Itô's Lemma and Stochastic Calculus › Itô's Lemma and Stochastic Calculus
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In the previous chapter, we explored Brownian motion, the mathematical foundation for modeling random price movements. We saw that Brownian motion has continuous but nowhere-differentiable paths, creating a fundamental problem: how do we apply calculus to functions of processes that cannot be differentiated in the classical sense?
Stochastic calculus, developed by Kiyoshi Itô in the 1940s, extends ordinary calculus to random processes. Itô's Lemma is the chain rule for this framework. Just as the ordinary chain rule tells us how to differentiate composite functions, Itô's Lemma tells us how functions of stochastic processes evolve over time.
Quantitative finance uses Itô's Lemma to derive derivative price dynamics. For an asset following geometric Brownian motion, Itô's Lemma shows how options and other derivatives behave. It will be the key tool we use in the next chapter to derive the Black-Scholes-Merton partial differential equation.