Barrier and Path-Dependent Options in Exponential Lévy Models

Under GBM, the reflection principle gives exact analytic formulas for barrier options. Under an exponential Lévy model, this approach fails: jumps allow the stock to overshoot the barrier — to jump from above $B$ to well below $B$ in a single step, skipping the barrier level entirely. The probability of overshooting depends on the Lévy measure near the barrier, and the overshoot distribution is not a simple reflection.

Two approaches handle barrier options in Lévy models: the PIDE (developed in the incomplete markets lesson) and the Wiener-Hopf factorisation (exact for the Kou double-exponential model, approximate for general Lévy processes). The gap between GBM and Lévy barrier prices is the jump gap risk — the P&L loss when the stock jumps through the barrier without the hedge position being unwound.

Learning objectives. By the end of this note you should be able to: