The Evaluation of American Option Prices Under Stochastic Volatility and Jump-Diffusion Dynamics Using the Method of Lines

Abstract

This paper considers the problem of numerically evaluating American option prices when the dynamics of the underlying are driven by both stochastic volatility following the square root process of Heston (1993), and by a Poisson jump process of the type originally introduced by Merton (1976). We develop a method of lines algorithm to evaluate the price as well as the delta and gamma of the option, thereby extending the method developed by Meyer (1998) for the case of jump-diffusion dynamics. The accuracy of the method is tested against two numerical methods that directly solve the integro-partial differential pricing equation. The first is an extension to the jump-diffusion situation of the componentwise splitting method of Ikonen & Toivanen (2007). The second method is a Crank-Nicolson scheme that is solved using projected successive over relaxation and which is taken as the benchmark for the price. The relative efficiency of these methods for computing the American call option price, delta, gamma and free boundary is analysed. If one seeks an algorithm that gives not only the price but also the delta and gamma to the same level of accuracy for a given computational effort then the method of lines seems to perform best amongst the methods considered.

High-Order Splitting Methods for Forward PDEs and PIDEs

“…A seven-point stencil for discretization of the mixed-derivative operator that preserves the positivity of the solution was proposed in Chiarella et al (2008), Toivanen (2010) for correlations ρ < 0, and in Ikonen & Toivanen (2007, 2008 for positive correlations. However, in their schemes the mixed derivative term was treated implicitly (that is the reason they needed a discretized matrix to be an Mmatrix).…”

A fast Fourier transform technique for pricing American options under stochastic volatility

“…They develop a …xed boundary adjustment algorithm such that the corresponding sequence of value functions monotonically converges to an American option price. Chiarella et al (2008) propose a method of lines, in which the original p.d.e. is replaced with an equivalent system of simpler di¤erential equations to be solved using a stabilized FD scheme.…”

“…Rather, I let the calibration procedure select the parameters of the actual measure utilized by the market for me. Chockalingam and Muthuraman (2007) and Chiarella et al (2008) discuss the issue of nonuniqueness in more detail.…”

Componentwise Splitting Methods for Pricing American Options Under Stochastic Volatility

“…2. A similar stencil has been described in [7]. For a positive correlation ρ, a suitable seven-point stencil is given in [21], [22].…”

“…The treatment of the jumps in the Merton and Kou models have been studied in [2], [3], [9], [10], [26], [32]. Pricing under the Bates model has been considered in [7] and under the correlated jump model in [13].…”