Oleg Reichmann

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.

Pricing electricity derivatives on an hourly basis

The purpose of this paper is to develop a framework for pricing electricity derivatives on an hourly basis. We do not in contrast to most current approaches focus exclusively on spot models which primarily reflect empirical spot price dynamics, but also ensure a straightforward applicability to the valuation of electricity derivatives. We show that a model with a jump and a spike component can be calibrated to both the time-series of hourly spot prices and the cross-section of futures prices, once we allow for time-dependent jump and spike parameters. Furthermore, we illustrate the importance of derivative pricing in electricity markets and present some examples of options on futures and hourly spot-options, such as operating reserves and physical transmission rights.

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.

Numerical analysis of additive, Lévy and Feller processes with applications to option pricing

We review the design and analysis of multiresolution (wavelet) methods for the numerical solution of the Kolmogorov equations arising, among others, in financial engineering when Levy and Feller or additive processes are used to model the dynamics of the risky assets. In particular, the Dirichlet and free boundary problems connected to barrier and American style contracts are specified and solution algorithms based on wavelet representations of the Feller processes’ Dirichlet forms are presented. Feller processes with generators that give rise to Sobolev spaces of variable differentiation order (corresponding to a state-dependent jump intensity) are considered. A copula construction for the systematic construction of parametric multivariate Feller-Levy processes from univariate ones is presented and the domains of the generators of the resulting multivariate Feller-Levy processes is identified. New multiresolution norm equivalences in such Sobolev spaces allow for wavelet compression of the matrix representations of the Dirichlet forms. Implementational aspects, in particular the regularization of the process’ Dirichlet form and the singularity-free, fast numerical evaluation of moments of the Dirichlet form with respect to piecewise linear, continuous biorthogonal wavelet bases are addressed. Monte Carlo path simulation techniques for such processes by FFT and symbol localization are outlined. Numerical experiments illustrate multilevel preconditioning of the moment matrices for several exotic contracts as well as for Feller-Levy processes with variable order jump intensities. Model sensitivity of Levy models embedded into Feller classes is studied numerically for several types of plain vanilla, barrier and exotic contracts.

hp-DGFEM FOR KOLMOGOROV–FOKKER–PLANCK EQUATIONS OF MULTIVARIATE LÉVY PROCESSES

We analyze the discretization of nonlocal degenerate integrodifferential equations arising as so-called forward equations for jump-diffusion processes. Such equations arise in option pricing problems when the stochastic dynamics of the markets is modeled by Levy driven stochastic volatility models. Well-posedness of the arising equations is addressed. We develop and analyze stable discretization schemes, in particular the discontinuous Galerkin Finite Element Methods (DG-FEM). In the DG-FEM, a new regularization of hypersingular integrals in the Dirichlet form of the pure jump part of infinite variation processes is proposed, allowing in particular a stable DG discretization of hypersingular integral operators. Robustness of the stabilized discretization with respect to various degeneracies in the characteristic triple of the stochastic process is proved. We provide in particular an hp-error analysis of the DG-FEM. Numerical experiments for model equations confirm the theoretical results.

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.

Multidimensional Feller Processes

In this chapter, we extend the setting of Chap. 14 to a more general class of processes. We consider a large class of Markov processes in the following. Under certain assumptions we can apply the theory of pseudodifferential operators in order to analyse the arising pricing equations. The dependence structure of the purely discontinuous part of the market model X is described using Levy copulas. Wavelets are used for the discretization and preconditioning of the arising PIDEs, which are of variable order with the order depending on the jump state.