The risk-neutral section motivates a change of measure rather than assuming investors literally expect the risk-free rate. Discounted tradable prices should be martingales under a pricing measure \\(\\mathbb Q\\). Under that measure, a payoff \\(H_T\\) is valued as
\n$$V_t=\\mathbb E^{\\mathbb Q}\\left[e^{-r(T-t)}H_T\\mid\\mathcal F_t\\right].$$
\nGirsanov's theorem explains how the drift changes when the Brownian motion is shifted. The new measure is equivalent to the original one, so zero-probability events stay zero; only the weights of possible paths change. The notes use the density process and the discounted asset to connect this measure change to no-arbitrage pricing.
","contentMarkdown":"The risk-neutral section motivates a change of measure rather than assuming investors literally expect the risk-free rate. Discounted tradable prices should be martingales under a pricing measure \\(\\mathbb Q\\). Under that measure, a payoff \\(H_T\\) is valued as\n\n$$V_t=\\mathbb E^{\\mathbb Q}\\left[e^{-r(T-t)}H_T\\mid\\mathcal F_t\\right].$$\n\nGirsanov's theorem explains how the drift changes when the Brownian motion is shifted. The new measure is equivalent to the original one, so zero-probability events stay zero; only the weights of possible paths change. The notes use the density process and the discounted asset to connect this measure change to no-arbitrage pricing.","dataUrl":"https://sharifhsn.dev/api/posts/risk-neutral-measures.json","date":"2024-11-07","datePublished":"2024-11-07","description":"The risk-neutral section motivates a change of measure rather than assuming investors literally expect the risk-free rate. Discounted tradable prices should be martingales under a …","site":"https://sharifhsn.dev","slug":"risk-neutral-measures","source":"FE-610 | Stochastic Calculus","sourceUrl":null,"tags":["Stochastic Calculus","Risk-Neutral Pricing","Girsanov"],"title":"Risk-Neutral Measures","url":"https://sharifhsn.dev/blog/risk-neutral-measures/","version":"1","wordCount":94}