# Abstracts

**An explicit Euler scheme with strong rate of convergence for non-Lipschitz SDEs**

*Ivo Mihaylov (Imperial College, UK)*

Thursday June 5, 11:00-11:30 | session P5 | Poster session | room lobby

We consider the approximation of stochastic differential equations (SDEs) with non-Lipschitz drift or diffusion coefficients. We present a modified explicit Euler-Maruyama discretisation scheme that allows us to prove strong convergence, with a rate. Under some regularity conditions, we obtain the \textit{optimal} strong error rate of $1/2$. We consider SDEs popular in the mathematical finance literature, including the Cox-Ingersoll-Ross, the $3/2$ and the Ait-Sahalia models, as well as a family of mean-reverting processes with locally smooth coefficients.

**Valuation and Hedging Efficiency of Ratchet GMWBs for Life with Early Surrender via Least-Squares Monte Carlo**

*Georg Mikus (Frankfurt School of Finance and Management, Germany)*

Thursday June 5, 11:00-11:30 | session P5 | Poster session | room lobby

We analyse the technique based on the regression-based numerical approach that was applied to options e.g. in the seminal paper of Longstaff and Schwartz (2001) for pricing and hedging of Variable Annuities with ratchet GMWB riders for life with the option of early partial or full surrender. We compare our results with earlier studies of e.g. Holz, Kling and Ruß (2006) or Coleman, Kim, Li and Patron (2006) and propose an extension to the structure set up in Bauer, Bergmann and Kiesel (2010). We investigate Hedging programs based on the above method and arrive at somehow different results than previous studies.

**A Stochastic Free Boundary Problem and Limit Order Book Model**

*Marvin Müller (Technische Universität Berlin, Germany)*

Thursday June 5, 11:00-11:30 | session P5 | Poster session | room lobby

We introduce a continuous model for the limit order book density with infinitesimal tick size, where the evolution of buy and sell side is described by a semilinear second-order SPDE. The mid price process defines a free boundary separating buy and sell side. Price changes are assumed to be determined by the bid-ask imbalance. Following empirical observations by Lipton, Pesavento and Sotiropoulos (2013) we allow this dependency to be nonlinear. The resulting limit order book model can be considered as a generalization of the linear stochastic Stefan problem introduced by Kim, Sowers and Zheng (2012).

In order to show existence of a solution we transform the problem into a stochastic evolution equation, where the boundary interaction leads to an additional non-Lipschitz drift. There is no chance to control this term in any reasonable Banach space, however, smoothing properties of the heat semigroup allow to make use of the mild formulation of the equation. Despite of the non-standard setting for the stochastic evolution equation, we show existence of a unique maximal mild solution of the general model; extending results of Kim, Sowers and Zheng. We show that this solution is continuous, and, up to a stopping time, solves the equation even in the analytically strong sense. Additional assumptions on the boundary interaction then yield non-explosion and global existence. Finally, we obtain that a Nagumo-type condition is sufficient for positivity of the order volume, a natural property any limit order book model should satisfy.

**Multi-asset risk measures with applications to conical markets, risk sharing, and superhedging**

*Walter Farkas (University of Zurich, Switzerland)*

Thursday June 5, 11:00-11:30 | session P5 | Poster session | room lobby

We discuss risk measures for financial positions in a multi-asset setting, representing the minimum amount of capital to raise and invest in admissible portfolios in order to meet a prescribed acceptability constraint. We investigate the interplay between the acceptance set and the set of admissible portfolios, with a special focus on continuity properties and dual representations. To avoid degenerate representations, we need to ensure the existence of specific extensions of the underlying pricing functional. We characterize when such extensions exist. Applications to conical market models, set-valued risk measures, optimal risk sharing, and superhedging with shortfall risk are provided.

