· 7 min
How an option is priced by hedging away its risk. The Black-Scholes partial differential equation derived from Ito's formula and a delta hedge, its risk-neutral solution as a discounted expected payoff, and the closed-form Black-Scholes formula for a European call.
· 7 min
Equations driven by noise, and the theorem that they have a unique solution. The strong solution, Gronwall's inequality, uniqueness via the Lipschitz condition and the Ito isometry, and existence by Picard iteration with the factorial decay that makes the iterates converge.
· 8 min
The chain rule of stochastic calculus. The second-order Taylor expansion that the quadratic variation of Brownian motion forces, Ito's formula for a function of a Brownian motion and of an Ito process, the integration-by-parts rule, and the solution of geometric Brownian motion.
· 6 min
Why Brownian motion needs its own calculus. The quadratic variation of a path, the theorem that Brownian motion accumulates quadratic variation equal to elapsed time, the resulting infinite total variation, and the covariation that gives the chain rule its extra term.