The Wiener-Hopf Technique and Discretely Monitored Path-Dependent Option Pricing

Abstract

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Spitzer identity, Wiener-Hopf factorization and pricing of discretely monitored exotic options

“…Unfortunately he did not present a constructive procedure for the determination of the quantities J + (ξ, q) and J − (ξ, q) in Equation (16). The problem was later solved in the Gaussian case by Green et al [28,Section 2.4]. Here we generalize the latter construction to Lévy processes.…”

“…Similar techniques were developed later for path-dependent derivatives [e.g. 11,19,28]. Our paper provides a unified framework and a fast operational method for pricing barrier and lookback (or hindsight) options when the underlying asset evolves as an exponential Lévy process.…”

“…Here we generalize the latter construction to Lévy processes. In particular, Green et al [28] proved that J + (ξ, q) and J − (ξ, q) are the solution of the coupled integral equations…”

“…with a = max(l, k), b = u for a call option and a = min(k, u), b = l for a put option [28,Equation (3.26)], the main problem in evaluating pathdependent options is the computation of the characteristic functions of the (joint) probability densities defined in Equations (5)- (8). Here we exploit the Spitzer identity and the factorization procedure previously described.…”

“…6,7,28,46] and the present paper is mainly focused on this kind of applications, our method goes well beyond option pricing and opens up the way to a more extensive use of the Spitzer identity and the WH factorization in a variety of non-financial fields; for physics, see a recent review by Bray et al [9]. In this regard we would like to mention the applicability to queuing theory due to the strict connection between random walks and queues, see Lindley [47] for pioneering contributions as well as Cohen [14], Prabhu [50] and Asmussen [2,3].…”