# Abstracts

**Convergence of the Modified Craig-Sneyd scheme for multi-dimensional convection-diffusion equations with application to the Heston PDE**

*Maarten Wyns (UAntwerpen, Belgium)*

Tuesday June 3, 16:30-17:00 | session 3.1 | Computational Finance | room AB

In contemporary finance, partial differential equations (PDEs) with mixed spatial derivative terms play an important role in the pricing of option products. For the majority of these PDE problems no exact solution is available and one has to resort to numerical methods to find an approximate solution. Here, Alternating Direction Implicit (ADI) splitting schemes are very popular amongst practitioners as they are highly efficient in comparison to classical (non-splitted) methods. However, for practical applications, a structural analysis of their convergence properties is of paramount importance. To the best of our knowledge, no such analysis has been performed in the literature up to now for problems with mixed derivative terms. The Modified Craig-Sneyd (MCS) scheme is an often used ADI scheme, and in this talk we shall consider its convergence properties in the numerical solution of multi-dimensional convection-diffusion problems containing mixed spatial derivatives. We prove that, under some natural stability assumptions, the MCS scheme has an order of convergence equal to two uniformly in the spatial mesh width. Next, we apply the theory in the case of the Heston PDE which plays an important role in financial option pricing theory. Our numerical experiments point out that all the stability assumptions are fulfilled and we observe indeed a second-order convergence behavior that holds uniformly in the spatial mesh width.

**Analytical Option Pricing under an Asymmetrically Displaced Double Gamma Jump-Diffusion Model**

*Matthias Thul (University of New South Wales, Australia)*

Tuesday June 3, 17:00-17:30 | session 3.1 | Computational Finance | room AB

One of the central trade-offs in the option pricing literature is between the generality of the asset price dynamics and the computational tractability of the corresponding solutions for European plain vanilla options. While pricing through quasi-analytical Fourier inversion methods is possible as long as the characteristic function of the logarithmic terminal asset prices is known, only few models admit genuine closed-form solutions. The latter are desirable due to their superior computational speed and numerical robustness across the full parameter space.

We propose to generalize the Kou (2002) double exponential (DE) jump-diffusion model by letting the jump sizes follow an asymmetrically displaced double gamma (AD-DG) distribution. This extension is motivated economically and we conjecture that it provides a significantly better fit to the time series of historical logarithmic returns across various asset classes. Our key objective is to obtain closed-form solutions for European plain vanilla options.

Our proposed dynamics are supported by a Naik and Lee (1990) type equilibrium model. This economy also induces a risk-neutral probability measure which corresponds to the Esscher transform of the logarithmic return process. The AD-DG jump sizes are closed under this measure transformation.

Starting from the Kou (2002) DE model, we first derive analytical expressions for the cumulative distribution function of the logarithmic returns when the two exponential tails are symmetrically displaced away from zero. Second, we generalize these results to asymmetrically displaced exponential tails by considering the displacements as being symmetric with respect to a different origin. Finally, the extension to AD-DG tails follows from the relationship between the exponential and gamma distributions.

To asses the empirical goodness of fit, we estimate the physical parameters through maximum likelihood and for a diverse sample of equity indices, commodities and exchange rates. We simultaneously compute the likelihood function through the Baily and Swarztrauber (1991, 1994) fractional Fourier transform and numerically solve the non-convex optimization problem using the Storn and Price (1997) differential evolution algorithm.

To our knowledge, the AD-DG model represents the most general jump-diffusion dynamics yet under which European plain vanilla options can still be valued in closed-form. Our empirical test confirm that the newly introduced asymmetric displacement terms are not only academically interesting but also reflect some statistical properties of asset returns. In all cases, both displacement terms are individually and jointly significant at the 1\% level thus allowing us to reject the DE model in favor of the AD-DG dynamics. We further find that the special case of two displaced exponential tails provides the best fit for almost all assets.