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Correlation and the
pricing of risks
Marc Atlan, Helyette Geman, Dilip B. Madan, Marc Yor
It is shown that reliance on direct correlation as an indicator
for the pricing of a risk can be misleading. Examples are given of
risks correlated with the pricing kernel that are not priced while
uncorrelated risks may be priced. These examples lead to new
definitions for non-priced risks in correlation terms. Additionally
we show that the density is definitely priced.
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A Comparison of
Markov-Functional and Market Models: The One-Dimensional Case
Michael N. Bennett and Joanne E. Kennedy
The LIBOR Markov-functional model is an efficient arbitrage-free pricing
model suitable for callable interest rate derivatives. We demonstrate that
the one-dimensional LIBOR Markov-functional model and the separable
one-factor LIBOR market model are qualitatively very similar. Consequently,
the intuition behind the familiar SDE formulation of the LIBOR market model
may be applied to the LIBOR Markov-functional model.
The application of a drift approximation to a separable one-factor LIBOR
market model results in an approximating model driven by a one-dimensional
Markov process, permitting efficient implementation. For a given
parameterisation of the driving process we find the distributional structure
of this model and the corresponding Markov-functional model are numerically
virtually indistinguishable over a wide variety of market conditions and
both are very similar to the market model. A theoretical uniqueness result
shows that any approximation to a separable market model that reduces to a
function of the driving process is effectively an approximation to the
analogous Markov-functional model. Therefore, our conclusions are not
restricted to our particular choice of driving process.
Under stress-testing, although these models continue to exhibit similar
behavior, the higher dimensionality of the market model becomes apparent and
the use of drift approximations introduces arbitrage. In this situation, we
argue the Markov-functional model is a more appropriate choice for pricing.
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Classification Using
Optimization: Application to Credit Ratings of Bonds
Vladimir Bugera, Stanislav Uryasev, Grigory Zrajevsky
We proposed a new classification method utilizing mathematical
programming techniques. The method concludes in minimizing a penalty
function measuring misclassification. The penalty function is
constructed with quadratic separating surfaces dividing the space
into several areas. Elements from one area are supposed to belong to
the same class. We formulated an optimization problem for finding
optimal coefficients of quadratic separating functions. The
optimization problem reduces to linear programming. To adjust
flexibility of the model and to avoiding overfitting we imposed
various types of constraints. The classification procedure includes
two phases. In the first phase, a classification rule is developed
based on "in-sample" information by using a dataset with known
classification. In the second phase, we apply the classification
rule based on computed separating surfaces and validate is with an
"out-of-sample" dataset. We applied the suggested approach to the
problem of rating of bonds. We considered Standard and Poor's credit
rating. The bonds are rated according to their risk characteristics.
We demonstrated that the classification procedure reliably extract
information from a given dataset with known classification
(in-sample classification) and then used this information to
classify new objects (out-of-sample classification). The approach is
quite general and can be applied in many other areas.
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Australian Yield
Curves and GARCH Modelling
Shuling Chen
Australian Yield Curves and GARCH Modelling Shuling Chen,
William Dunsmuir and Ben Goldys School of Mathematics University of
New South Wales Sydney, New South Wales, Australian In this paper we
describe two types of term structure interest rates for Australia.
The first are the generic yield curves produced by the Reserve Bank
of Australia depending on the bonds on issue and the second the
constructed yield curves of the Commonwealth Bank Australia from
swap rates. Statistics of the yield curves are discussed, and it is
shown that the dynamics of the yields curves from the two different
sources are very similar. Modelling the dynamics of the yield
curves, based on the Australian treasury yields 1996-2001, with
different maturity levels, is investigated using generalised
autoregressive conditional heteroskedastic models. We found that the
univariate GARCH (1, 1) models, with some exogenous innovation
variables and residuals in student's t distribution, are quite
adequate for the middle-to-long-term bond yield returns, while
inadequate with the short-term yield returns. Moreover, it is shown
that the parameters of the GARCH (1, 1) models, for
middle-to-long-term bond yield returns, are logarithmic functionally
dependent on the length of time to maturity. This result can be
applied to predict the bond yield returns for any
middle-to-long-term maturity.
