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]?

E[W_t²] = t

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 ⟹ (dW_t)² = dt

[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.

dM_t = 2W_t dW_t + dt − dt = 2W_t dW_t

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.

P(max W_s ≥ a) = 2(1 − Φ(a/√t))

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}).

d(e^{W_t}) = e^{W_t} dW_t + ½ e^{W_t} dt

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).

d(W_t³) = 3W_t² dW_t + 3W_t dt

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.

S_t = S₀ exp[(μ − σ²/2)t + σW_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).

∫₀ᵀ W_t eᵗ dt = W_T e^T − ∫₀ᵀ eᵗ dW_t

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.

E[(∫₀ᵀ H_t dW_t)²] = E[∫₀ᵀ H_t² dt]

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.

X_t = θ + (X₀ − θ)e^{−κt} + σ∫₀ᵗ e^{−κ(t−s)} dW_s

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

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

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.

u(t,x) = E^{t,x}[e^{−r(T−t)} g(X_T)]

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.

W̃_t = W_t + ∫₀ᵗ θ_s ds is BM under Q

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 = S₀N(d₁) − Ke^{−rT}N(d₂)

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.

M_t = M₀ + ∫₀ᵗ H_s dW_s

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.

∂V/∂t + ½σ²S²·∂²V/∂S² + rS·∂V/∂S − rV = 0

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.