Picture the whiteboard question a trader slides across the table at Optiver or Akuna: you’re long a 100-strike call, the stock sits at 100, and there’s a week to expiry. “I lift the stock to 101. What happens to your P&L, and what’s your delta now?” The interviewer isn’t checking whether you memorized Black-Scholes. They want to watch you reason about a position the way someone who has to hedge it in real time does.
The options rounds at trading firms reward a specific skill: turning a price into its sensitivities, then reasoning about how those sensitivities move as the market moves. Get comfortable with the four main Greeks and one identity and you can talk your way through most of what they throw at you.
Put-call parity is the one identity you cannot fumble
If there’s a single equation the interviewer expects instantly, it’s put-call parity. For a European call and put on the same underlying, same strike K, same expiry T:
C - P = S - K * e^(-rT)Read it as a statement about replication, not algebra. Long a call and short a put at the same strike gives you a synthetic forward: you buy the stock at K no matter what, so the payoff is S_T – K. Discount that and you get the right side. Interviewers love this because it lets them test whether you can spot an arbitrage. Suppose the call trades at 6, the put at 2, stock at 100, strike 100, rates near zero. Parity says C – P should sit near 0, but the market quotes 4. So the call is rich or the put is cheap. You sell the call, buy the put, and buy the stock, and you’ve boxed in a riskless profit. Walk that loop out loud and you’ve answered a question they might have budgeted five minutes for.
Parity also hands you a shortcut people miss. Because C – P is pinned, the delta of a call minus the delta of a put equals 1. An at-the-money call runs a delta near 0.5, which forces the at-the-money put to a delta near -0.5. And vega is identical for the call and put at the same strike and expiry, since the S – K * e^(-rT) term carries no volatility. Interviewers will ask “is a put’s vega positive or negative?” hoping you reason from parity instead of guessing.
Delta is a hedge ratio before it’s a derivative
Every candidate can recite that delta is the change in option price per unit change in the underlying. The ones who get the offer treat it as the number of shares they’d short to neutralize direction. Long one call with delta 0.4? Short 40 shares per contract and you’re flat, at least for a small move. The moment the stock moves, though, your delta changes, and that second-order effect is where the real conversation goes.
Expect the follow-up: “The stock rallies. Does your call’s delta go up or down?” Up, toward 1, because the option is drifting into the money. “Now you’re delta-hedged and short the shares. Are you happy the stock moved?” That’s the gamma question wearing a disguise.
Gamma is why market makers care about big moves
Gamma measures how fast delta changes. Long options carry positive gamma, which means your delta grows in your favor: as the stock rallies your long call gets longer, as it falls the call gets shorter, so you’re always leaning the right way. Being long gamma and delta-hedged is a comfortable place to sit when the stock whips around, because you rebalance by buying low and selling high. That rebalancing profit is what people mean by gamma scalping.
The catch is theta, and interviewers will make you feel it. Long gamma costs you time decay every day the stock sits still. Short gamma is the mirror image: you collect premium, but a violent move can run your losses faster than you can hedge. Gamma peaks for at-the-money options close to expiry, which is exactly when a market maker’s book is most dangerous. A common prompt is “you’re short a straddle that expires tomorrow and the stock is pinned at the strike. How do you feel?” The answer is that you’re nervous, because your gamma is enormous and a gap either way hurts.
Vega and the volatility conversation
Vega is sensitivity to implied volatility, and it’s where options interviews separate people who grasp that an option is a bet on movement, not direction. Higher implied vol means a wider distribution of outcomes, which makes both calls and puts worth more. Vega is largest for at-the-money options with plenty of time left, since those have the most room for volatility to matter. A one-week option has little vega; a one-year at-the-money option has plenty.
Good interviewers push into the vol surface. “Why does a 25-delta put trade at a higher implied vol than the at-the-money?” Because equity markets crash down more violently than they melt up, so out-of-the-money puts carry a skew premium. You don’t need to derive the smile, but you should know it exists and roughly why. Claim implied vol is flat across strikes and you’ve told them you’ve never looked at a real options screen.
