this model. To determine prices from this model we develop a robust analytic approximation. To better fit observed options prices, we generalise the model Hobson and Rogers (1998) by the addition of two additional parameters. This model is able to match option prices on the British Pound/US Dollar across both the striking price dimension (smiles) and across different maturities (the term structure of implied volatility). By use of Mavillian calculus, we are able to determine partial derivatives of the generalised model and compute hedging ratios. Superreplication of options on several underlying assets Johan Tysk, Erik Ekström, Svante Janson We investigate when a hedger who over-estimates the volatility with a time- and level-dependent volatility model will superreplicate a convex claim on several underlying assets. It is shown that the classical Black-Scholes model is the only model, within a large class, for which over-estimation of the volatility yields the desired superreplication property for any convex claim. This is in contrast to the one-dimensional case, in which it is known that over-estimation of the volatility with any model guarantees superreplication of convex claims. The proof is based on the fact that preservation of convexity of solutions to parabolic partial differential equations, with no lower order terms, is a generic property in one spatial dimension but is very rare in higher dimensions. A new jump-diffusion model and performances of affine stochastic volatility models for equity emerging markets Rosanna Pezzo and Mariacristina Uberti Most traditional models fail short of completely capturing several volatility properties, such as time variation, clustering, mean reversion and volatility smile. A good volatility model should be able to reflect the above properties as well as take into account the fact that positive and negative shocks in returns have rather different impact on the volatility. This leads us to involve affine stochastic volatility models with jumps. While in literature similar models are usually applied to option pricing, Pezzo-Uberti (2002) showed how Duffie-Pan-Singleton's affine jump-diffusion model with jumps in returns and volatility outperforms the stochastic volatility model without jumps in volatility forecasting. In this paper we propose a new affine jump-diffusion model for volatility, in which returns and volatility are generated by two independent stochastic processes and shocks are driven by separate jump processes. We model the volatility jumps with a different kind of positive distribution: an Inverted-Gamma. Some interesting theoretical results involve the jump joint process that turns out to follow a Cauchy distribution. We calibrate and use the model to simulate the volatility behaviour of assets in very volatile markets. A good match with real data is pointed out and in several occasions the model performs better than the existing models. Criterions for absolute continuity and singularity of measures via separating times Mikhail Urusov and A. Cherny We introduce a notion of separating time for a pair of measures P and Q on a filtered space. This notion is convenient for describing the mutual arrangement of P and Q from the viewpoint of absolute continuity and singularity. Furthermore, we find an explicit form of the separating time for the case, where P and Q are distributions of Levy processes, solutions of stochastic differential equations, and distributions of Bessel processes. The obtained results yield, in particular, criteria for local absolute continuity, absolute continuity, and singularity of P and Q. Portfolio Analysis with General Deviation Measures Michael Zabarankin, R. Tyrrell Rockafellar, Stan Uryasev Generalized measures of deviation, as substitutes for standard deviation, are considered in a framework like that of classical portfolio theory for coping with the uncertainty inherent in achieving rates of return beyond the risk-free rate. Such measures, associated for example with conditional value-at-risk and its variants, can reflect the different attitudes of different classes of investors. They lead nonetheless to generalized one-fund theorems as well as to covariance relations which resemble those commonly used in capital asset pricing models (CAPM), but have wider interpretations. A more customized version of portfolio optimization is the aim, rather than the idea that a single "master fund" might arise from market equilibrium and serve the interests of all investors. Through techniques of convex analysis, they deal rigorously with a number of features that have not been given much attention in this subject, such as solution nonuniqueness, or nonexistence, and a potential lack of differentiability of the deviation expression with respect to the portfolio weights. Moreover they address in detail the previously neglected phenomenon that, if the risk-free rate lies above a certain threshold, a master fund of the usual type will fail to exist and need to be replaced by one of an alternative type, representing a "net short position" instead of a "net long position" in the risky instruments. Pricing of arithmetic Asian options and basket options by conditioning on more than one variable Michèle Vanmaele, Deelstra Griselda, Liinev Jan Pricing of arithmetic Asian options and basket options where the underlying is modelled in a Black-Scholes setting, boils down to computing stop-loss premia of sums of dependent lognormal random variables. In earlier work we derived lower and upper bounds for these types of options based on comonotonicity results and by conditioning