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A Structural Model
with Jump-Diffusion Processes
Binh Dao
In this paper, we extend the framework of Leland's 94 by
examining corporate debt, equity and firm values with jump-diffusion
processes. We choose two kinds of jumps such as the uniform and
double exponential jumps to modelize the distribution of the jump
sizes. Through this choice, we are able to derive closed-form
results in both models for perpetual American put, equity, debt and
firm values. Our results have the same forms as those of Leland's
94. However, in both our models, the spreads are modified
significantly in comparison with those of Leland due to jumps'
assumption.
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Convexity of the
optimal stopping boundary for the American put option
Erik Ekström
We show that the optimal stopping boundary for the American put
option is convex in the standard Black-Scholes model. The methods
are adapted from ice-melting problems and rely upon studying the
behavior of level curves of solutions to parabolic di erential
equations. In particular, we de ne and study a function v that in
the continuation region satis es the equation vt = vxx + hvx for
some function h, and that on the optimal stopping boundary f(t; x) :
x = x(t)g satis es v =
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Backward Stochastic
Differential Equations with Enlarged Filtration - Option hedging of an
insider trader in a financial market with Jumps
Anne Eyraud-Loisel
Insider trading consists in having an additional information,
unknown from the common investor, and using it on the financial
market,in order to improve the wealth of a portfolio. Mathematical
modeling can study such behaviors, by modeling this additional
information within the market, and comparing the investment
strategies of an insider trader and a non informed investor.
Research on this subject has already been carried out by A. Grorud
and M. Pontier since 1996, studying the problem in a wealth
optimization point of view. This work focuses more on option hedging
problems. We have chosen to study wealth equations as backward
stochastic differential equations (BSDE), and we use Jeulin's method
of enlargement of filtration to model the information of our insider
trader. We will try to compare the strategies of an insider trader
and a non insider one. Different models are studied: at first prices
are driven only by a Brownian motion, and in a second part, we add
jump processes (Poisson point processes) to the model.
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An analysis of the
doubling strategy: The countable case
Mark Fisher and Christian Gilles
We analyze the doubling strategy in static and dynamic settings
with a countable state space. We apply the no-arbitrage and
no-free-lunch definitions of Kreps (1981), which (in the dynamic
setting) put the focus on the gain produced by a self-financing
trading strategy, rather than on the strategy itself. By applying
the Krepsian notions of no arbitrage and no free lunches to dynamic
models, instead of the notions common in standard practice, we avoid
the situation where there are no free lunches at the same time there
are arbitrage opportunities. Depending on the topological space one
adopts, the doubling strategy is either (i) not in the space of
payouts (and hence not a free lunch), (ii) in the space and a free
lunch, or (iii) in the space but not a free lunch. In the latter
case, which requires `near risk-neutrality', the doubling strategy
has a bubble component in the sense of Gilles and LeRoy (1997).
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The Lookback American
Option with Finite Horizon
Pavel Gapeev
We present a solution to the problem of pricing fixed-strike
lookback American option in the Black-Scholes model with finite time
horizon. For solving the initial problem we construct an equivalent
optimal stopping problem for a three-dimensional Markov process and
show that the continuation region for the price process is
determined by a continuous increasing curved stochastic boundary
depending on the maximum process. In order to find analytic
expressions for the boundary we formulate an equivalent parabolic
free-boundary problem and then derive a nonlinear Volterra integral
equation of the second kind following from the early exercise
premium representation. Using the change-of-variable formula (being
an extension of Ito-Tanaka formula) containing the local time of a
diffusion process (geometric Brownian motion) on continuous curved
stochastic boundaries of bounded variation we show that this
equation is sufficient to determine the optimal stopping boundary
uniquely.
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Bridging the Gap
between Financial and Actuarial Pricing
Nicolas Gaussel and Marie Pascale Leonardi
In this paper we aim at giving new insights to the problem of
pricing an option given on an untraded asset and hedged by a traded
liquid proxy. Building on previous results on indifference pricing
and using an exponential preference framework, we first illustrate
how the option price can be expressed as a certainty equivalent with
a modified risk aversion under some particular probability measure.
This measure is characterized by (i) the choice of a premium
associated to the non-hedgeable risk and (ii) the fact that all
forward prices of traded assets become martingales. As for the
modified risk aversion, it depends on the risk aversion parameter
and the correlation between both assets. Second, we discuss the
economic interpretation of this price, illustrating Frittelli's
(Frittelli 2000b) result in that particular case. These theoretical
results are illustrated by a detailed study of the pricing and
hedging of both a vanilla put and an exotic barrier put.
