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 [18], and by a Poisson jump process of the type originally introduced by Merton [25]. 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 [26] 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 and Toivanen [21]. 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 * Corresponding author. 393 Int. J. Theor. Appl. Finan. 2009.12:393-425. Downloaded from www.worldscientific.com by THE UNIVERSITY OF CHICAGO on 12/26/14. For personal use only. 394 C. Chiarella et al.of accuracy for a given computational effort then the method of lines seems to perform best amongst the methods considered.

The evaluation of American options in a stochastic volatility model with jumps: An efficient finite element approach

“…In fact, if the simulations are carried out on a computer with a Pentium Dual Core E 2140 Processor 1.6 GHz 2 GB RAM, the American option price, the hedge parameters ∆ and Γ , and the early exercise boundary are obtained with relative errors of order 10 −4 or smaller in a time equal to or smaller than 16.3 s. In addition, the computer time necessary for evaluating the jump-integral term and the computer time necessary for evaluating (all) the differential terms are approximately the same, which indicates that the discretization of the jump-integral operator has been carried out efficiently. Furthermore, the experiments performed also reveal that the numerical method proposed in this paper is considerably faster than the numerical method presented in [24] and in [25].…”

“…However, to the best of our knowledge, a thorough investigation on the boundary conditions at y = 0 and y → +∞ is still lacking. In this paper following a common approach (see for instance [24]) we will circumvent the problem by extrapolating the solution at y = 0 and y → +∞ from the numerical solution obtained in the interior of the (S, y) computational domain (see Section 4).…”

“…In particular, according to the componentwise splitting employed in [25], the matrix obtained from the discretization of the mixed second-order derivatives is split into three submatrices. As shown in [25], this amounts to changing the coefficients of the diffusive operators in the partial integro-differential equation (15), and, according to numerical experiments carried out in [24], can cause serious accuracy and stability issues. The operator splitting employed in this paper is as follows:…”

“…In particular, a finite element method for European Call and Put options has been developed in [23], in order to establish a basis for pricing more complex exotic products. Instead, in the case of American options, a numerical method has recently been presented by Chiarella et al [24]. In particular, in this latter work, a finite difference scheme and the method of lines are used to reduce the American option pricing problem to a system of second-order ordinary differential equations.…”

“…Moreover, the componentwise splitting technique employed in [25] requires a special treatment of the mixed secondorder derivatives, which may cause some loss of accuracy, and imposes severe restrictions on the choice of the grid step sizes when a non-uniform mesh is employed (see [21,22]). Finally it is worth noticing that in [24] Chiarella et al test the componentwise splitting employed in [25], and highlight serious accuracy and stability issues.…”