Not every quant interview covers derivatives — but many do, especially at banks, market makers, and any team touching risk. The good news is that the tested material is a small, stable set of concepts that reward understanding over memorization.
Black-Scholes: know the assumptions, not just the formula
Interviewers rarely ask you to reproduce the formula. They ask what has to be true for it to hold: constant volatility, continuous hedging, no transaction costs,European exercise, no arbitrage, and a lognormal underlying with constant rates. Then they ask what breaks when each assumption fails — and the interesting discussion is always about volatility, which is not constant and not known.
Put-call parity and no-arbitrage thinking
Put-call parity is the single most useful identity in interviews: it turns "price this option" questions into "build a replicating portfolio" questions. Practice reasoning no-arbitrage first and formulas second — most pricing questions reduce to constructing a position that pays the same thing in every state of the world.
The Greeks, conceptually
- Delta: hedge ratio. How much of the underlying do you hold per option. Be able to explain delta-hedging out loud.
- Gamma: how fast delta moves. Long gamma earns a profit from price movement; short gamma bleeds. This is the intuition behind most vol trading discussions.
- Theta: time decay. Connect it to the gamma-theta tradeoff: option P&L comes from realized versus implied volatility.
- Vega: sensitivity to volatility. Note that it is not a Greek letter — interviewers sometimes notice whether you know that.
Volatility: the smile and why it exists
The volatility smile is the observation that implied vol varies by strike and maturity, contradicting Black-Scholes's constant-vol assumption. Be ready to discuss why: crash risk and negative correlation between spot and vol in equities, jumps in returns, and supply-and-demand for downside protection. Understand that this means options with the same underlying are priced on a surface, not a number.
Question shapes to expect
- "Delta-hedge a long call position. What are you exposed to?" — gamma and, second-order, vol moves.
- "Why is an option's expected payoff not simply its fair value?" — talks directly at risk neutrality versus real-world probability.
- "How does implied vol change when a stock crashes?" — skew dynamics, and volatility clustering.
- "Price a simple exotic: barrier, Asian, or lookback" — usually wants the payoff structure and hedging intuition rather than closed-form math.
Preparation that pays off
- Derive Black-Scholes once by hand, or at least follow the derivation line by line and reconstruct the intuition — the hedge portfolio grows at the risk-free rate.
- Run one backtest: delta-hedge a simulated option position and plot the P&L distribution against realized versus implied volatility. Nothing teaches the Greeks faster.
- Learn the vocabulary: contango and backwardation, forward versus spot, funding, basis — the surrounding market language that separates candidates
If you can draw the payoff diagram, explain who holds the risk at every point, and name the assumption that breaks under stress, you have covered the majority of what these interviews test.