upon one variable. A natural extension is to condition on more than one variable, because one intuitively expects in this way to improve those bounds. We derive analytical expressions for the comonotonic bounds of stop-loss premia of sums of dependent random variables when conditioning on more variables. As expected, the dimension of the integrals in the comonotonic bounds increases with the number of conditioning variables. However in case of two conditioning variables, the lower bound uses only one integration if the conditional density function is known. We specify this lower bound in the case of a sum of lognormal variables. Numerical results show that conditioning on two variables leads to very sharp lower bounds. Comparison Theorem and Option Pricing in the Presence of Jumps Jan  Vecer and Mingxin Xu It is well known that in diffusional framework, the process with the largest volatility term in absolute value has the largest expectation if evaluated in some convex function (under certain mild technical conditions). This result has immediate applications to option pricing and stochastic optimal control problems. Since option prices are computed as expectations of convex payoffs corresponding to the stock price process, one expects that the price will be higher for stocks with higher volatility. The aim of this talk is to explore possibilities of generalizing this result to processes which exhibit jumps. One might expect that the larger is the jump size, the larger would be the convex payoff, but this turns out to be not true in general. This talk will discuss these issues and the consequences for option pricing and other areas of stochastic analysis. (Joint work with Mingxin Xu). Expanding the Universe of Exotic Options Closed Pricing Formulas in the Black Sholes Framework Carlos Veiga A pricing method resulting in a closed formula is proposed for a large class of options such as Best Of and Rainbow based on the analysis of the return profile of the option. We assume that returns follow a brownian motion and the usual hypotheses of the Black-Scholes model extended for the multi-underlying/multi-currency case. The result states that, if the pay-of of an option is a linear combination of the prices at maturity of traded assets multiplied by an indicator function generated by an exercise condition, then, the pricing formula is also a linear combination of the current market prices of the traded assets multiplied by a probability expressed in the risk-neutral measure where the asset is the risk-free asset. The proof of the result follows the self-financing portfolio reasoning for the multi-underlying/multi-currency case and uses the change of numeraire technique. We show that the well-known results by Black and Scholes (1973), Magrabe (1978) and Johnson (1987) follow as a particular case of our result and we show how to price certain new options. Comparison results of simulation are also presented. Pricing Power Derivatives: a two-factor jump-diffusion approach Pablo Villaplana We propose a two-factor jump-diffusion model with seasonality for the valuation of electricity future contracts. The model we propose is an extension of Schwartz and Smith (Management Science, 2000) long-term / short-term model. One of the main contributions of the paper is the inclusion of a jump component, with a non-constant intensity process (probability of occurrence of jumps), in the short-term factor. We model the stochastic behaviour of the underlying (unobservable) state variables by Affine Diffusions (AD) and Affine Jump Diffusions (AJD). We obtain closed form formulas for the price of futures contracts using the results by Duffie, Pan and Singleton (Econometrica, 2000). We provide empirical evidence on the observed seasonality in risk premium, that has been documented in the PJM market. This paper also complements the results provided by the equilibrium model of Bessembinder and Lemmon (Journal of Finance, 2002), and provides an easy methodology to extract risk-neutral parameters from forward data. Liquidity Risk and Corporate Demand for Hedging and Insurance Stephane Villeneuve and Rochet Jean-Charles We analyze the demand for hedging and insurance by a corporation that faces liquidity risk. Namely, we consider a firm that is solvent (i.e. exploits a technology with positive expected net present value) but potentially illiquid (i.e. that may face a borrowing constraint). As a result, the firm's optimal liquidity management policy consists in accumulating reserves up to some threshold and distribute dividends to its shareholders whenever its reserves exceed this threshold. We study how this liquidity management policy interacts with two types of risk: a Brownian risk that can be hedged through a financial derivative, and a Poisson risk that can be insured by an insurance contract. We derive individual demand functions for hedging and insurance by corporations. We show that there is a finite price above which both demand functions are zero. Surprisingly we find that the patterns of insurance and hedging decisions as a function of liquidity are pole apart: cash poor firms should hedge but not insure, whereas the opposite is true for cash rich firms. We also find non monotonic effects of profitability and leverage. This may explain the mixed findings of empirical studies on corporate demand for hedging and insurance: linear specifications are bound to miss the impact of profitability and leverage on risk management decisions. Monte Carlo Static Replication of Barrier Options Antonio Vulcano The static hedge for barrier options, initially