Fourier Space Time-stepping (FST) framework a generic framework based on the Fourier transform for pricing and hedging of various options in equity, commodity, currency, and insurance markets. The pricing problem can be reduced to solving a partial integro-differential equation (PIDE). The FST framework circumvents the problems associated with the existing finite difference methods by utilizing the Fourier transform to solve the PIDE, with resulting method being highly efficient and rapidly convergent.

The Fourier transform can be applied to the pricing PIDE to obtain a linear system of ordinary differential equations that can be solved explicitly. Solving the PIDE in Fourier space allows for the integral term to be handled efficiently and avoids the asymmetrical treatment of diffusion and jump terms arising in the finite difference schemes found in literature. For path-independent options, prices can be obtained for a range of stock prices in one iteration. For exotic, path-dependent options, a time-stepping algorithm is developed to handle barriers, free boundaries, and exercise policies.

The FST framework-based methods are applied to a wide range of option pricing problems. Pricing of single- and multi-asset, European and path-dependent options under independent-increment exponential Levy stock price models, common in equity and insurance markets, can be done efficiently via the cornerstone FST method. Mean-reverting Levy spot price models, common in commodity markets, are handled by introducing a frequency transformation, which can be readily computed via scaling of the option value function. Generating stochastic volatility, to match the long-term equity options market data, and stochastic skew, observed in currency markets, is addressed by introducing a non-stationary extension of multi-dimensional Levy processes using regime-switching. Finally, codependent jumps in multi-asset models are introduced through copulas.

The FST methods are computationally efficient, running in O(MN log N) time on a M time steps and N space points grid. The methods achieve second-order convergence in space; for American options, a penalty iteration is used to attain second-order convergence in time. Graphics processing units are utilized to reduce the computational time of FST methods.

The code is now available on SourceForge.net