Karel in ’t Hout

ADI schemes for pricing American options under the Heston model

Abstract In this article, a simple, effective adaptation of Alternating Direction Implicit time discretization schemes is proposed for the numerical pricing of American-style options under the Heston model via a partial differential complementarity problem. The stability and convergence of the new methods are extensively investigated in actual, challenging applications. In addition, a relevant theoretical result is proved.

ADI Finite Difference Discretization of the Heston‐Hull‐White PDE

This paper concerns the efficient numerical solution of the time‐dependent, three‐dimensional Heston‐Hull‐White PDE for the fair prices of European call options. The numerical solution method described in this paper consists of a finite difference discretization on non‐uniform spatial grids followed by an Alternating Direction Implicit scheme for the time discretization and extends the method recently proved effective by In’t Hout & Foulon (2010) for the simpler, two‐dimensional Heston PDE.

Convergence of the Modified Craig-Sneyd scheme for two-dimensional convection-diffusion equations with mixed derivative term

We consider the Modified Craig-Sneyd (MCS) scheme which forms a prominent time stepping method of the Alternating Direction Implicit type for multidimensional time-dependent convection-diffusion equations with mixed spatial derivative terms. Such equations arise often, notably, in the field of financial mathematics. In this paper a first convergence theorem for the MCS scheme is proved where the obtained bound on the global temporal discretization errors has the essential property that it is independent of the (arbitrarily small) spatial mesh width from the semidiscretization. The obtained theorem is directly pertinent to two-dimensional convection-diffusion equations with mixed derivative term. Numerical experiments are provided that illustrate our result.

ADI Schemes with Ikonen—Toivanen Splitting for Pricing American put Options in the Heston Model

The numerical valuation of American put options under the Heston stochastic volatility model is considered. We investigate in this paper the potential of combining the recent splitting approach of Ikonen & Toivanen (2004, 2009) with Alternating Direction Implicit schemes to obtain more efficient numerical methods.

Application of Operator Splitting Methods in Finance

Financial derivatives pricing aims to find the fair value of a financial contract on an underlying asset. Here we consider option pricing in the partial differential equations framework. The contemporary models lead to one-dimensional or multidimensional parabolic problems of the convection-diffusion type and generalizations thereof. An overview of various operator splitting methods is presented for the efficient numerical solution of these problems. Splitting schemes of the Alternating Direction Implicit (ADI) type are discussed for multidimensional problems, e.g. given by stochastic volatility (SV) models. For jump models Implicit-Explicit (IMEX) methods are considered which efficiently treat the nonlocal jump operator. For American options an easy-to-implement operator splitting method is described for the resulting linear complementarity problems. Numerical experiments are presented to illustrate the actual stability and convergence of the splitting schemes. Here European and American put options are considered under four asset price models: the classical Black-Scholes model, the Merton jump-diffusion model, the Heston SV model, and the Bates SV model with jumps.

A Stability Result for the Modified Craig‐Sneyd Scheme Applied to 2D and 3D Pure Diffusion Equations

The stability of ADI type schemes is considered when applied to two‐ and three‐dimensional pure diffusion equations with mixed derivative terms. We present a first stability result that takes the actual size of the mixed derivative coefficients into account.

Convergence of the Hundsdorfer–Verwer scheme for two-dimensional convection-diffusion equations with mixed derivative term

Alternating Direction Implicit (ADI) schemes are popular in the numerical solution of multidimensional time-dependent partial differential equations (PDEs) arising in various contemporary application fields such as financial mathematics. The Hundsdorfer–Verwer (HV) scheme is an often used ADI scheme. A structural analysis of its fundamental properties, notably convergence, is of main interest. Up to now, however, a convergence result is only known in the literature relevant to the case of one-dimensional PDEs. In this paper we prove that, under natural stability and smoothness conditions, the HV scheme has a temporal order of convergence equal to two, uniformly in the spatial mesh width, whenever it is applied to two-dimensional convection-diffusion equations with mixed derivative term.

Stability of central finite difference schemes for the Heston PDE

This paper deals with stability in the numerical solution of the prominent Heston partial differential equation from mathematical finance. We study the well-known central second-order finite difference discretization, which leads to large semidiscrete systems with nonnormal matrices A. By employing the logarithmic spectral norm we prove practical, rigorous stability bounds. Our theoretical stability results are illustrated by ample numerical experiments. We also apply the analysis to obtain useful stability bounds for time discretization methods.

ADI Schemes in the Numerical Solution of the Heston PDE

This paper deals with ADI type schemes in the numerical solution of the large systems of stiff ODEs that arise after spatial discretization of the Heston PDE from financial option pricing. A feature of this well‐known two‐dimensional PDE is the presence of a mixed spatial derivative term, and ADI schemes were not originally developed to handle such terms. We discuss how to adapt three ADI schemes and next we review theoretical results and perform numerical experiments.

An adjoint method for the exact calibration of Stochastic Local Volatility models

This paper deals with the exact calibration of semidiscretized stochastic local volatility (SLV) models to their underlying semidiscretized local volatility (LV) models. Under an SLV model, it is common to approximate the fair value of European-style options by semidiscretizing the backward Kolmogorov equation using finite differences. In the present paper we introduce an adjoint semidiscretization of the corresponding forward Kolmogorov equation. This adjoint semidiscretization is used to obtain an expression for the leverage function in the pertinent SLV model such that the approximated fair values defined by the LV and SLV models are identical for non-path-dependent European-style options. In order to employ this expression, a large non-linear system of ODEs needs to be solved. The actual numerical calibration is performed by combining ADI time stepping with an inner iteration to handle the non-linearity. Ample numerical experiments are presented that illustrate the effectiveness of the calibration procedure.