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Accelerating Monte
Carlo Pricing of Path Dependent Options
Andreas Grau
One of the most challenging problems in option pricing is the
efficient pricing and hedging of path dependant options. This paper
presents a method which can increase the convergence of Monte Carlo
pricing significantly. The method can be extended such that Monte
Carlo simulation and PDE solver are combined to a consistent
framework. As an example for the efficiency of the framework, the
computational effort for different types of Parisian and Asian style
options, especially a moving window Parisian option (delayed barrier
option) is compared with the cost of classical Monte Carlo and PDE
pricing.
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Valuation of European
Call Options with Transaction Costs under Jump Diffusion
Process
Aurele Houngbedji
This paper discusses extensions of the transaction costs model
of Leland (1985) for geometric Brownian motion to jump diffusion
processes. We derive an equation for European call options and give
an expression for the value of the call prices when the underlying
asset follow the jump diffusion process in the presence of small
proportional transaction. We also address the issue of discrete time
hedging and the associated hedgin error.
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Modelling Term
Structures of Default Probability by Structural Model with Time-dependent
Target Leverage Ratios
Ming-Xi Huang, C.H. Hui, C.F. Lo
Stationary-leverage-ratio models of modelling credit risk based
on constant target leverage ratios cannot generate probabilities of
default which replicate empirically observed default rates. This
paper presents a structural model to address this problem. The main
feature of the model is that a firm's leverage ratio is
mean-reverting to a time-dependent target leverage ratio. The
time-dependent target leverage ratio reflects the firm's intention
of moving its initial target ratio toward a long-term target ratio
over time. We derive a closed-form solution of the probability of
default based on the model as a function of the firm value,
liability and short-term interest rate. The numerical results
calculated from the solution with simple time-dependent functions of
the target leverage ratios show that the model is capable of
producing term structures of probabilities of default that are
consistent with some empirical findings. This model could provide
new insight for future research on corporate bond analysis and
credit risk measurement.
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VaR and ES for linear
Portfolios with mixture of elliptically distributed risk
factors
Sadefo Kamden Jules
The particular subject of this paper, is to give an explicit
formulas that will permit to obtain the linear VaR or Linear ES,
when the joint risk factors of the Linear portfolios, changes with
mixture of t-Student distributions. Note that, since one shortcoming
of the multivariate t- distribution is that all the marginal
distributions must have the same degrees of freedom, which implies
that all risk factors have equally heavy tails, the mixture of
t-Student will be view as a serious alternatives, to a simple
t-Student-distribution. Therefore, the methodology proposes by this
paper seem to be interesting to controlled thicker tails than the
standard Student distribution.
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Capital Stock
Assessment with Three Equation Dynamic Model
Jan Kodera and Vaclava Pankova
The aim of the article is to make an assessment of the capital
stock using Keynesian model formulated as a dynamic three equations
system. The system is introduced as a continuous description of
three blocks of Keynesian economy. The first block describes
production, the second one shows interest rate dynamics. The third
block includes wage rate movement. The production dynamics is caused
by the disequilibrium in commodities market, money market generates
interest rate movement, wage rate plays a role of equalizing factor
in labour market. The theoretical model is transformed in a linear
continuous model and then re-formulated as a discrete dynamic model.
Applying an econometric approach we specify a VAR model estimated
for the Czech economy. The prognosis of production, wage rates and
interest rates is carried out. These projections are necessary for
computing the value of the capital stock of the Czech economy by
discounted returns method. The computation of value of capital stock
in the Czech economy is presented in closing part of the
contribution. The method is proposed as an alternative to a direct
computation of rate of returns.
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Intraday options trading and liquidation scenarios
Igor Kliakhandler and Asitha Kodippili
Conventional measurements of risk and losses in scenario
analysis usually use some averaged parameters, such as
volatility, spread, etc. In particular, liquidation
scenarios are used to estimate the impact of market stress.
During the time when such scenarios are activated, however,
volatility rises and spreads widen, exacebrating losses.
Using high-frequency extensive database with more than 4
millions option quotes for each underlyings, we show that
intraday trading risks are substantially higher and losses
are deeper than those that are obtained from averaged
volatilities and spreads values.
