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Refined and enhanced FFT techniques, with applications to pricing barrier options and their sensitivities

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Many mathematical methods of option pricing rely on one’s ability to calculate the action of certain integro-differential operators and convolution operators quickly and efficiently. In turn, the latter computations are based on FFT techniques. However, in many important cases, a straightforward application of FFT and iFFT leads to errors of several kind, which cannot be made simultaneously small (uncertainty principle) unless grids with too many points are used. We explain an approach to using FFT techniques that gives one more flexibility in controlling the aforementioned errors, and, at the same time, yields fast and efficient algorithms. As applications, using Carr’s randomization, we compute the prices and sensitivities of barrier options and first-touch digital options on stocks whose log-price follows a Levy process. The numerical results obtained via our approach are demonstrated to be in good agreement with the results obtained using other (sometimes fundamentally different) approaches that exist in the literature. However, our method is computationally much faster (often, dozens of times faster). Moreover, our technique has the advantage that its application does not entail a detailed analysis of the underlying Levy process: one only needs an explicit analytic formula for the characteristic exponent of the process. Thus our algorithm is very easy to implement in practice. Finally, our method yields accurate results for a wide range of values of the spot price, including those that are very close to the barrier, regardless of whether the maturity period of the option is long or short. A natural extension of the method gives similar results for double-barrier options.

This talk is part of the Probability series.

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