Fast deterministic pricing of options on Lévy driven assets

Abstract

Abstract. Arbitrage-free prices u of European contracts on risky assets whose log-returns are modelled by Lévy processes satisfy a parabolic partial integro-differential equationThis PIDE is localized to bounded domains and the error due to this localization is estimated. The localized PIDE is discretized by the θ-scheme in time and a wavelet Galerkin method with N degrees of freedom in log-price space. The dense matrix for A can be replaced by a sparse matrix in the wavelet basis, and the linear systems in each implicit time step are solved approximatively with GMRES in linear complexity. The total work of the algorithm for M time steps is bounded by O (MN(log(N )) 2 ) operations and O(N log(N )) memory. The deterministic algorithm gives optimal convergence rates (up to logarithmic terms) for the computed solution in the same complexity as finite difference approximations of the standard Black-Scholes equation. Computational examples for various Lévy price processes are presented.Mathematics Subject Classification. 65N30, 60J75.

Integro-differential equations for option prices in exponential Lévy models

“…Such numerical methods are discussed in a companion paper [11] and make a key use of the comparison principle for semicontinuous solutions [2]. The use of viscosity solutions allows to obtain pointwise convergence of option prices [11], which is more relevant for approximating option prices than L 2 -type convergence obtained using the notion of weak solution in Sobolev spaces [22].…”

“…Such partial integro-differential equations (PIDEs) have been used by several authors to price options in models with jumps [3,8,22,13] but the derivation of these equations is omitted in these works. We explore in this paper the precise link between option prices in exponential Lévy models and the related partial integro-differential equations (PIDEs) in the case of European and barrier options in exponential Lévy models.…”

Numerical valuation of options with jumps in the underlying

“…To overcome these problems, a number of alternative models have appeared in the financial literature: Stochastic volatility models [20,18]; deterministic local volatility functions [9,13]; jump-diffusion models [22,23,26]; Lévy models [2,7,14,25,28] amongst others. Jump-diffusion models and Lévy based models are attractive because they explain the jump patterns exhibited by some stocks.…”

“…In [27], the case of American options with Poisson jumps is treated numerically by a method of lines. More general models based on Lévy processes are also solved numerically in [1] by the ADI finite difference method combined with the fast Fourier transform and in [25] by a finite element method that gives a compressed sparse matrix in a convenient wavelet basis. An explicit method was used in [6] to solve Merton's model and a convergence theory for explicit schemes and CFL conditions were given for a general family of integro-differential Cauchy problems.…”

“…Recently, we came across [12], where the value of American options using Merton's model is found implicitly by the penalty method. Here, we intend to simplify some ideas from [25] for the jump-diffusion case, while keeping the algorithm fast.…”