proposed by Derman.et.al (1995), is theoretically very appealing, since no adjustment is required once the replicating portfolio is in place. However, their procedure is practically flawed in assuming a perfect knowledge of the future implied volatility surface. As a possible solution, in this paper we propose a statistical static hedge, also called Monte Carlo static replication, consisting in generating risk-neutral dynamics for the volatility surface and in determining the portfolio minimizing the tracking error, i.e. the difference between the portfolio value and the barrier option price. This minimization is accomplished by a least square approach along the barrier level, where the dependent variable is the barrier option price, while the independent variable is the replicating portfolio value. Through our statistical approach, we can use the R² coefficient in order to have a preliminary measure of the goodness of the Monte Carlo static hedge; check the convergence of the statistical replicating portfolio to the barrier option price obtained by direct simulation of the underlying; construct confidence intervals around the estimated portfolio weights and hence around the barrier option price; evaluate the performance of our methodology when employing realistic dynamics of implied volatility surfaces. Nonlinear Term Structure Dependence: Copula Functions, Empirics, and Risk Implications Niklas Wagner, Markus Junker, Alex Szimayer This paper documents nonlinear cross-sectional dependence in the term structure of U.S. Treasury yields and points out risk management implications. The analysis is based on a Kalman filter estimation of a two-factor affine model which specifies the yield curve dynamics. We then apply a broad class of copula functions for modeling dependence in factors spanning the yield curve. Our sample of monthly yields in the 1982 to 2001 period provides evidence of upper tail dependence in yield innovations; i.e., large positive interest rate shocks tend to occur under increased dependence. In contrast, the best fitting copula model coincides with zero lower tail dependence. This asymmetry has substantial risk management implications. We give an example in estimating bond portfolio loss quantiles and report the biases which result from an application of the normal dependence model. Adjusting the measure change function in Lévy markets Jens Wannenwetsch This article focuses on the problem of incompleteness in a financial market where the underlying security is modeled by an exponential Lévy process. Incompleteness entails non-uniqueness of the martingale measure, which again means that contingent claims cannot be priced uniquely unless further constraints are imposed on the model. Starting from an estimation of the distribution under the historical probability measure we choose a change of measure which is able to reproduce the observed volatility, skewness, and kurtosis of the risk-neutral distribution while at the same time remaining as close as possible to the historical measure as measured by relative entropy. Doing this amounts to reformulating the problem of finding a suitable martingale measure into a finite-dimensional non-linear minimisation problem with linear constraints. The results are compared with Black-Scholes prices and Lévy prices obtained by using the prevalent Esscher change of measure. By construction this method explains very well different shapes of the volatility smile. Optimal portfolio choice with discontinuous price processes and multiple regimes Andrew Lim, Thaisiri Watewai This paper concerns the problem of optimal investment and consumption, with power utility, discontinuous price processes, and regime switching. Regime switching is modelled by a finite state Markov chain, and unlike traditional regime switching models, changes in regime may be accompanied by jumps in the asset price at the instant of transition, where the distribution of the jump sizes are conditional on the regime before and after the transition. This enables us to model a situation where a transition from a `good' regime to a `bad' one (for instance) is likely to be accompanied by a downward jump in the price, while transitions from a `bad' regime to a `good' one is likely to be accompanied by an upward jump. Expressions for the optimal investment portfolio and consumption policy are obtained using stochastic control methods. It is shown that regime switching models with jumps at the instant of transition have optimal solutions that are significantly different from those associated with traditional regime switching models where changes in regime are typically not accompanied by jumps. An Asset Based Model of Defaultable Convertible Bonds with Endogenised Recovery Nick Webber and Ana Bermudez We describe a two factor valuation model for convertible bonds when the firm may default. The underlying state variables are the asset value of the firm and the short riskless interest rate. Default can occur exogenously, or endogenously at a time a cash payment is made by the bond. We endogenize the recovery value of a defaulted bond through assumptions concerning the character of the reorganization period following default. We use a tailored Lagrange-Galerkin discretization, coupled with a Lagrange multiplier method for free boundaries, to value convertibles in the model. Our framework enables us to specify numerically and financially consistent boundary conditions and inequality constraints. We investigate the affect of changing the default, recovery and loss specification. The affect of introducing a stochastic interest rate