{"categories":["Stochastic Calculus"],"contentHtml":"<p>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.</p>\n<p>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.</p>\n<p>These products are a natural reason to study stochastic calculus: their value depends on the whole path, so a terminal-price shortcut is insufficient.</p>","contentMarkdown":"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.\n\nThe 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.\n\nThese products are a natural reason to study stochastic calculus: their value depends on the whole path, so a terminal-price shortcut is insufficient.","dataUrl":"https://sharifhsn.dev/api/posts/exotic-options.json","date":"2024-12-12","datePublished":"2024-12-12","description":"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 expi…","site":"https://sharifhsn.dev","slug":"exotic-options","source":"FE-610 | Stochastic Calculus","sourceUrl":null,"tags":["Stochastic Calculus","Exotic Options","Barrier Options"],"title":"Exotic Options","url":"https://sharifhsn.dev/blog/exotic-options/","version":"1","wordCount":101}