The wiener-hopf technique and discretely monitored path-dependent option pricing

Fusai, Abrahams, and Sgarra (2006) employed the Wiener-Hopf technique to obtain an exact analytic expression for discretely monitored barrier option prices as the solution to the Black-Scholes partial differential equation. The present work reformulates this in the language of random walks and extends it to price a variety of other discretely monitored path-dependent options. Analytic arguments familiar in the applied mathematics literature are used to obtain fluctuation identities. This includes casting the famous identities of Baxter and Spitzer in a form convenient to price barrier, first-touch, and hindsight options. Analyzing random walks killed by two absorbing barriers with a modified Wiener-Hopf technique yields a novel formula for double-barrier option prices. Continuum limits and continuity correction approximations are considered. Numerically, efficient results are obtained by implementing Padé approximation. A Gaussian Black-Scholes framework is used as a simple model to exemplify the techniques, but the analysis applies to Lévy processes generally