is quantified, and asset and interest rate delta and gammas are found. We find that the value of the convertible bond is more sensitive to the initial asset value when its conversion rate is higher. Its sensitivity to interest rate changes is about one tenth that of a corresponding defaultable straight bond, chiefly due to the presence of the conversion feature. Distribution-Invariant Dynamic Risk Measures Stefan Weber The paper provides an axiomatic characterization of dynamic risk measures for multi-period financial positions. For the special case of a terminal cash flow, we require that risk depends on its conditional distribution only. We prove a representation theorem for dynamic risk measures and investigate their relation to static risk measures. Two notions of dynamic consistency are proposed. A key insight of the paper is that dynamic consistency and the notion of "measure convex sets of probability measures" are intimately related. Measure convexity can be interpreted using the concept of compound lotteries. We characterize the class of static risk measures that represent consistent dynamic risk measures. It turns out that these are closely connected to shortfall risk. Under weak additional assumptions, static convex risk measures coincide with shortfall risk, if compound lotteries of acceptable respectively rejected positions are again acceptable respectively rejected. This result implies a characterization of dynamically consistent convex risk measures. Liquidation Triggers and the Valuation of Equity and Debt Zvi Wiener, Dan Galai, Alon Raviv Net-worth covenants provide the firm's bondholders with the right to force reorganization or liquidation if the value of the firm falls below a certain threshold. In the event of default, however, many bankruptcy codes stipulate an automatic stay of assets that prevent bondholders from triggering liquidation. To consider this impact on the valuation of corporate securities we develop a model where liquidation is driven by a state variable that accumulates with time and severity of distress. In addition, current distress periods may have greater weight than old ones. The liquidation trigger can be adjustable to a wide array of bankruptcy codes and jurisdictions. On the Martingale Property of Stochastic Exponentials Bernard Wong and C. C. Heyde We present a necessary and sufficient condition for a stochastic exponential to be a true martingale. It is proved that the criteria for the true martingale property is related to whether an auxiliary process explodes. An alternative and interesting interpretation of this result is that the stochastic exponential is a true martingale if and only if under a `candidate measure' the integrand process does not explode. Applications of our theorem to problems arising in mathematical finance are also given LIBOR Market Model: from Deterministic to Stochastic Volatility Lixin Wu and Fan Zhang LIBOR market model is the benchmark model for interest-rate derivatives. It has been a challenge to extend the standard LIBOR market model so as to cope with the volatility smiles and/or skews that are pronounced in the swaption markets. In this talk we extend the standard LIBOR market model, which takes forward rate or swap rate as state variables, by adopting stochastic volatility. Specifically, we adopt a multiplicative stochastic factor for the volatility functions of all relevant forward rates. The stochastic factor follows a squared-root diffusion process, and it can be correlated with the forward-rate processes. We derive approximate processes for swap rates after the change to forward swap measures, and develop a closed-form formula for swaption prices in terms of Fourier integrals. We then develop a fast Fourier transform algorithm for the implementation of the formula. The approximations are well supported by pricing accuracy. By adjusting the correlation between the forward rates and the volatility in a way consistent with intuition, we can generate volatility smiles or skews of the swaption prices similar to those observed in the markets. Calibration of the model will also be discussed. FX Instalment Options: Pricing, Applications, Risk Management Uwe Wystup and Susanne Griebsch We compare pricing techniques, present a new closed form solution and analyze the limiting case. Joint work with Susanne Griebsch and Christoph Kühn, Goethe University Minimizing Shortfall Risk Using Duality Approach Mingxin Xu and Steven Shreve Option pricing and hedging in a complete market are well-studied with nice results using martingale theories. However, they remain as open questions in incomplete markets. In particular, when the underlying processes involve jumps, there could be infinitely many martingale measures which give an interval of no-arbitrage prices instead of a unique one. Consequently, there is no martingale representation theorem to produce a perfect hedge. The question of picking a particular price and executing a hedging strategy according to some reasonable criteria becomes a non-trivial issue and an interesting question. In this paper, we study the duality approach in minimizing the shortfall risk proposed by Föllmer and Leukert (2000). First we extend the duality results in Kramkov and Schachermayer (1999) to utility functions which are state dependent and not necessarily strictly concave, as our model requires, and in the generality of a semimartingale setting. Then we specialize the duality results to the problem of minimizing shortfall. We next focus on the mixed diffusion case where we explicitly characterize the primal and dual sets in