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

Abstract

a b s t r a c tWe consider the problem of pricing American options in the framework of a well-known stochastic volatility model with jumps, the Bates model. According to this model the asset price is described by a jump-diffusion stochastic differential equation in which the jump term consists of a Lévy process of compound Poisson type, while the volatility is modeled as a CIR-type process correlated with the asset price. Pricing American options under the Bates model requires us to solve a partial integro-differential equation with the final condition and boundary conditions prescribed on a free boundary. In this paper a numerical method for solving such a problem is proposed. In particular, first of all, using a Richardson extrapolation technique, the problem is reduced to a problem with fixed boundary. Then the problem obtained is solved using an ad hoc finite element method which efficiently combines an implicit/explicit time stepping, an operator splitting technique, and a nonuniform mesh of right-angled triangles. Numerical experiments are presented showing that the option pricing algorithm developed in this paper is extremely accurate and fast. In particular it is significantly more efficient than other numerical methods that have recently been proposed for pricing American options under the Bates model.

Pricing European and American Options Using a Very Fast and Accurate Scheme: The Meshless Local Petrov–Galerkin Method

“…The option price Vðs; tÞ satisfies, for s 2 ½0; þ1Þ and t 2 ½0; TÞ, the following partial differential problem: In this work, the price of American option is computed by Richardson extrapolation of the price of Bermudan option [9]. In essence the Richardson extrapolation reduces the free boundary problem and linear complementarity problem to an easily solvable fixed boundary problem.…”

“…Then, the obtained approximation, which is only first-order accurate, is improved by Richardson extrapolation. In particular, we manage to obtain second-order accuracy by extrapolation of two solutions computed using M and 2M time steps [9,10,25]. We here restrict our attention to first stage of the Richardson extrapolation procedure, where M time steps are employed, and the fact that the partial differential problems considered are also solved with 2M time steps.…”

“…In particular, we manage to obtain second-order accuracy by extrapolation of two solutions computed using M and 2M time steps [9,10,25]. …”

“…Analytical and numerical methods have been used for European and American option pricing such as finite difference (and the others mesh methods) [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] and the binomial/triniomial trees [20][21][22][23], nevertheless some authors have also proposed the use of meshfree algorithms based on radial basis functions [24][25][26][27][28] and on quasi radial basis functions [29]. To solve the dynamical systems, analytical solutions are the first choice, but are applicable to sample cases, not for American options.…”

A stable local radial basis function method for option pricing problem under the Bates model

“…The results are mostly reported at some certain points of the domain which are, also known as hot zones, (i.e., (100, 0.04)) throughout this section. Actually, these values are deemed significant from the financial standpoint . Furthermore, if the point that we are computing for, is not a node of the mesh, the approximate solution at such a point is obtained by a built‐in interpolation command in the applied programming package Mathematica.…”