Stochastic Calculus for Quant Interviews: Brownian Motion, Itô, Black-Scholes

Stochastic Calculus for Quant Interviews: Brownian Motion, Itô’s Lemma, and Real Applications

Stochastic calculus is the mathematical machinery behind continuous-time financial models. Black-Scholes, the Heston model, interest rate term-structure models (Vasicek, CIR, Hull-White), and most option-pricing theory live in this framework. For quant-research candidates targeting D. E. Shaw, Two Sigma, Citadel’s quant arm, the Strats groups at Goldman Sachs and Morgan Stanley, or any firm with serious derivatives pricing, fluency with stochastic calculus is expected.

This guide covers what’s actually tested and what’s just background knowledge. You don’t need a PhD-level understanding of measure theory to pass quant interviews; you need the working knowledge that lets you set up Itô’s lemma correctly, understand what Brownian motion does and doesn’t represent, and explain how Black-Scholes follows from these primitives.

The Core Objects

Brownian motion (Wiener process)

A continuous-time stochastic process W(t) with these properties:

• W(0) = 0
• W has independent increments: W(t+s) – W(t) is independent of all prior values
• W(t+s) – W(t) ~ N(0, s): increments are normally distributed with variance equal to time elapsed
• Paths are continuous but nowhere differentiable

The “nowhere differentiable” property is the source of the unusual rules of stochastic calculus. Brownian motion is too rough for ordinary calculus to apply.

Itô integral

An integral against Brownian motion: ∫ f(t, W(t)) dW(t). Defined as a limit of Riemann-style sums where the integrand is evaluated at the left endpoint of each partition interval (this matters; alternative conventions like Stratonovich give different rules).

Key property: the Itô integral is a martingale. E[∫ f dW] = 0 if f is adapted (depends only on past information).

Stochastic differential equations (SDEs)

Equations of the form: dX(t) = μ(t, X) dt + σ(t, X) dW(t). The drift μ controls the deterministic trend; the diffusion σ controls the volatility. Almost every continuous-time financial model is an SDE.

The most famous: geometric Brownian motion, dS = μS dt + σS dW. Solution: S(t) = S(0) exp((μ – σ²/2)t + σW(t)). This is the model underlying Black-Scholes.

Itô’s Lemma

The chain rule of stochastic calculus. For a smooth function f(t, X) where X follows the SDE dX = μ dt + σ dW:

df = (∂f/∂t + μ ∂f/∂x + (1/2) σ² ∂²f/∂x²) dt + σ ∂f/∂x dW

The “extra” second-derivative term (1/2) σ² ∂²f/∂x² is the signature of Itô calculus. It comes from the fact that (dW)² = dt (a non-zero quadratic variation), unlike ordinary calculus where (dt)² = 0.

Canonical application

“X follows geometric Brownian motion. What SDE does ln(X) follow?”

Let f(X) = ln(X). Then ∂f/∂x = 1/X, ∂²f/∂x² = -1/X². With dX = μX dt + σX dW:

d(ln X) = (μ – σ²/2) dt + σ dW

This says that log returns over time t are normally distributed with mean (μ – σ²/2)t and variance σ²t. This is why log returns are the natural quantity in financial modeling.

Another canonical application

“What’s the SDE for X²?” Let f(X) = X². Then ∂f/∂x = 2X, ∂²f/∂x² = 2. With dX = μ dt + σ dW:

d(X²) = (2X μ + σ²) dt + 2X σ dW

From SDE to Black-Scholes

The Black-Scholes derivation outline:

1. Stock follows GBM: dS = μS dt + σS dW.
2. Option price V(t, S) is a function of t and S. Apply Itô’s lemma to find dV.
3. Construct a portfolio Π = V – Δ S where Δ = ∂V/∂S. The portfolio is delta-hedged: its value doesn’t depend instantaneously on dW.
4. Because the portfolio is risk-free (no dW exposure), it must earn the risk-free rate r in the absence of arbitrage: dΠ = rΠ dt.
5. Substituting and simplifying yields the Black-Scholes PDE: ∂V/∂t + (1/2) σ² S² ∂²V/∂S² + r S ∂V/∂S – rV = 0.
6. Solving the PDE with boundary condition V(T, S) = max(S – K, 0) for a call gives the Black-Scholes formula.

You don’t need to derive this in 30 minutes during an interview; you need to be able to outline it and answer follow-up questions about each step.

Risk-Neutral Measure

A subtle but important concept. Under the “physical” probability measure, stocks have expected return μ (typically > r). Under the “risk-neutral” measure, all assets have expected return r.

Why? Pricing derivatives by replication doesn’t depend on individual investors’ risk preferences (that’s the no-arbitrage assumption). Equivalent martingale measures exist that price all derivatives consistently as discounted expectations. Under this measure, the stock SDE becomes:

dS = rS dt + σS dW̃

where W̃ is Brownian motion under the risk-neutral measure (related to the original W by the Girsanov theorem). All option prices are expectations under this measure.

For interview purposes: you should know risk-neutral pricing means “price = discounted expected payoff under risk-neutral measure,” and you should be able to explain why μ disappears from option prices (it gets absorbed into the change of measure).

Important SDEs in Finance

Vasicek interest rate model

dr = a(b – r) dt + σ dW. Mean-reverting; r tends toward b. Allows negative rates (a feature in 2010s, less relevant in 2020s).

Cox-Ingersoll-Ross (CIR)

dr = a(b – r) dt + σ √r dW. Mean-reverting and bounded below by zero (the √r term makes the diffusion vanish as r → 0).

Heston model

Stock with stochastic volatility:

• dS = μS dt + √v S dW₁
• dv = κ(θ – v) dt + σ √v dW₂

where W₁ and W₂ have correlation ρ. Captures volatility clustering and the implied-volatility smile better than Black-Scholes.

Common Interview Questions

Apply Itô’s lemma

“X follows dX = X dt + 2X dW. Find the SDE for X².” This is mechanical Itô; the candidate’s job is to compute ∂f/∂t, ∂f/∂x, ∂²f/∂x² and assemble the result.

Compute expected value under risk-neutral measure

“Compute E[max(S(T) – K, 0)] under risk-neutral pricing for a stock following GBM.” This walks through Black-Scholes by hand. Strong candidates set up the integral, recognize the form, and either compute or invoke the BS formula.

Discuss path properties of Brownian motion

“Why is W(t) nowhere differentiable but everywhere continuous?” The answer involves the scaling property: increments of size √dt over intervals of size dt give effective slope 1/√dt → ∞. This makes BM continuous (small increments) but with infinite slope (no derivative).

Explain quadratic variation

“What’s the quadratic variation of Brownian motion over [0, t]?” Answer: t. This is why (dW)² = dt heuristically, which is the source of the Itô correction term.

Discuss change of measure

“Why is the drift of the stock different under physical and risk-neutral measures?” Equivalent martingale measure transforms the drift; under risk-neutral measure, all assets earn r so derivatives can be priced as discounted expectations without specifying individual risk preferences.

What Interviewers Don’t Test

Quant interviews almost never ask measure-theoretic foundations: σ-algebras, filtrations, Radon-Nikodym derivatives in detail. You should be able to use these concepts in spoken language (“under the risk-neutral measure,” “adapted process,” “martingale”) but not derive them from scratch.

You also don’t need to memorize complete derivations of every named model. You need to recognize them, sketch the structure, and discuss applications.

Frequently Asked Questions

See also: Options Pricing for Quant Interviews • Linear Algebra for Quant Interviews • Random Walks and Stopping Times