terms of the characteristics. We provide upper bounds for the value function using duality results. Each upper bound produced in this way corresponds to a dual element. For lower bounds, we pick a particular strategy which we call the 'bold strategy' and compute the corresponding value function. In the cases of bonds and call options and constant parameters, closed form solutions for the upper and lower bounds are computed and numerical examples given. This research provides for the first time a method of checking the quality of a hedging strategy according to the principle of minimizing shortfall in an incomplete market model. Multifractal Spectral Analysis of the 1987 Stock Market Crash Rossitsa Yalamova, Cornelis A. The multifractal model of ssset returns captures the fat tails and volatility persistence of many financial time series. The multifractal spectrum computed from wavelet transform modulus maxima lines provides information on the irregularity of the higher moments of the distribution of market returns, in particular on the kinds of singularities that occur in a market. We found that changes in the multifractal spectrum display distinctive patterns around substantial market "drawdowns" or "crashes." In other wordsm the kind of singularites and the kinds of irregularity or "randomness" changes in a distinct fashion in th periods immediately preceding major market drawdowns. This paper focuses on this identifiable multifractal spectral patterns preceding the stock market crash of 1987. Although we were not able to find a uniquely iodentifiable irregularity pattern within the same market preceding different crashes, we do find the same uniquely identifiable in various stock markets experiencing the same crash at the same time. Morover, our results suggest that crashes are preceded by a gradual increase in the weighted average of the values of the irregularity exponents. At a crash this weighted average irregularity value drops, while the dispersion of the spectrum of irregularity coefficients jumps up. We find that the Wavelet Transform Modulus Maxima (WTMM) methodology provides a non-sampled complete measure of the changes in the risk patterns preceding the stock market crash. Estimation of Value-at-Risk and Conditional Value-at-Risk for Dynamic Hedging with Jumps Yuji Yamada and James A. Primbs In this paper, we present a Value-at-Risk (VaR) and conditional Value-at-Risk (CVaR) estimation technique for dynamic heding with jumps in the underlying asset model. At first, we approximate a jump-diffusion process through its first four moments including skewness and kurtosis using a general parameterization of multinomial lattices, and solve the mean square optimal hedging problem. Then our recently developed technique is applied to extract the hedging loss distributions in option hedge positions. Finally, we investigate how the hedging error distribution changes with respect to non-zero kurtosis and skewness in the underlying through numerical experiments, and examine the relation between VaR and CVaR of the hedging loss distributions and kurtosis of the underlying. Asset Allocation with Regime-Switching: Discrete-Time Case Hailiang Yang and Ka Chun Cheung In this paper, we study the optimal asset allocation problem under a discrete regime switching model. Under the short-selling and leveraging constraints, the existence and uniqueness of the optimal trading strategy are obtained. We also obtain some natural properties of the optimal strategy. In particular, we show that if there exists a stochastic dominance order relationship between the random returns at different regimes, then we can order the optimal proportions we should invest in such regimes. Modeling Credit Risk Yildiray Yildirim Modeling Credit Risk This project provides an alternative approach to the structural credit risk models. The first-passage-time approach extends the original Merton model by accounting the default may occur not only at the debt's maturity, but also prior to this date. Default happens when the underlying process hits a barrier. We also define default the first time the underlying process hits a barrier as in the first passage time models, and the liquidation time as the area of this process under the barrier is greater then a constant barrier. We use this technique to price risky debt, and separate default from insolvency. Completeness of Security Markets and Backward Stochastic Differential Equations with Unbounded Coefficients Jiongmin Yong For a standard Black-Scholes type security market, completeness is equivalent to the solvability of a linear backward stochastic differential equation (BSDE, for short). If the interest rate is bounded, there exists a bounded risk premium process, and the volatility matrix has certain surjectivity, then the BSDE will be solvable and the market will be complete. However, if the risk premium process and/or the interest rate is not bounded, one gets a BSDE with unbounded coefficients to solve. In this paper, we will discuss such a situation and will present some solvability results for the BSDE which will lead to the completeness of the market in a broad sense. American Option Pricing with Transaction Costs Valeri Zakamouline In this paper we examine the problem of finding investors' reservation option prices and corresponding early exercise policies of American-style options in the market with proportional transaction costs using the utility based approach proposed by Davis and Zariphopoulou (1995). We present a model, where investors have a CARA utility, and derive some properties of reservation option