Stochastic calculus and partial differential equations (PDEs) are not just academic tools—they’re the backbone of derivatives pricing, risk management, and algorithmic trading. If you’re interviewing for quant research roles, mastering these concepts (and the typical stochastic calculus quant interview questions) is non-negotiable. This comprehensive guide covers the must-know questions, concepts, derivations, and numerical techniques you’re almost guaranteed to encounter. Whether you’re prepping for bulge-bracket banks or boutique quant funds, understanding the intuition and applications behind the math will put you ahead of the pack.

Must-Know Stochastic Calculus & PDE Questions for Quant Research Roles

Why Stochastic Calculus Is the Language of Derivatives Pricing

Stochastic calculus is to quant finance what grammar is to language. It’s the foundation of pricing derivatives, modeling market risks, and devising alpha-generating strategies. What separates junior from senior quant candidates is not rote memorization of formulas, but a deep, intuitive grasp of how randomness, risk-neutral valuation, and PDEs come together in financial markets.

Typical Quant Interview Expectation

• Derive and manipulate SDEs for asset prices.
• Apply Ito’s Lemma in various contexts (including multidimensional cases).
• Connect SDEs to PDEs via the Feynman-Kac theorem.
• Justify modeling assumptions and numerical schemes.

1. Brownian Motion & Ito Calculus Fundamentals

What Is a Wiener Process (Brownian Motion)?

A Wiener process (or standard Brownian motion) is the fundamental building block of stochastic calculus and almost every quant interview begins here.

• Definition: \((W_t)_{t\geq0}\) is a Wiener process if:
  • \(W_0 = 0\) almost surely.
  • For \(0 \leq s < t\), \(W_t - W_s \sim \mathcal{N}(0, t-s)\) (independent, stationary increments).
  • Paths are almost surely continuous (but nowhere differentiable).

Key Properties to Remember

• Quadratic variation: \(\langle W \rangle_t = t\)
• No drift: \(\mathbb{E}[W_t] = 0\)
• Variance grows linearly: \(\text{Var}(W_t) = t\)

Ito’s Lemma: The Workhorse of Stochastic Calculus

Every quant interview will ask you to apply Ito’s Lemma—it’s the “product rule” for stochastic calculus, accounting for the randomness in Brownian motion.

Question: Apply Ito’s Lemma to find \(dF\) for \(F = f(t, W_t)\) and \(F = f(t, S_t)\).

• For \(F = f(t, W_t)\), where \(f\) is twice differentiable,
  \[ df(t, W_t) = \frac{\partial f}{\partial t} dt + \frac{\partial f}{\partial W_t} dW_t + \frac{1}{2} \frac{\partial^2 f}{\partial W_t^2} (dW_t)^2 \] But in Ito calculus, \((dW_t)^2 = dt\), so:
  \[ df(t, W_t) = \frac{\partial f}{\partial t} dt + \frac{\partial f}{\partial W_t} dW_t + \frac{1}{2} \frac{\partial^2 f}{\partial W_t^2} dt \]
• For \(F = f(t, S_t)\), where \(dS_t = \mu S_t dt + \sigma S_t dW_t\),
  \[ df(t, S_t) = \frac{\partial f}{\partial t} dt + \frac{\partial f}{\partial S_t} dS_t + \frac{1}{2} \frac{\partial^2 f}{\partial S_t^2} (dS_t)^2 \] Substitute in \(dS_t\), and recall \((dW_t)^2 = dt\), \(dt^2 = dW_t dt = 0\):
  \[ (dS_t)^2 = (\sigma S_t dW_t)^2 = \sigma^2 S_t^2 (dW_t)^2 = \sigma^2 S_t^2 dt \] Thus, \[ df = \frac{\partial f}{\partial t} dt + \frac{\partial f}{\partial S_t} (\mu S_t dt + \sigma S_t dW_t) + \frac{1}{2} \frac{\partial^2 f}{\partial S_t^2} \sigma^2 S_t^2 dt \] Or, collecting terms: \[ df = \left(\frac{\partial f}{\partial t} + \mu S_t \frac{\partial f}{\partial S_t} + \frac{1}{2} \sigma^2 S_t^2 \frac{\partial^2 f}{\partial S_t^2}\right)dt + \sigma S_t \frac{\partial f}{\partial S_t} dW_t \]

What’s the Difference: \(dW_t\) vs. \((dW_t)^2\)?

