TL;DR

The chain rule for stochastic calculus. For a twice-differentiable function \( f \) of an Ito process \( X_t \): \( df = f'dX + \frac{1}{2}f''(dX)^2 \), where \( (dW)^2 = dt \).

By Valenke Exam Prep Team·Last updated 2026-06-03

Ito's Lemma

The chain rule for stochastic calculus. For a twice-differentiable function \( f \) of an Ito process \( X_t \): \( df = f'dX + \frac{1}{2}f''(dX)^2 \), where \( (dW)^2 = dt \).

Why it matters for interviews

The single most important result in mathematical finance. Used to derive the Black-Scholes PDE, transform stochastic processes, and compute dynamics of portfolio values. Tested extensively in quant interviews.

Definition and Mathematical Foundation

The chain rule for stochastic calculus. For a twice-differentiable function \( f \) of an Ito process \( X_t \): \( df = f'dX + \frac{1}{2}f''(dX)^2 \), where \( (dW)^2 = dt \).

Application in Quantitative Finance

Related Concepts

Brownian Motion (Wiener Process)

Related Terms

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Frequently Asked Questions

Why does Ito's lemma have an extra term compared to the ordinary chain rule?▼

Because Brownian motion has quadratic variation: \( (dW)^2 = dt \neq 0 \). This non-zero quadratic variation produces the \( \frac{1}{2}f'' dt \) correction term that has no analog in ordinary calculus.

How is Ito's lemma used to derive Black-Scholes?▼

Apply Ito's lemma to \( f(S,t) \) (option price as a function of stock price and time) with \( dS = \mu S dt + \sigma S dW \). Setting the coefficient of \( dW \) to zero via delta hedging yields the Black-Scholes PDE.

What is the multidimensional version of Ito's lemma?▼

For \( f(X^1_t, \ldots, X^n_t, t) \), the formula includes all partial derivatives and cross-terms \( dX^i dX^j \), using the rules \( dW^i dW^j = \rho_{ij} dt \) and \( dW dt = 0 \).