Stochastic Calculus Quant Interview Cheat Sheet: Key Formulas Template

Stochastic Calculus Quant Interview Cheat Sheet: Key Formulas Template

TL;DR

The interviewer expects you to recite a concise, correct set of stochastic calculus formulas, demonstrate when each applies, and expose hidden assumptions. Anything less signals superficial preparation. Memorize the Itô‑Taylor expansions, Girsanov’s theorem, and the Black‑Scholes PDE, then rehearse the decision‑tree for “when do I need a martingale argument?” The cheat sheet must be a one‑page PDF that you can annotate in five minutes before any interview.

Who This Is For

You are a senior mathematics or computer‑science graduate with two to three years of market‑data engineering experience, aiming for a quantitative analyst role at a hedge fund or prop‑trading shop. Your current CTC is $165 k base + 20 % bonus, and you have survived the coding screen but stalled on the finance round. You need a formula‑driven weapon that translates theory into the language of the hiring committee in under a minute.

What stochastic calculus formulas must I have at my fingertips for a quant interview?

The answer is a short list of five families: Itô’s lemma (first and second order), Itô‑Taylor expansions up to order 1.5, Girsanov’s change‑of‑measure, the Feynman‑Kac representation, and the Black‑Scholes PDE. In a recent third‑round interview at a top‑tier hedge fund, the hiring manager challenged a candidate on the drift‑adjustment term of Girsanov. The candidate stumbled because he treated the drift as optional, not as a necessary signal of the equivalent martingale measure. The lesson is that the formulas are not decorative; they are decision triggers. The first counter‑intuitive truth is that the “most obvious” formula—Itô’s lemma—is rarely the answer; the interviewers look for the right lemma, not the most famous one. When the prompt mentions a “log‑normal asset”, you must instantly reach for the Black‑Scholes PDE, not the generic Itô formula.

How do I recognize when to apply Girsanov’s theorem versus a simple martingale argument?

The answer: if the question asks you to price under a risk‑neutral measure, you must invoke Girsanov; if it merely asks for an expectation under the physical measure, a plain martingale argument suffices. In a debrief after the interview, the hiring committee noted that the candidate incorrectly swapped the two, saying “the discounted price is a martingale under the physical measure”. The panel’s judgment was that the candidate displayed a textbook‑level understanding but lacked practical discrimination. The second counter‑intuitive truth is that the problem is not “knowing the theorem” — it is “knowing when the theorem is the only tool that closes the loop.” The decision tree is: does the pay‑off depend on the measure? If yes → Girsanov; if no → martingale.

When should I write out the Itô‑Taylor expansion instead of using Itô’s lemma directly?

The answer: use the Itô‑Taylor series whenever the interview question involves a multi‑step discretisation or a higher‑order scheme, such as a Milstein method. In a recent interview at a proprietary trading desk, the candidate was asked to derive the strong order‑1.0 scheme for a stochastic differential equation with state‑dependent volatility. The candidate defaulted to Itô’s lemma, which only yields a first‑order expansion, and the interviewers marked the response as incomplete. The third counter‑intuitive truth is that the problem is not “the formula is complex” — it is “the formula is the minimal representation of the discretisation you need”. Remember: a single‑step Euler scheme is insufficient when the diffusion term is non‑linear; you must present the full Itô‑Taylor terms up to the mixed \(dW\,dt\) component.

Why does the Black‑Scholes PDE appear so often even when the interview problem is framed in terms of options on commodities?

The answer: the PDE emerges whenever you have a tradable underlying with geometric Brownian motion, regardless of the asset class. In a debrief from a commodities quant interview, the hiring manager pushed back when the candidate tried to treat the commodity forward price as a deterministic function. The panel’s judgment was that the candidate ignored the stochastic drift term, a classic “not deterministic, but stochastic” misstep. The interview is testing whether you can abstract the underlying’s dynamics into a risk‑neutral diffusion; if you can, the Black‑Scholes PDE follows automatically. The fourth counter‑intuitive truth is that the problem is not “the asset class matters” — it is “the diffusion structure matters”. Use the PDE as a template, then adjust boundary conditions for storage costs or convenience yields.

How can I structure my cheat sheet so that I retrieve the right formula under pressure?

The answer: partition the sheet into three columns—(1) formula name, (2) canonical statement, (3) “when to use” cue. In a four‑hour interview day at a quant shop, a candidate flipped through a dense textbook and lost 15 minutes before the interview started. The hiring committee recorded the delay as a red flag for time‑management. The fifth counter‑intuitive truth is that the problem is not “having more formulas”—it is “having a retrieval map”. The cue column should read “risk‑neutral pricing → Black‑Scholes PDE”, “measure change → Girsanov”, “path‑dependent payoff → Feynman‑Kac”. Keep the sheet to one side of an A4 page; annotate with a single example per formula for instant recall.

Preparation Checklist

• Review each formula’s derivation once to confirm you understand the assumptions.
• Write a one‑sentence “when to use” cue beside every formula on the cheat sheet.
• Practice translating a random interview prompt into the cue column within 30 seconds.
• Simulate a full interview loop: 45‑minute technical, 30‑minute case, 15‑minute rapid‑fire; time each response.
• Work through a structured preparation system (the PM Interview Playbook covers stochastic‑calculus interview frameworks with real debrief examples).
• Memorize the exact coefficient values for the Black‑Scholes PDE (σ²/2 term, r‑q drift) to avoid sign errors.
• Keep a printable PDF of the cheat sheet on your laptop and a laminated copy on your desk.

Mistakes to Avoid

BAD: Memorizing formulas without linking them to decision cues. GOOD: Pair each formula with a concrete “when” scenario, such as “Girsanov → risk‑neutral pricing”. The interview panel penalizes pure memorisation because it cannot differentiate depth from breadth.

BAD: Using a generic Itô lemma when the problem demands an Itô‑Taylor expansion. GOOD: Identify the discretisation order required and write the mixed‑term explicitly. The hiring manager’s notes often highlight “incorrect expansion level” as a decisive flaw.

BAD: Treating deterministic drift as a placeholder and ignoring the need for a measure change. GOOD: Explicitly state the need for Girsanov when the question shifts from physical to risk‑neutral expectation. The committee’s debriefs repeatedly flag “missing measure‑change justification” as a deal‑breaker.

FAQ

What if I forget a formula during the interview? The judgment is to admit the gap, reference the related cue, and walk the interviewer through the derivation. Concealing ignorance is a bigger red flag than a brief pause.

How many interview rounds typically involve stochastic calculus? Most quant hiring pipelines contain three technical rounds over a 12‑day window, each lasting 45 minutes, plus a final case study. Expect stochastic calculus in at least two of those rounds.

Should I bring a handwritten cheat sheet into the interview room? No, bring a printed one‑page PDF; the interviewers will view a handwritten sheet as unprofessional and a sign of insufficient preparation.amazon.com/dp/B0GWWJQ2S3).