• \(dW_t\): Infinitesimal increment of Brownian motion. Has mean zero, variance \(dt\).
• \((dW_t)^2\): In Ito calculus, \((dW_t)^2 = dt\). This is a key rule—it’s why the second derivative term appears in Ito’s Lemma.

Interview tip: If you can explain why \((dW_t)^2 = dt\) (from the quadratic variation), you’ll impress your interviewer.

2. Essential Stochastic Differential Equations (SDEs)

Quant researchers must be comfortable solving and interpreting core SDEs. Expect to see these models and their properties in interviews.

Arithmetic Brownian Motion (ABM)

The simplest SDE: \[ dX_t = \mu dt + \sigma dW_t \] where \(\mu\) is drift and \(\sigma\) is volatility.

• Solution: \[ X_t = X_0 + \mu t + \sigma W_t \] It can go negative—unlike stock prices!

Geometric Brownian Motion (GBM)

The canonical model for asset prices: \[ dS_t = \mu S_t dt + \sigma S_t dW_t \]

• Solution: (via Ito’s Lemma) \[ S_t = S_0 \exp\left((\mu - \frac{1}{2}\sigma^2)t + \sigma W_t\right) \] Ensures \(S_t > 0\) (no negative prices).

Ornstein-Uhlenbeck Process

A mean-reverting SDE, often used for interest rates or spreads: \[ dX_t = \theta(\mu - X_t) dt + \sigma dW_t \] where \(\theta\) is the mean-reversion speed.

• Solution: \[ X_t = X_0 e^{-\theta t} + \mu(1 - e^{-\theta t}) + \sigma \int_0^t e^{-\theta(t-s)} dW_s \]

Classic Interview Question: Log-Price SDE for GBM

Question: What SDE does the log price follow if the price follows GBM? Why is this useful?

• If \(S_t\) follows GBM: \[ dS_t = \mu S_t dt + \sigma S_t dW_t \] Apply Ito’s Lemma to \(X_t = \log S_t\):
  \[ dX_t = d(\log S_t) = \frac{1}{S_t} dS_t - \frac{1}{2} \frac{1}{S_t^2} (dS_t)^2 \] Recall \((dS_t)^2 = \sigma^2 S_t^2 dt\), so: \[ dX_t = \frac{1}{S_t}(\mu S_t dt + \sigma S_t dW_t) - \frac{1}{2} \sigma^2 dt = (\mu - \frac{1}{2}\sigma^2)dt + \sigma dW_t \] Why is this useful? It shows log-returns are normally distributed and simplifies option pricing and risk calculations.

Martingale Representation Theorem

• What is a martingale?
  A process \(M_t\) such that \(\mathbb{E}[M_t | \mathcal{F}_s] = M_s\) for \(t \geq s\), i.e., its expectation doesn’t change given the past. No predictable drift.
• Why is the discounted asset price a martingale under risk-neutral measure?
  Under risk-neutral measure \(\mathbb{Q}\), if \(S_t\) is the stock price and \(r\) is the risk-free rate, then \[ e^{-rt} S_t \] is a martingale. This is fundamental to no-arbitrage pricing, since the expected discounted payoff equals the current price.

3. The Bridge to PDEs: Feynman-Kac Theorem

The Feynman-Kac Theorem is the magical bridge between stochastic processes (SDEs) and PDEs. It allows us to express the value of derivatives as solutions to PDEs, and vice versa.

Feynman-Kac Theorem (Intuitive Statement)

If \(u(t, x)\) is the solution to a certain PDE, it can be represented as the expected value of a function of a stochastic process starting at \(x\) at time \(t\). Formally:

\[ u(t, x) = \mathbb{E}^{\mathbb{Q}}\left[\left. e^{-\int_t^T r(s) ds} g(X_T) \right| X_t = x \right] \] where \(X_t\) solves an SDE, and \(u(t, x)\) solves a backward parabolic PDE with terminal condition \(g(x)\).

Classic Interview Question: Deriving the Black-Scholes PDE via Feynman-Kac

Question: Using Feynman-Kac, derive the Black-Scholes PDE for a European call option.

• The risk-neutral SDE for the stock is: \[ dS_t = rS_t dt + \sigma S_t dW_t^{\mathbb{Q}} \] The discounted price of the option is: \[ V(t, S_t) = \mathbb{E}^{\mathbb{Q}}\left[e^{-r(T-t)} \text{Payoff}(S_T) \mid S_t\right] \]
• By Feynman-Kac, \(V(t, S)\) solves: \[ \frac{\partial V}{\partial t} + rS \frac{\partial V}{\partial S} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} - rV = 0 \] with terminal condition \(V(T, S) = \max(S - K, 0)\).

