Christoph Winter

On Kolmogorov Equations for Anisotropic Multivariate Levy Processes

For d-dimensional exponential Lévy models, variational formulations of the Kolmogorov equations arising in asset pricing are derived. Well-posedness of these equations is verified. Particular attention is paid to pure jump, d-variate Lévy processes built from parametric, copula dependence models in their jump structure. The domains of the associated Dirichlet forms are shown to be certain anisotropic Sobolev spaces. Singularity-free representations of the Dirichlet forms are given which remain bounded for piecewise polynomial, continuous functions of finite element type. We prove that the variational problem can be localized to a bounded domain with explicit localization error bounds. Furthermore, we collect several analytical tools for further numerical analysis.

Computational Methods for Quantitative Finance

1.Introduction.- Part I.Basic techniques and models: 2.Notions of mathematical finance.- 3.Elements of numerical methods for PDEs.- 4.Finite element methods for parabolic problems.- 5.European options in BS markets.- 6.American options.- 7.Exotic options.- 8.Interest rate models.- 9.Multi-asset options.- 10.Stochastic volatility models-. 11.Levy models.- 12.Sensitivities and Greeks.- Part II.Advanced techniques and models: 13.Wavelet methods.- 14.Multidimensional diffusion models.- 15.Multidimensional Levy models.- 16.Stochastic volatility models with jumps.- 17.Multidimensional Feller processes.- Apendices: A.Elliptic variational inequalities.- B.Parabolic variational inequalities.- References.- Index.

Numerical methods for Lévy processes

We survey the use and limitations of some numerical methods for pricing derivative contracts in multidimensional geometric Lévy models.

Finite Element Methods for Parabolic Problems

The finite element methods are an alternative to the finite difference discretization of partial differential equations. The advantage of finite elements is that they give convergent deterministic approximations of option prices under realistic, low smoothness assumptions on the payoff function as, e.g. for binary contracts. The basis for finite element discretization of the pricing PDE is a variational formulation of the equation. Therefore, we introduce the Sobolev spaces needed in the variational formulation and give an abstract setting for the parabolic PDEs.

Two-scale finite element discretizations for integro- differential equations

Some two-scale finite element discretizations are introduced for a class of linear partial differential equations. Both boundary value and eigenvalue problems are studied. Based on the two-scale error resolution techniques, several two-scale finite element algorithms are proposed and analyzed. It is shown that this type of two-scale algorithms not only significantly reduces the number of degrees of freedom but also produces very accurate approximations.

Wavelet Galerkin Schemes for Multidimensional Anisotropic Integrodifferential Operators

We consider a wavelet Galerkin scheme for solving partial integrodifferential equations arising from option pricing in multidimensional Levy models. Sparse tensor product spaces are applied for the discretization to reduce the complexity in the number of degrees of freedom, and wavelet compression methods are used to decrease the number of nonzero matrix entries. We focus on algorithmic details of the scheme, in particular on the numerical integration of the matrix coefficients.

Multi-asset Options

In Chap. 6, we considered exotic options written on a single underlying. Further examples of exotic options are given by the so-called multi-asset options. These are options derived from d≥2 underlying risky assets, whose price movement can be described by a system of SDEs. The pricing functions of multi-asset options are multivariate functions satisfying a parabolic partial differential equation in d dimensions, together with an appropriate terminal value depending on the type of the option. We distinguish between different types of European multi-asset options. We distinguish between different types of multi-asset options like basket, rainbow or quanto options.

Interest Rate Models

We consider options on interest rates and present commonly used short rate models to model the time-evolution of the interest rate. Many interest rate derivatives in fixed income markets can then be priced numerically using the computational techniques described in the previous chapter, i.e. they can be interpreted as compound options on bonds.

Notions of Mathematical Finance

The present notes deal with topics of computational finance, with focus on the analysis and implementation of numerical schemes for pricing derivative contracts. There are two broad groups of numerical schemes for pricing: stochastic (Monte Carlo) type methods and deterministic methods based on the numerical solution of the Fokker–Planck (or Kolmogorov) partial integro-differential equations for the price process. Here, we focus on the latter class of methods and address finite difference and finite element methods for the most basic types of contracts for a number of stochastic models for the log returns of risky assets. We cover both, models with (almost surely) continuous sample paths as well as models which are based on price processes with jumps. Even though emphasis will be placed on the (partial integro)differential equation approach, some background information on the market models and on the derivation of these models will be useful particularly for readers with a background in numerical analysis.

Elements of Numerical Methods for PDEs

In this chapter, we present some elements of numerical methods for partial differential equations (PDEs). The PDEs are classified into elliptic, parabolic and hyperbolic equations, and we indicate the corresponding type of problems that they model. PDEs arising in option pricing problems in finance are mostly parabolic. Occasionally, however, elliptic PDEs arise in connection with so-called “infinite horizon problems”, and hyperbolic PDEs may appear in certain pure jump models with dominating drift.