Theta, and the tradeoff nobody escapes
Theta is the daily bleed. Long an option, you pay theta; short it, you collect. The clean way to frame it in an interview is as the price of gamma. You’re renting convexity, and theta is the rent. When someone asks “why would anyone sell options if the market can gap against them?” the answer is that they’re paid to, every day, and most days the gap doesn’t come. That’s the whole business of a lot of vol desks: sizing that trade so the theta they collect outruns the gamma losses they eat.
A question that looks like math and is really about reasoning
Here’s one that SIG and Jane Street style desks like: “A stock is at 100. A one-year call struck at 100 costs 10. Rates are zero, no dividends. Roughly what implied vol does that imply?” You’re not expected to invert Black-Scholes on a whiteboard. Use the at-the-money approximation a lot of traders keep in their head:
ATM call price ~= 0.4 * S * sigma * sqrt(T)Plug in: 10 ~= 0.4 * 100 * sigma * 1, so sigma ~= 0.25, about 25% vol. The interviewer watches whether you know that 0.4 * S * sigma * sqrt(T) shortcut and whether you can rearrange it under mild pressure. Miss the formula and you can still reason that a 10% premium on an at-the-money one-year option is a fairly ordinary vol, not 5% and not 80%.
The five sensitivities an options interviewer expects you to sign correctly, for a long position in a single option:
| Greek | Measures sensitivity to | Sign, long call | Sign, long put | Largest when |
|---|---|---|---|---|
| Delta | Underlying price (first derivative) | 0 to +1 | -1 to 0 | Deep in the money, approaching plus or minus 1 |
| Gamma | Rate of change of delta (curvature) | Positive | Positive | At the money, near expiry |
| Vega | Implied volatility | Positive | Positive | At the money, long time to expiry |
| Theta | Passage of time (per day) | Negative, you pay | Negative, you pay | At the money, near expiry |
| Rho | Interest rates | Positive | Negative | Long-dated options |
How the options round fits the rest of the loop
At the pure trading firms the options theory conversation is one piece of a day built around speed and probability. A typical new-grad trader loop runs a timed mental-math test (Optiver’s 80-questions-in-8-minutes drill, Akuna’s arithmetic sprint), a probability and expected-value round with dice and coin problems, a market-making game where you quote a bid and ask on some invented asset and manage the position as the interviewer trades against you, and then the options round where the Greeks show up. Engineering candidates at the same firms get more systems and low-latency questions and lighter options theory, though knowing parity and delta still helps you talk to the traders you’d support.
The market-making game is where options intuition quietly pays off even when nobody says the word “option.” When the interviewer keeps hitting your bid, your inventory grows and your risk with it, and the right move is to shade your quotes to get flat. Same instinct as hedging a delta. Firms like SIG lean hard on this, and they’re candid that they care more about how you update on new information than whether you nailed the arithmetic.
A few prompts phrased close to how they actually land:
- “You’re long a call and short a put, same strike and expiry. What position is that, and what’s its delta?”
- “Vol just doubled. What happens to a deep out-of-the-money option’s price in percentage terms versus an at-the-money one?”
- “You have a book that’s long gamma and short vega. What market do you want?”
- “Price me a bet that pays $1 if the stock is above 100 at expiry. What’s that worth relative to a regular call?”
That last one trips people up, and it’s a good note to leave on. A digital that pays $1 above the strike is close to the derivative of a call’s payoff with respect to strike, which ties its price to the slope of the call curve and, through that, to the skew. You don’t have to produce that on the spot. The candidates who get called back are the ones who, instead of freezing, start reasoning from what they do know: a digital is like a very tight call spread, a call spread’s value comes from the difference of two option prices, and prices come from the distribution. Reach for the position and the payoff, not the memorized formula, and the Greeks stop being trivia and start being the language the desk actually speaks.
