A Fast and Accurate FFT-Based Method for Pricing Early-Exercise Options Under Levy Processes

Abstract

Abstract. A fast and accurate method for pricing early exercise and certain exotic options in computational finance is presented. The method is based on a quadrature technique and relies heavily on Fourier transformations. The main idea is to reformulate the well-known risk-neutral valuation formula by recognising that it is a convolution. The resulting convolution is dealt with numerically by using the Fast Fourier Transform (FFT). This novel pricing method, which we dub the Convolution method, CONV for short, is applicable to a wide variety of payoffs and only requires the knowledge of the characteristic function of the model. As such the method is applicable within many regular affine models, among which the class of exponential Lévy models. For an M -times exercisable Bermudan option, the overall complexity is O(M N log 2 (N )) with N grid points used to discretise the price of the underlying asset. American options are priced efficiently by applying Richardson extrapolation to the prices of Bermudan options.

Monte Carlo Simulation of the CGMY Process and Option Pricing

“…In recent years, numerical integration/transform techniques have proved very fast and accurate on pricing a wide range of single‐asset derivative products with path‐dependence and early‐exercise features written on Lévy driven underlying price processes with known characteristic functions (e.g., see Chung et al, ; Lord et al, ; Feng & Linetsky, ; Černý & Kyriakou, ). However, these techniques often become impractical when tackling high dimensional problems, such as multi‐asset contracts with path‐dependent payoffs (like, e.g., Asian basket options); in such cases, Monte Carlo simulation is generally the method of choice.…”

Accounting for earnings announcements in the pricing of equity options

“…The same observations have been made in Lord et al (2008), where (3.4) is implemented as part of a new pricing algorithm. Jackson et al (2008) also derive the same formula by analyzing the associated pricing PIDE in Fourier space.…”

“…For example, the methods of Carr and Madan (1999), Duffie and Singleton (2000), Lee (2004), and Raible (2000), among others, can be used to price European options. Alternatively, the methods developed in Jackson et al (2008) or Lord et al (2008) can be adapted to price both European and American options in models incorporating the jump Z e .…”

Valuing Early-Exercise Interest-Rate Options With Multi-Factor Affine Models

“…For path-dependent options a time-stepping algorithm must be employed to apply boundary conditions or impose early exercise constraints. As previously mentioned, [16,20] make use of a similar time-stepping approach for equity options on exponential Lévy models without mean-reversion. Further, [17] extended the approach to mean-reverting jump-diffusion models in the context of commodity derivatives.…”

“…Ballotta and Kyriakou [2] have developed a Fourier transform technique for pricing convertible bonds under Vasicek dynamics for the short rate and jump-diffusion for equity. Their method is based on the CONV method of [20] for pricing equity derivatives (see also [16]). …”