Stochastic Calculus Interview Questions: 20 Must-Know Problems

“State the four defining properties of a standard Brownian motion W_t.”

W₀ = 0 (starts at zero) · Independent increments: W_t − W_s ⊥ W_s − W_r for r < s < t · Stationary increments: W_t − W_s ~ N(0, t−s) · Continuous sample paths almost surely.

“What is E[W_t²]?”

Since W_t ~ N(0,t), we have E[W_t²] = Var(W_t) = t. This result, [W,W]_t = t, is the foundation of Itô's lemma.

“Compute the quadratic variation of Brownian motion and explain why it matters.”

[W,W]_t = t, captured by the heuristic (dW_t)² = dt. This non-zero quadratic variation is why stochastic calculus differs from ordinary calculus — Taylor expansions must include second-order terms.

“Show that M_t = W_t² − t is a martingale.”

Apply Itô's lemma to f(t,x) = x² − t: dM_t = 2W_t dW_t + dt − dt = 2W_t dW_t. No drift term remains, so M_t is a martingale.

“Compute P(max_{0≤s≤t} W_s ≥ a) for a > 0.”

By the reflection principle: P(max_{0≤s≤t} W_s ≥ a) = 2P(W_t ≥ a) = 2(1 − Φ(a/√t)). This result appears in barrier option pricing and first-passage-time problems.

“Find d(e^{W_t}).”

With f(x) = eˣ: d(e^{W_t}) = e^{W_t} dW_t + ½e^{W_t} dt. The ½e^{W_t}dt 'Itô correction' has no ordinary-calculus analogue. It ensures E[e^{W_t}] = e^{t/2}, not 1.

“Find d(W_t³).”

With f′ = 3x², f″ = 6x: d(W_t³) = 3W_t² dW_t + 3W_t dt.

“Solve dS_t = μS_t dt + σS_t dW_t.”

Apply Itô's lemma to ln S_t: d(ln S_t) = (μ − σ²/2) dt + σ dW_t. Integrating: S_t = S₀ exp[(μ − σ²/2)t + σW_t]. The −σ²/2 Itô correction ensures E[S_t] = S₀e^{μt}.

“Compute ∫₀ᵀ W_t d(eᵗ).”

By Itô's product rule with deterministic eᵗ: d(W_t eᵗ) = eᵗ dW_t + W_t eᵗ dt. Integrating: ∫₀ᵀ W_t eᵗ dt = W_T e^T − ∫₀ᵀ eᵗ dW_t.

“State the Itô isometry and apply it.”

For adapted H_t: E[(∫₀ᵀ H_t dW_t)²] = E[∫₀ᵀ H_t² dt]. Example: E[(∫₀ᵀ W_t dW_t)²] = ∫₀ᵀ E[W_t²] dt = ∫₀ᵀ t dt = T²/2.

“Solve dX_t = κ(θ − X_t) dt + σ dW_t.”

Using integrating factor e^{κt}: X_t = θ + (X₀ − θ)e^{−κt} + σ∫₀ᵗ e^{−κ(t−s)} dW_s. X_t is Gaussian; the mean reverts to θ at speed κ. Used for interest rates (Vasicek) and spread trading.

“Write the CEV SDE and explain when it reduces to GBM.”

dS_t = μS_t dt + σS_t^γ dW_t. When γ = 1: this is GBM. When γ < 1: volatility rises as S falls (leverage effect), generating an implied volatility skew absent from standard GBM.

“What does the Vasicek model predict about future rates, and what is its key limitation?”

dr_t = κ(θ − r_t) dt + σ dW_t implies normally distributed future rates with mean reversion. Key limitation: rates can turn negative — which became empirically relevant in EUR and JPY markets after 2014.

“State the conditions for a strong solution to an SDE.”

The SDE dX_t = b(t,X_t) dt + σ(t,X_t) dW_t has a unique strong solution if b and σ are (1) Lipschitz continuous in x and (2) of at most linear growth in x. These conditions prevent the solution from exploding in finite time.

“State Feynman-Kac and give a finance application.”

If u(t,x) solves ∂u/∂t + b·∂u/∂x + (σ²/2)·∂²u/∂x² − ru = 0 with terminal condition u(T,x) = g(x), then u(t,x) = E^{t,x}[e^{−r(T−t)}g(X_T)]. In finance: the Black-Scholes PDE solution equals the discounted risk-neutral expected payoff — letting quants choose between PDE solvers and Monte Carlo.

“State Girsanov's theorem and explain its role in option pricing.”

Define the measure change dQ/dP = exp(−∫₀ᵀ θ_s dW_s − ½∫₀ᵀ θ_s² ds). Then W̃_t = W_t + ∫₀ᵗ θ_s ds is a Brownian motion under Q. Setting θ = (μ−r)/σ removes the equity risk premium — under Q, option prices depend only on observable parameters, not on investors' expected return μ.

“Write the risk-neutral pricing formula for a European call.”

C₀ = e^{−rT} E^Q[max(S_T − K, 0)] = S₀N(d₁) − Ke^{−rT}N(d₂), where d_{1,2} = (ln(S₀/K) + (r ± σ²/2)T) / (σ√T).

“State the martingale representation theorem and give its finance implication.”

Every ℱ^W-martingale M_t can be written as M_t = M₀ + ∫₀ᵗ H_s dW_s for some adapted H. Finance implication: every contingent claim can be replicated by a self-financing dynamic portfolio — the mathematical foundation of delta hedging and market completeness.

“Explain the change of numeraire technique.”

Switching from the risk-neutral measure (numeraire = money market account) to the T-forward measure (numeraire = zero-coupon bond maturing at T) makes the forward price F_t = E^T[S_T | ℱ_t] a martingale. This eliminates the stochastic discount factor, simplifying pricing of swaptions, bond options, and caplets.

“Derive the Black-Scholes PDE via delta hedging.”

Construct portfolio Π = V − ΔS. Apply Itô's lemma to V(t,S) and choose Δ = ∂V/∂S to cancel the dW_t risk term. The remaining riskless portfolio must earn rΠ dt by no-arbitrage.