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Dry Markets and
Superreplication Bounds of American Derivatives
Ana Lacerda, João Amaro de Matos
This paper studies the impact of dry markets for underlying
assets on the pricing of American derivatives, using a discrete time
framework. Dry markets are characterized by the possibility of
non-existence of trading at certain dates. Such non-existence may be
certain or probabilistic. Using superreplicating strategies, we
derive expectation representations for the range of the
arbitrage-free value. In the probabilistic case, if an enlarged
filtration, resulting from the price process and the market
existence process, is considered, ordinary stopping times are
required. If not, randomized stopping times are required. Several
comparisons of the ranges obtained with the two market restrictions
are performed. Finally, we conclude that arbitrage arguments are not
enough to define the optimal exercise policy.
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A General Pricing
Model for Time-changed Levy Processes
Seng
Yuen Leung
Normality assumption of asset returns is used in many financial
modeling. A number of empirical findings show that returns have
fatter than normal tails. One direct result of non-normality in
asset returns is volatility smirk. Stochastic volatility and jumps
are the most common sources to describe these discrepancies, and
time-changed Levy processes are evidenced as good models to
incorporate these sources in option pricing. In this paper the
relationship between time-changed Levy processes and pricing models
are established. In the setting of n-dimensional time-changed Levy
processes, we present an analytical treatment of pricing models by
means of Fourier transform. With time changes driven by a
superposition of log money market account and other macroeconomic
activity times, our model provides a wide range of applications
which include the valuation of options and zero-coupon bonds. We
also propose a Levy term structure which can accommodate our
framework to price a variety of interest rate derivatives.
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Measuring Provisions
for Collateralised Retail Lending
Po
Kong Man, C.H. Hui, C.F. Lo, T.C. Wong
This paper develops a simple model based on an options approach
to measure provisions covering expected losses of collateralised
retail lending due to default. A closed-form formula of the model is
derived and used to calculate the required provision for a pool of
retail loans with the same collateral type. The numerical results
show that the loan-to-value ratio, correlation between the
collateral value and the probability of default of borrowers in the
pool, volatility of the collateral value, mean-reverting process of
the probability of default and time horizon are the important
factors for measuring provisions. As the parameters associated with
these factors are in general available in banks' databases of their
retail loan portfolios, the model could be a useful quantitative
tool for measuring provisions.
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The Discrete
Black-Scholes Partial Differential Equation
Miklavz Mastinsek
The well known Black-Scholes partial differential equation was
derived under the assumption of continuous trading. If trading takes
place at discrete time points (e.g. when transaction costs are
considered), then the continuous time hedging is a good
approximation for small trading time intervals. In many articles on
options with transaction costs the modified continuous-time
Black-Scholes equation is considered. The objective of this paper is
to develop and study a partial functional differential equation
suitable for describing the option values when the trading is in
discrete time. Since the framework is not continuous the Ito's lemma
is not used. In the case of discrete trading the hedging ratio at
time t can be approximated by the partial derivative of the option
value V(t,S) with respect to the stock value S at time t+dt, where
dt is a oninfinitesimal length of the revision interval. By the
development analogous to that for the continuous-time Black-Scholes
equation a partial differential equation for V(t,S) can be obtained.
After the transformation from the backward to the forward equation a
linear partial functional differential equation is obtained. The
equation is uniquely solved. The difference between the solution of
the discrete-time and the continuous-time Black-Scholes equation is
given.
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Portfolio Selection
with Transaction Costs and Delays
Tim Maull and Jussi Keppo
We analyze the effects of transaction costs and trading delays
on the optimal portfolio strategy of individual investors. Trading
delays occur with large block orders in liquid markets and any
orders in illiquid markets. The investor attempts to maximize
expected terminal utility by allocating wealth between risk-free and
risky assets. The optimal strategy is given by no trading and
trading regions that depend on the transaction costs and delays. We
analyze the effect of these frictions on the optimal trading
strategy. Under high volatility the affect of delay is significant.