**An expansion-based closed form approximation for pricing basket and spread options**

*Ciprian Necula (Bucharest University of Economic Studies, Romania)*

Thursday June 5, 11:00-11:30 | session P5 | Poster session | room lobby

The purpose of the paper consists in developing a novel closed form approximation scheme for pricing European basket and spread options when the joint characteristic function of the corresponding log-prices is known in closed form. The idea consists in expanding the risk neutral density of the terminal underlying value in a Gauss-Hermite series around the normal density. Recently, Necula, Drimus and Farkas (2013, A General Closed Form Option Pricing Formula, SSRN id 2210359) have pointed out that the Gauss-Hermite expansion is more appropriate for heavy tailed log-return distributions than the Gram-Charlier expansion and, at the same time, it allows for a closed form European option pricing formula. In this paper, we obtain closed form pricing formulas for basket and spread options that depend on the Gauss-Hermite expansion coefficients of the risk neutral distribution of the terminal underlying value. We take advantage of the fact that the terminal underlying value in the case of these options is a linear combination of asset prices and provided that the joint characteristic function of the corresponding log-prices is known in closed form, we devise an efficient method for obtaining these Gauss-Hermite expansion coefficients. Our approach has several advantages over existing approximations. First, it is valid both for basket and spread options since no restriction is imposed that the terminal underlying value should be positive. Second, it can be employed beyond the simple case when log-prices are normally distributed since many heavy-tailed distributions have closed form characteristic functions. Finally, our approximation method is feasible for basket/spread options with more than two assets and hence it is appropriate for pricing energy spreads between three commodities such as clean spark spreads. We also conduct a study of the performance of the new approximation method.

Moreover, the option pricing formulas obtained in this paper can be interpreted as generalized Bachelier option pricing formulas, since the risk neutral density of the terminal underlying value (and not that of log-returns) is expanded around the normal density. As a curiosity, the method also allows to obtain a generalized Bachelier formula for pricing European options in the context of the Black-Scholes model, by expanding the log-normal density of the terminal price in a Gauss-Hermite series around the normal density.

**Heston-Type Stochastic Volatility with a Markov Switching Regime**

*Katsumasa Nishide (Yokohama National University, Japan)*

Thursday June 5, 11:00-11:30 | session P5 | Poster session | room lobby

We propose an extension of Heston (1993)'s stochastic volatility model to the case where the mean reverting level of the volatility is modulated by a Markov chain. An explicit formula is obtained for the option price which includes the solution of a matrix ODE. It is demonstrated that our model is able to induce a wide variety of volatility surfaces with both flat and steep smiles/smirks for the same parameter values.

**Maximum likelihood estimation in special forward interest rate models**

*Balazs Nyul (University of Debrecen, Hungary)*

Thursday June 5, 11:00-11:30 | session P5 | Poster session | room lobby

J. Gáll, Gy. Pap and M. V. Zuijlen (2004) described a special interest rate model which driven by a geometric spatial AR sheet ([1]) and introduce a new type of Heath-Jarrow-Morton forward interest rate model. In this model we give the no-arbitrage criteria and we estimate parameters of the model (for example volatility) on special samples by maximum likelihood estimation. Finally we observe the asymptotic behaviour of the maximum likelihood estimator in each cases.

**A Black-Scholes equation for illiquid markets**

*Carlos Oliveira (CEMAT and Instituto Superior Técnico, Portugal)*

Thursday June 5, 11:00-11:30 | session P5 | Poster session | room lobby

Perfect liquidity of risky assets is a strong assumption in the Black-Scholes model. Several authors proposed alternative models accounting for imperfect liquidity. One such model was proposed and extensively discussed by Schönbucher and Wilmott (2000).

In the present contribution, we argue that the definition of self-financed strategy used in that paper does not take into full account the effects of imperfect liquidity introduced in the model. We propose one alternative formulation and discuss some properties of the resulting price process.

We use the modified model to discuss the effect of collective behaviour by a large number of small hedgers. If a large number of small traders use similar strategies wrongly assuming perfect liquidity, then synchronized trading of large quantities may have a significant impact in the strategy outcome.

We show that in such circumstances, the expected outcome of the classical Black-Scholes hedging strategy for an European put option can diverge significantly from the perfect hedging obtained under perfect liquidity. The effect of illiquidity can be described by a nonlinear Black-Scholes equation having some very unusual features.

**Pricing and Hedging Basket Options with Exact Moment Matching**

*Tommaso Paletta (University of Kent, UK)*

Thursday June 5, 11:00-11:30 | session P5 | Poster session | room lobby

Basket options are contingent claims on a group of assets such as equities, commodities, currencies and other vanilla derivatives. From a modelling point of view, the framework to price and hedge these options ought to be multidimensional since baskets of 15 to 30 assets are frequently traded. Many pricing methods that seem to work well for single assets cannot be easily expanded to a multidimensional set-up, mainly due to computational difficulties.

