On the acceleration of explicit finite difference methods for option pricing

Abstract

Implicit finite difference methods are conventionally preferred over their explicit counterparts for the numerical valuation of options. In large part the reason for this is a severe stability constraint known as the Courant-Friedrichs-Lewy (CFL) condition which limits the latter class's efficiency. Implicit methods, however, are difficult to implement for all but the most simple of pricing models, whereas explicit techniques are easily adapted to complex problems. For the first time in a financial context, we present an acceleration technique, applicable to explicit finite difference schemes describing diffusive processes with symmetric evolution operators, called Super-Time-Stepping. We show that this method can be implemented as part of a more general approach for non-symmetric operators. Formal stability is thereby deduced for the exemplar cases of European and American put options priced under the Black-Scholes equation. Furthermore, we introduce a novel approach to describing the efficiencies of finite difference schemes as semi-empirical power laws relating the minimal real time required to carry out the numerical integration to a solution with a specified accuracy. Tests are described in which the method is shown to significantly ameliorate the severity of the CFL constraint whilst retaining the simplicity of the underlying explicit method. Degrees of acceleration are achieved yielding comparable, or superior, efficiencies to a set of benchmark implicit schemes. We infer that the described method is a powerful tool, the explicit nature of which makes it ideally suited to the treatment of symmetric and non-symmetric diffusion operators describing complex financial instruments including multi-dimensional systems requiring representation on decomposed and/or adaptive meshes.Numerical methods for option pricing, Black-Scholes model, Computational finance, Equity options, American options, Exotic options,

Pricing European and American Options in the Heston Model With Accelerated Explicit Finite Differencing Methods

“…Reference [33] find no evidence of any such instability influencing their solutions for cases with Nsts < ∼ 30.…”

“…These include: nonlinear degenerate convectiondiffusion [13]; electromagnetic wave scattering [37]; isotropic and anisotropic diffusion on biological membranes [35]; and magnetic field diffusion in astrophysics [32,29]. Recently, STS has been applied in finance and rigorously tested for the Black-Scholes pricing model by [33] where it was shown to be comparable, and in many cases superior, to the CN-(P)SOR scheme in terms of accuracy and computational speed for European (American) put options.…”

“…The former may then be treated via STS while the latter is efficiently integrated via a suitable scheme in a procedure introduced by [31,32]. In this way, reference [33] demonstrated formal stability for the accelerated scheme and showed that the splitting procedure is unnecessary in the case of a weakly non-symmetric operator. In this work, we follow that precedent and do not split the operator.…”

“…In this paper, the restriction that the CFL constraint imposes is reduced significantly using an acceleration technique for explicit algorithms known as Super-Time-Stepping (STS). STS was recently introduced to the finance literature for the first time in the one-factor Black-Scholes setting by [33] for vanilla European and American put options.…”

“…While the behavior of STS has only been analytically established for symmetric operators [2], a novel splitting approach was used in [33] to deal with non-symmetric operators under STS. This splitting method is based on the unique decomposition of the operator into its symmetric and skewsymmetric parts.…”