Interview tip: Be prepared to walk through each step, and explain why risk-neutrality means we replace drift \(\mu\) with \(r\) in the SDE. Understanding the economic intuition behind the PDE is just as important as the math.

4. Numerical Methods & "So What?" Questions

Stochastic calculus and PDEs aren’t just theory for theory’s sake—they underpin practical numerical methods used day-in, day-out by quants. Interviewers often ask about simulation, approximation, and pricing methods to gauge your applied understanding.

How Would You Simulate These SDEs?

To implement SDEs on a computer, we discretize them. Two of the most common schemes are Euler-Maruyama and Milstein.

• Euler-Maruyama Scheme:
  For SDE: \(dX_t = a(X_t, t) dt + b(X_t, t) dW_t\), discretize as:
  \[ X_{t+\Delta t} = X_t + a(X_t, t)\Delta t + b(X_t, t)\sqrt{\Delta t}\ \varepsilon \] where \(\varepsilon \sim \mathcal{N}(0,1)\).
• Milstein Scheme:
  Improves upon Euler by adding a correction term for the stochastic integral:
  \[ X_{t+\Delta t} = X_t + a(X_t, t)\Delta t + b(X_t, t)\sqrt{\Delta t}\ \varepsilon + \frac{1}{2}b(X_t, t) b_x(X_t, t) \left[ (\varepsilon^2 - 1)\Delta t \right] \] where \(b_x(X_t, t)\) is the derivative of \(b\) with respect to \(X_t\).

import numpy as np

def euler_maruyama(mu, sigma, S0, T, N):
    dt = T / N
    S = np.zeros(N+1)
    S[0] = S0
    for i in range(1, N+1):
        dW = np.sqrt(dt) * np.random.randn()
        S[i] = S[i-1] + mu * S[i-1] * dt + sigma * S[i-1] * dW
    return S

When to use which? Euler is simpler, but Milstein reduces error for SDEs with non-constant diffusion terms.

Monte Carlo & American Option Pricing: The "So What?" Question

Question: When pricing an American option with Monte Carlo, why can’t we just use the SDE? What methods do we need?

• Key issue: American options can be exercised at any time before expiry. The optimal stopping problem can’t be solved by simply simulating the SDE and discounting the payoff at maturity.
• Solution: Need to estimate the continuation value at each time step to decide whether early exercise is optimal. This requires regression or backward induction.
• Longstaff-Schwartz Method: The most famous Monte Carlo method for American options. It uses least-squares regression on simulated paths to estimate the expected payoff (“continuation value”) at each exercise date.
  • Simulate many paths for the underlying (using Euler or Milstein).
  • At each timestep, use regression to estimate the continuation value as a function of state variables.
  • If the immediate exercise value exceeds continuation value, choose to exercise; otherwise, continue.

# Simplified Longstaff-Schwartz (conceptual)
for t in reversed(time_grid):
    # For all paths where option is alive:
    # 1. Compute immediate exercise value
    # 2. Regress realized payoffs (discounted) on features (e.g., stock price)
    # 3. Exercise if immediate value > estimated continuation value

Interview tip: Know why vanilla Monte Carlo fails for American options (can’t model early exercise), and be able to sketch the Longstaff-Schwartz approach.

Conclusion: Study Advice and Core References

Learning stochastic calculus and PDEs for quant interviews is about more than memorizing formulas. It’s about understanding how randomness, risk, and pricing are deeply connected in modern markets. Focus on building intuition: know the stories behind Ito’s Lemma, the risk-neutral measure, and the Feynman-Kac theorem. Practice deriving core results, but also think about how you’d implement or extend them in code.

Recommended Core Texts

• Joshi, Mark S. – “The Concepts and Practice of Mathematical Finance”
  Excellent for intuition, practical examples, and interview prep.
• Baxter, M. & Rennie, A. – “Financial Calculus: An Introduction to Derivative Pricing”
  The classic, concise intro to stochastic calculus in finance.
• Shreve, S. – “Stochastic Calculus for Finance I & II”
  For a more thorough, mathematical treatment.

To ace stochastic calculus quant interview questions, always relate the math back to market intuition and practical applications, not just technical proofs. Use this as your checklist, and you’ll be confident in any quant research interview.

Summary Table: Must-Know Interview Topics

Good luck with your quant research interviews—and remember, the math tells a story. Make sure you can tell it!

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