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Stochastic Foundations for Finance

The probability and stochastic calculus behind continuous-time asset pricing.

Forty gray Brownian paths starting at zero spread out over time inside a dashed band that widens like the square root of time. One path, in red, ends near 1. A side panel shows a histogram of the values at time 1 matching a bell-shaped standard normal density, with a red line marking where the highlighted path ends.

Forty simulated paths of a standard Brownian motion on [0, 1], with one highlighted, inside the band \pm 1.96\sqrt{t} that contains W_t with probability 95% at each date. On the right, the distribution of W_1 across 20,000 paths and the standard normal density.

This site collects my notes on stochastic processes for finance, which I use across several courses. They start from basic probability and build up to Poisson processes, Brownian motion, Itô calculus, and the pricing of derivatives in continuous time, with and without jumps. Each step is motivated by the finance problem it will be used for.

The notes contain the full derivations and worked examples, and each chapter is available in HTML and as a PDF for printing.