Stochastic Calculus & Itô's Lemma

The chain rule for random processes, and the foundation of modern derivatives pricing.

Calculus for Randomness

Standard calculus deals with smooth, predictable changes. But what about processes that have a random component, like the path of a stock price? Stochastic calculus, and specifically Itô's Lemma, provides the mathematical framework for handling this.

Itô's Lemma is essentially the chain rule of stochastic calculus. It tells us how to find the differential of a function of a random process. The surprising result is an extra term that arises purely from the randomness, a term that is zero in normal calculus. This term is the key to almost all of modern quantitative finance.

Itô's Lemma
If a variable xx follows an Itô process of the form dx=a(x,t)dt+b(x,t)dWtdx = a(x,t)dt + b(x,t)dW_t, then for some function G(x,t)G(x,t), its differential is:
dG=(∂G∂t+a∂G∂x+12b2∂2G∂x2)dt+b∂G∂xdWtdG = \left( \frac{\partial G}{\partial t} + a \frac{\partial G}{\partial x} + \frac{1}{2} b^2 \frac{\partial^2 G}{\partial x^2} \right) dt + b \frac{\partial G}{\partial x} dW_t
  • The first part (in parentheses) is the deterministic "drift" of G.
  • The second part is the random "diffusion" of G.
  • The term 12b2∂2G∂x2\frac{1}{2} b^2 \frac{\partial^2 G}{\partial x^2} is the **Itô term**. It's non-zero because in a random walk, the variance grows with time ((dWt)2=dt(dW_t)^2 = dt), unlike in deterministic calculus where squared differentials vanish.

This lemma is used to derive the Black-Scholes equation, the foundational equation for pricing options.

Simulation vs. Reality

An interactive simulation for Itô's Lemma is currently under development. Stay tuned!