APPROXIMATING LEVY PROCESSES WITH A VIEW TO OPTION PRICING

Abstract. We examine how to approximate a Levy process by a hyperexponential
jump-diffusion (HEJD) process, composed of Brownian motion and of an arbitrary number
of sums of compound Poisson processes with double exponentially distributed jumps.
This approximation will facilitate the pricing of exotic options since HEJD processes
have a degree of tractability that other Levy processes do not have. The idea behind
this approximation has been applied to option pricing by [Asmussen et al.:2007]
and [Jeannin and Pistorius:2008]. In this paper we introduce a more systematic
methodology for constructing this approximation which allow us to compute the intensity
rates, the mean jump sizes and the volatility of the approximating HEJD process (almost)
analytically. Our methodology is very easy to implement. We compute vanilla option prices
and barrier option prices using the approximating HEJD process and we compare our results
to those obtained from other methodologies in the literature. We demonstrate that our
methodology gives very accurate option prices and that these prices are more accurate than
those obtained from existing methodologies for approximating Levy processes by HEJD
processes.

An Excel spreadsheet that contains in tabular format some of the results from sections 4 and 5
is available here.

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