{"categories":["Stochastic Calculus"],"contentHtml":"<p>The Black–Scholes notes start with a stock following geometric Brownian motion,</p>\n<p>$$dS_t=\\mu S_t\\,dt+\\sigma S_t\\,dW_t.$$</p>\n<p>Constructing a delta-hedged portfolio removes the Brownian shock. Applying Itô's formula to the option value and matching the remaining drift produces the Black–Scholes PDE. With terminal payoff \\(g(S_T)\\), the problem is a boundary-value problem: the price is determined backward from the payoff.</p>\n<p>The notes work through calls, puts, and the Black–Scholes–Merton form with a continuously compounded rate. The important modeling move is the hedge: risk is removed locally, so the option's drift is tied to the financing rate rather than to the stock's expected return.</p>","contentMarkdown":"The Black–Scholes notes start with a stock following geometric Brownian motion,\n\n$$dS_t=\\mu S_t\\,dt+\\sigma S_t\\,dW_t.$$\n\nConstructing a delta-hedged portfolio removes the Brownian shock. Applying Itô's formula to the option value and matching the remaining drift produces the Black–Scholes PDE. With terminal payoff \\(g(S_T)\\), the problem is a boundary-value problem: the price is determined backward from the payoff.\n\nThe notes work through calls, puts, and the Black–Scholes–Merton form with a continuously compounded rate. The important modeling move is the hedge: risk is removed locally, so the option's drift is tied to the financing rate rather than to the stock's expected return.","dataUrl":"https://sharifhsn.dev/api/posts/black-scholes.json","date":"2024-10-17","datePublished":"2024-10-17","description":"The Black–Scholes notes start with a stock following geometric Brownian motion,","site":"https://sharifhsn.dev","slug":"black-scholes","source":"FE-610 | Stochastic Calculus","sourceUrl":null,"tags":["Stochastic Calculus","Black–Scholes","Option Pricing"],"title":"Black–Scholes","url":"https://sharifhsn.dev/blog/black-scholes/","version":"1","wordCount":99}