Exotic Options

The final FE-610 notes apply the earlier stopping-time and maximum results to path-dependent payoffs. A barrier option depends on whether the underlying crosses a level before expiry; a lookback option depends on the running maximum or minimum.

The running maximum \(M_t=\max_{0\leq u\leq t}W_u\) is not determined by the terminal value alone. The pair \((W_t,M_t)\) is the useful Markov state. Reflection arguments relate events involving the maximum to ordinary Brownian probabilities, which makes joint distributions and first-passage calculations possible.

These products are a natural reason to study stochastic calculus: their value depends on the whole path, so a terminal-price shortcut is insufficient.