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 exponential Lévy models, including the exponentially affine jump-diffusion models. For an M-times exercisable Bermudan option, the overall complexity is O(MN log(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.
This paper has been presented at the Computational Finance minisymposium of the Jahrestagung der Deutschen Mathematiker-Vereinigung 2006 in Bonn, the Fourth World Congress of the Bachelier Finance Society in Tokyo, the AMaMeF workshop on financial modelling with jump processes in Palaiseau and the Frankfurt MathFinance Workshop 2007.