prices. We discuss the numerical algorithm and propose a new formulation of the problem in terms of quasi-variational HJB inequalities. Based on our formulation, we suggest original discretization schemes for computing reservation prices of American-style option. The discretization schemes are then implemented for computing prices of American put and call options. We examine the effects on the reservation option prices and the corresponding early exercise policies of varying the investor's ARA and the level of transaction costs. We find that in the market with transaction costs the holder of an American-style option exercises this option earlier as compared to the case with no transaction costs. This phenomenon concerns both put and call options written on a non-dividend paying stock. The higher level the transaction costs is, or the higher risk avers the option holder is, the earlier an American option is exercised. Pricing a class of exotic options via moments and SDP relaxations Mihail Zervos, J.B.Lasserre, T.Prieto We present a new methodology for the numerical pricing of a class of exotic derivatives such as Asian or barrier options when the underlying asset price dynamics are modelled by a geometric Brownian motion or a number of mean-reverting processes of interest. This methodology identifies derivative prices with infinite-dimensional linear programming problems involving the moments of appropriate measures, and then develops suitable finite-dimensional relaxations that take the form of semi-definite programs indexed by the number of moments involved. By maximising or minimising appropriate criteria, monotone sequences of both upper and lower bounds are obtained. Numerical investigation shows that very good results are obtained with only a small number of moments. Theoretical convergence results are also established. A Tale of Two Time Scales: Determining Integrated Volatility with Noisy High-Frequency Data Lan Zhang, Per A. Mykland, Yacine Ait-Sahalia It is a common practice in finance to estimate volatility from the sum of frequently-sampled squared returns. However market microstructure poses challenges to this estimation approach, as evidenced by recent empirical studies in finance. This work attempts to lay out theoretical grounds that reconcile continuous-time modeling and discrete-time samples. The present study incorporates microstructure noise in the modeling of the returns. We propose a subsampling-averaging approach which takes advantage of the rich sources in tick-by-tick data while delivering a consistent, asymptotically normal estimator for the integrated volatility. Under our framework, it becomes clear why and where the "usual" volatility estimator fails when the returns are sampled at the highest frequency. Our procedure is implementable regardless of the magnitude of the noise. In the event that the noise is "almost negligible ", our work provides an approach to finding an optimal sampling frequency if one wishes to use the classical "realized volatility", and to optimizing the subsample frequencies if one wishes to incorporate more data into the analysis. Mean--Risk Portfolio Selection Models in Continuous Time Xun Yu Zhou and George Yin This paper is concerned with continuous-time portfolio selection models where the objective is to minimize the risk subject to a prescribed expected payoff at the terminal time. The risk is measured by the expectation of a certain function of the deviation of the terminal payoff from its mean. First of all, a model where the risk has different weights on the upside and downside variance is solved explicitly. The limit of this weighted mean--variance problem, as the weight on the upside variance goes to zero, is the mean--semivariance model which is shown to admit no optimal solution. This negative result is further generalized to a mean--downside-risk portfolio selection problem where the risk has non-zero value only when the terminal payoff is lower than its mean. Finally, a general model is investigated where the risk function is convex. Sufficient and necessary conditions for the existence of optimal portfolios are given. Moreover, optimal portfolios are obtained when they do exist. Arbitrage Pricing Simplified William Ziemba and M. Kallio This paper derives fundamental arbitrage pricing results in finite dimensions in a simple unified framework using Tucker's theorem of the alternative. Frictionless results plus those with dividends, periodic interest payments, transaction costs, different interest rates for lending and borrowing, shorting costs and constrained short selling are presented. While the results are mostly known and appear in various places, our contribution is to present them in a coherent and comprehensive fashion with very simple proofs. The analysis yields a simple procedure to prove new results and some are presented for cases with frictions. Utility Maximization with a Stochastic Clock and an Unbounded Gordan Zitkovic We introduce a linear space of finitely additive measures to treat the problem of optimal expected utility from consumption under a stochastic clock and an unbounded random endowment process. In this way we establish existence and uniqueness for a large class of utility maximization problems including the classical ones of terminal wealth or consumption, as well as the problems depending on a random time-horizon or multiple consumption instances. As an example we treat explicitly the problem of maximizing the logarithmic utility of a consumption stream, where the local time of an Ornstein-Uhlenbeck process acts as a stochastic clock.