Black-Scholes interview questions — what they are really asking

Black-Scholes questions sort candidates into two groups very quickly: those who treat the formula as a black box that outputs a price, and those who understand it as a replication argument. The second group gets the offer.

The one sentence worth memorising

The model does not predict where the stock is going. It says: if I can continuously trade the underlying and a risk-free bond, I can build a portfolio that exactly reproduces the option payoff. The option must therefore cost what that replication strategy costs, or someone earns a riskless profit.

Everything else follows from that. It is also why the expected return of the stock does not appear in the formula — a fact interviewers love to probe. If your hedge kills the directional exposure, your view on direction cannot affect the price.

What each input actually does

• Spot and strike — together they set moneyness. Only their relationship matters, not their absolute level.
• Volatility — the width of the distribution of outcomes. More width means more value for both calls and puts, because the payoff is one-sided and the downside is already capped at the premium.
• Time — works mostly through volatility. What matters is the total uncertainty between now and expiry, which scales with the square root of time, not with time itself.
• Rates and carry — these move the forward, and options are priced off the forward, not off spot. Dividends, repo and funding all enter here. Getting this right is what separates a textbook answer from a desk answer.

Implied volatility: say this, not that

Implied volatility is simply the volatility number you must put into Black-Scholes to make the model price match the price quoted in the market. That is the whole definition. It is a quoting convention — a way of expressing a price in units everyone can compare across strikes and maturities.

Do notdescribe it as the market's forecast of future volatility. If it were a single forecast, every strike on the same underlying and maturity would imply the same number. They do not: that is the volatility smile, and it is the standing counter-example. Candidates who call implied vol a forecast get walked straight into that contradiction.

The assumptions they will attack

Expect to be pushed on where the model fails. The honest answer is that every assumption is wrong in a useful way:

• Constant volatility. Plainly false — hence the smile, and hence local and stochastic volatility models.
• Continuous trading, no transaction costs. You cannot rehedge infinitely often, and each rehedge costs money. Real desks trade off hedging error against cost.
• No jumps. Prices gap on earnings and on news. A gap is precisely when a delta hedge fails, which is why short-gamma books lose badly on those days.

The strong close: the market knows all of this and uses the model anyway, because it is a consistent language for quoting and hedging, not a claim about reality. The smile is the market's way of patching the assumptions back in.