Removing the Correlation Term in Option Pricing Heston Model: Numerical Analysis and Computing

Abstract

This paper deals with the numerical solution of option pricing stochastic volatility model described by a time-dependent, two-dimensional convection-diffusion reaction equation. Firstly, the mixed spatial derivative of the partial differential equation (PDE) is removed by means of the classical technique for reduction of second-order linear partial differential equations to canonical form. An explicit difference scheme with positive coefficients and only five-point computational stencil is constructed. The boundary conditions are adapted to the boundaries of the rhomboid transformed numerical domain. Consistency of the scheme with the PDE is shown and stepsize discretization conditions in order to guarantee stability are established. Illustrative numerical examples are included.

Numerical valuation of two-asset options under jump diffusion models using Gauss–Hermite quadrature

“…Both facts favor potential drawbacks arising from the dominance of the convection 40 versus the diffusion terms [10, 11,14]. In order to avoid these numerical drawbacks it is convenient to transform the original PIDE problem (2) into another one where the cross derivative one disappears.…”

“…In order to avoid these numerical drawbacks it is convenient to transform the original PIDE problem (2) into another one where the cross derivative one disappears. This technique has been successfully developed in [14,15]. Another strategy to reduce the number of points in the stencil schemes is based on special approximation of the mixed derivative, 45 see [9,10].…”

Positive finite difference schemes for a partial integro-differential option pricing model

“…Other further extensions have been studied in [9,10,11]. 20 In this paper we deal with the Bates model that describes the behavior of the underlying asset S and its variance ν by the coupled stochastic differential equations: dS(t) = (r − q − λξ)S(t)dt + ν(t)S(t)dW 1 + (η − 1)S(t)dZ(t), dν(t) = κ(θ − ν(t))dt + σ ν(t)dW 2 , dW 1 dW 2 = ρdt, where W 1 and W 2 are standard Brownian motions, Z is the poisson process.…”

“…We transform the PIDE (1) into a new PIDE without mixed spatial derivative before the discretization, following the idea of [20], and avoiding the above quoted drawbacks. Furthermore, this strategy has additional computa-50 tional advantage of the reduction of the stencil scheme points, from nine [21,22] or seven [13,17] to just five.…”

“…Under the transformation (9) as it is shown in [20] the rectangle R is transformed into the rhomboid ABCD as shown in Fig. 1, where the sides are given by…”