In the paper, we present a general computational solution to the problem of multidimensional models which lack closed-form formulae or models that require burdensome numerical procedures, avoiding most of the strong assumptions made by recent techniques. We generalize the approach in [S. Borovkova, F. Permana and H.V. Weide. A closed form approach to valuing and hedging basket and spread options. JOD (2007) 14, 8-24] by employing the Hermite polynomial expansion that replaces the risk-neutral density implied by the model and matches exactly its first m moments. Our approach elaborates on some variants of the method in [A. Leccadito, P. Toscano and R. Tunaru. Hermite binomial trees: a novel technique for derivative pricing. IJTAF (2012) 15,1-36] to deal with baskets that may take on negative values. In particular, the moment matching is carried on three different return quantities and consequently three different methods are presented. The methods can even be applied to price and hedge multi-asset derivatives in situations when some assets follow one diffusion model and other assets follow different ones, with the only assumption being able to calculate the moments of the basket in closed form. We use, as exemplification, the shifted log-normal process with jumps in pricing basket options.

Furthermore, we propose a test to evaluate the performances of methods for pricing and hedging European-style contingent claims. Through this test, we show that no more than four moments of the underlying assets processes are needed in pricing and hedging options. Moreover, the test shows that one of the proposed methods returns the best price in 84\% of cases and has a pricing error smaller than 5\% in 95\% of the cases. On the hedging side, our methods tend to slightly sub-hedge but their average error is much smaller when compared to other methods. Consequently, the methods are shown to provide superior results not only with respect to pricing but also for hedging.

**To freeze or not to freeze the drift of Swap interest rates**

*Rita Pimentel (IST, Portugal)*

Thursday June 5, 11:00-11:30 | session P5 | Poster session | room lobby

We explore the low variance martingale (LVM) assumption and their implications. LVM assumption was proposed by Schrager and Pelsser (2006) for pricing interest rate derivatives, in particular swaptions, under affine term structure models (ATSMs). A similar assumption is also used in LIBOR market model (LMM) and it is usually known as freezing the drift. Once the method based on LVM assumption is easy and has an intuitive implementation it has been preferred by practitioners.

The LVM assumption assumes that the ratio between a zero coupon bond and a portfolio of discount bonds (based on a specific tenor) is a low variance martingale under the swap measure. So, their value at each time may be approximate by its conditional expected value, particularly by their time zero value.

We prove that the LVM assumption implies that the instantaneous forward rates are frozen at their initial values. Based on this consequence, we deduce the parameters values at each time for Nelson-Siegel and Svensson models.

We also provide an empirical study, based on real financial market. We use data available at European Central Bank (ECB). Hence, we investigate the periods where the LVM assumption is acceptable and the periods where there are substantial differences. Finally, we compare these periods with the global economic environment.

**Finite-state approximation of polynomial preserving processes**

*Sergio Pulido (EPFL, Switzerland)*

Thursday June 5, 11:00-11:30 | session P5 | Poster session | room lobby

Polynomial preserving processes are defined as time-homogeneous Markov jump-diffusions whose generator leaves the space of polynomials of any fixed degree invariant. The moments of their transition distributions are polynomials in the initial state. The coefficients defining this relationship are given as solutions of a system of nested linear ordinary differential equations. Polynomial processes include affine processes, whose transition functions admit an exponential-affine characteristic function. These processes are attractive for financial modeling because of their tractability and robustness. In this work we study approximations of polynomial preserving processes with finite-state Markov processes via a moment-matching methodology. This approximation aims to exploit the defining property of polynomial preserving processes in order to reduce the complexity of the implementation of such models. More precisely, we study sufficient conditions for the existence of finite-state Markov processes that match the moments of a given polynomial preserving process. We first construct discrete time finite-state Markov processes that match moments of arbitrary order. This discrete time construction relies on the existence of long-run moments for the polynomial process and cubature methods over these moments. In the second part we give a characterization theorem for the existence of a continuous time finite-state Markov process that matches the moments of a given polynomial preserving process. This theorem illustrates the complexity of the problem in continuous time by combining algebraic and geometric considerations. We show the impossibility of constructing in general such a process for polynomial preserving diffusions, for high order moments and for sufficiently many points in the state space. We provide however a positive result by showing that the construction is possible when one considers finite-state Markov chains on lifted versions of the state space.