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On testing for
duration clustering and diagnostic checking of models for irregularly
spaced transaction data
Maria Pacurar, Pierre Duchesne
We propose two classes of test statistics for duration
clustering and one class of test statistics for the adequacy of ACD
models, using a spectral approach. The tests for ACD effects of the
first class are obtained by comparing a kernel-based normalized
spectral density estimator and the normalized spectral density under
the null hypothesis of no ACD effects. The second class of test
statistics for ACD effects exploits the one-sided nature of the
alternative hypothesis. The class of tests for the adequacy of an
ACD model is obtained by comparing a kernel-based spectral density
estimator of the estimated standardized residuals and the null
hypothesis of adequacy. The resulting test statistics possess a
convenient asymptotic normal distribution under the null hypothesis.
We present a simulation study illustrating the merits of the
proposed procedures and an application with financial data is
conducted.
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A Model of Investment,
Production and Consumption
Tao Pang and Wendell Fleming
We consider a stochastic control model in which an economic unit
has productive capital and also liabilities in the form of debt. The
worth of capital changes over time through investment, and also
through random Brownian fluctuations in the unit price of capital.
Income from production is also subject to random Brownian
fluctuations. The goal is to choose investment and consumption
controls which maximize total expected discounted HARA utility of
consumption. Optimal control policies are found using the method of
dynamic programming. In case of logarithmic utility, these policies
have explicit forms.
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Semi-Lagrangian Time
Integration for PDE Models of Asian Options
Arthur
Kevin Parrott
A variety of methods can be applied to the pricing of Asian
options. Finite difference methods are very flexible with regard to
the asset price model, but encounter difficulty when applied to PDE
models of Asian options because of the parabolic degeneracy in the
average-price direction. Semi-Lagrange (S-L) time-integration,
developed for numerical weather forecasting, is an elegant choice of
technique which integrates out the average price term and simplifies
the finite difference equations into a parameterised Black-Scholes
form. Uniform meshes are not efficient, however the S-L method is
shown to be easily applied in conjunction with coordinate
transformations. The S-L time integration method has been shown to
dramatically simplify the finite difference approximation of Asian
options. Second order accuracy has been confirmed for Asian options
that must be held to maturity. Early exercise is also easily
incorporated and the resulting linear complimentarity problem can be
solved using a projection method. A comparison with published
results for continuous-average-rate Put and Call options, with and
without early exercise, shows that the method achieves basis point
accuracy with a high degree of efficiency.
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Analysis of
equilibrium financial markets in continuous time
Stefan
Alex Popovici
The aim of the paper is to introduce and analyse a financial
market in continuous time based on economic agents which: * starts
with economic agents having individual wealth preferences and acting
on a given financial market, * analyses the portfolio selection
problem for the agents employing the maximization of expected
utility technique, focusing on the importance of the market
portfolio numeraire, * gives a quantitative statement about the
behaviour of rational economic agents (i.e. optimal investment
strategies), * introduces the notion of equilibrium of a financial
market and gives a characterization of equilibrium markets, *
explains the creation of prices by offer and demand in equilibrium
markets, * establishes a connection to market models based on the
absence of arbitrage assumption, * provides an explicit construction
of a martingale measures Q, * explains the risk neutral pricing and
hedging of derivatives technique, * gives a causal dependence
between interest rate markets and equilibrium markets, * contains
the Black-Scholes model as well as any arbitrage-free market as a
special case of an equilibrium market, * establishes an explicit
connection to the "Capital Asset Pricing Model" of Sharpe and
Lintner, * gives an explicit formula for the mean expected return of
assets and provides the "Beta" coefficients and Sharpe ratios of
assets in explicit form. The main result of the paper is the proof
that a financial market is in equilibrium if and only if the asset
price process expressed in the market portaolio numeraire M is a
regular martingale under the historical probability measure P (the
market portfolio M is defined as the sum of all assets on the
market).
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How Useful are
Volatility Options for Hedging Vega Risk?
Dimitris Psychoyios, George Skiadopoulos
Motivated by the growing literature on volatility options and
their imminent introduction in major exchanges, this paper addresses
two issues. First, we examine whether volatility options are
superior to standard options in terms of hedging volatility risk.
Second, we investigate the comparative pricing and hedging
performance of various volatility option pricing models in the
presence of model error. Monte Carlo simulations within a stochastic
volatility setup are employed to address these questions.
Alternative dynamic hedging schemes are compared, and various
option-pricing models are considered. The results have important
implications for the use of volatility options as hedging
instruments, and for the robustness of the volatility option pricing
models.
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