**Modeling share returns - an empirical study on the Variance Gamma model**

*Andreas Rathgeber (University of Augsburg, Germany)*

Thursday June 5, 11:00-11:30 | session P5 | Poster session | room lobby

Due to the fact that there has been only little research on some essential issues of the Variance Gamma (VG) process, we have recognized a gap in literature as to the performance of the various estimation methods for modeling empirical share returns. While some papers present only few estimated parameters for a very small, selected empirical database, Finlay and Seneta (2008) compare most of the possible estimation methods using simulated data. In contrast to Finlay and Seneta (2008) we utilize a broad, daily, and empirical data set consisting of the stocks of each company listed on the DOW JONES over the period from 1991 to 2011. We also apply a regime switching model in order to identify normal and turbulent times within our data set and fit the VG process to the data in the respective period. We find out that the VG process parameters vary over time, and in accordance with the regime switching model, we recognize significantly increasing fitting rates which are due to the chosen periods.

**Contingent capital: a robust regression analysis**

*Tom Reynkens (KULeuven, Belgium)*

Thursday June 5, 11:00-11:30 | session P5 | Poster session | room lobby

Contingent convertibles (CoCos) are a new loss-absorbing bond class which receive high interest from financial regulators. When developing models to price these Cocos, it is important to see which factors drive the CoCo's value. Possible factors are the price of the stock on which the Coco is based, the interest rate and the credit default swap spread, but other factors can also be included. A way to look at this sensitivity is by fitting a regression model with multiple factors as covariates based on historical data. We perform a full regression analysis to see which factors are relevant when determining the CoCo price and should therefore be included in our pricing model. Hereby, it is also necessary to study the interaction between included factors. When developing models for different CoCo types, we can determine which types are sensitive to changes in certain factors (such as the volatility) which leads to a differentiation between CoCo types based on these market parameters. To compensate for possible outliers in the data, we also focus on robust regression techniques. The use of these techniques also allows us to identify deviating trading periods.

**The Impact of Skew on the Pricing of Coco Bonds**

*Ine Marquet (KU Leuven, Belgium)*

Thursday June 5, 11:00-11:30 | session P5 | Poster session | room lobby

This paper presents a Heston-based pricing model for contingent convertible bonds (CoCos). The main finding is that skew in the implied volatility surface has a significant impact on the CoCo price. Hence stochastic volatility models, like the Heston model, which incorporate smile and skew are appropriate in the context of pricing CoCos.

The financial crisis of 2007-2008 triggered an avalanche of financial worries for financial institutions around the globe. After the collapse of Lehman Brothers, governments intervened and bailed out banks using tax-payer's money. Preventing such bail-outs in the future and designing a more stable banking sector in general, requires both higher capital levels and regulatory capital of a higher quality. Bank debt needed therefore to be made absorbing. This is where CoCos come in. The Lloyds Banking Group introduced the first CoCo bonds as early as December 2009. Since then a lot of other banks followed Lloyds and the market of CoCos, currently around \$70bn, is expanding very rapidly.

CoCos are hybrid financial instruments that convert into equity or suffer a write-down of the face value upon the appearance of a trigger event, often in terms of the bank's CET1 level in combination with a regulatory trigger. The valuation of CoCos boils down to the quantification of the trigger probability and the expected loss suffered by the investors if such a trigger event eventually takes place. There are at least two schools of thought regarding valuation of CoCos. Structural models can be put at work or investors can rely on market implied models. The latter category uses market data (share prices, CDS levels and implied volatility, ...) in order to calculate the theoretical price of a CoCo bond. In De Spiegeleer and Schoutens (2012a), the pricing of CoCo notes has been worked out in a market implied Black-Scholes context.

In this paper we move away from the assumption of a constant volatility which is the back-bone of Black-Scholes based valuation and put the Heston model at work and study CoCos in a stochastic volatility context. The existence of a semi closed-form formula for European options pricing under the Heston model allows for a fast calibration of the model. In our approach we combined market quotes of listed option prices with CDS data. As a case study, the procedure was applied on the Tier 2 10NC CoCo issued by Barclays in 2012.