Options Pricing Interview Questions: Black-Scholes, Greeks & Put-Call Parity

The core question set at every options market maker — what gets asked, the formulas worth memorizing, and one worked example that covers half of them.

Updated July 3, 2026 · QuantVault

Why options market makers test this so hard

If you interview at an options shop — Optiver, SIG, Akuna, or the options desks at Citadel Securities — Greeks questions are not a screen for memorized formulas. They test whether you can reason about how an option's value moves, because that is literally the job: a market maker quotes thousands of options and manages the book's aggregate delta, gamma, and vega all day. Interviewers probe three layers, in roughly this order: no-arbitrage relationships (put-call parity), Black-Scholes intuition (what each input does), and Greeks behavior (signs, shapes, and limits). Almost nobody asks you to derive the Black-Scholes PDE; almost everybody asks what happens to gamma near expiry.

The core toolkit

Put-call parity is the single most-used relation, because it requires no model at all — just no-arbitrage:

$$C - P = S - Ke^{-rT}$$

For a European call and put with the same strike and expiry. A classic opener: "call is trading at 6, put at 4, stock at 100, strike 100, zero rates — what do you do?" Parity says $C-P$ should be 0; it is 2, so you sell the call, buy the put, buy the stock, and lock in \$2. If you cannot spot that in thirty seconds, expect the round to end early.

Then the Black-Scholes call price, $C = S N(d_1) - Ke^{-rT} N(d_2)$, with $d_{1,2} = \frac{\ln(S/K) + (r \pm \sigma^2/2)T}{\sigma\sqrt{T}}$. You rarely compute it exactly in the room, but you must know what each piece means: $N(d_2)$ is the risk-neutral probability the option finishes in the money, and $N(d_1)$ is the delta. The Greeks you need cold:

GreekDefinitionCall (long)What interviewers ask
Delta$\partial C/\partial S = N(d_1)$0 to 1ATM delta ≈ 0.5; why slightly above?
Gamma$\partial^2 C/\partial S^2 = \frac{\varphi(d_1)}{S\sigma\sqrt{T}}$PositiveWhere is it largest? (ATM, near expiry)
Theta$\partial C/\partial t$NegativeWhy does long gamma cost theta?
Vega$\partial C/\partial \sigma = S\varphi(d_1)\sqrt{T}$PositiveWhy do long-dated options have more vega?

One approximation earns its keep in every options interview: the ATM call value with low rates is roughly

$$C_{ATM} \approx 0.4\, S\, \sigma \sqrt{T}$$

which comes from $\varphi(0) \approx 0.4$. It lets you price ATM options mentally, which is exactly the kind of quick-quote skill our market-making game drills.

Worked example: price an ATM call and all its Greeks

Stock at \$100, strike 100, 3 months to expiry ($T = 0.25$), volatility 20%, rates zero. This one setup answers about half the standard Greeks questions.

Price. The approximation gives $C \approx 0.4 \times 100 \times 0.20 \times \sqrt{0.25} = \$4.00$. Exact Black-Scholes: with $S=K$ and $r=0$, $d_1 = \frac{\sigma\sqrt{T}}{2} = 0.05$ and $d_2 = -0.05$, so $C = 100\,(N(0.05) - N(-0.05)) = 100\,(0.520 - 0.480) = \$3.99$. The shortcut is accurate to a penny.

Delta. $\Delta = N(d_1) = N(0.05) \approx 0.52$. This is the answer to "why is ATM call delta slightly above 0.5?" — lognormal drift under the risk-neutral measure pushes $d_1$ above zero by $\sigma\sqrt{T}/2$.

Gamma. $\Gamma = \frac{\varphi(0.05)}{100 \times 0.20 \times 0.5} \approx \frac{0.398}{10} \approx 0.040$. A \$1 stock move changes delta by about 0.04.

Vega. $S\varphi(d_1)\sqrt{T} = 100 \times 0.398 \times 0.5 \approx 19.9$ per unit of vol, i.e. about \$0.20 per vol point. Sanity check against the shortcut: $\partial C/\partial\sigma \approx 0.4\,S\sqrt{T} = 20$. Consistent.

Theta. With $r=0$, $\Theta = -\frac{S\varphi(d_1)\sigma}{2\sqrt{T}} \approx -\frac{100 \times 0.398 \times 0.20}{1} \approx -8$ per year, or roughly 2 cents a day. Follow-up they love: verify the gamma-theta relation $\Theta \approx -\frac{1}{2}\Gamma S^2 \sigma^2 = -\frac{1}{2}(0.04)(10000)(0.04) = -8$. It checks — and it is the heart of why long-gamma positions bleed theta: the daily decay is exactly the rent you pay for convexity.

The traps that actually fail candidates

  • Delta is not the probability of finishing ITM. That is $N(d_2)$, which is smaller. Conflating them is the most common Greeks error we see, and interviewers bait it deliberately.
  • Gamma and theta near expiry. ATM gamma blows up as $T \to 0$ (note the $\sqrt{T}$ in the denominator), and theta with it. An ITM or OTM option's gamma goes to zero instead. Draw the picture before answering.
  • Vega scales with $\sqrt{T}$, so a 1-year option has twice the vega of a 3-month option, not four times. Candidates guess linear surprisingly often.
  • Put-call parity is for European options. With American options or dividends, it becomes an inequality — stating the clean equality for an American put on a dividend payer is a flag.
  • "Option prices are increasing in volatility" needs care. True for vanilla calls and puts (vega > 0), but not for every exotic — if the interviewer pushes, mention that a deeply capped payoff can lose value with more vol.

The probability machinery behind all of this — normal distributions, expectations, and risk-neutral pricing — gets tested alongside it, and stronger candidates connect the dots explicitly (e.g. "delta hedging works because the hedged portfolio is a martingale under the pricing measure"). If you want the continuous-time foundations properly, Shreve volume II is still the reference.

Practice the real thing

Reading about Greeks is not the same as computing them under a clock. Work through our options pricing question bank for graded problems with full solutions, warm up your quoting instincts in the make-me-a-market game, and round out the underlying math with the probability bank — the two topics are almost always interviewed together.

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Frequently asked questions

What Greeks do I need to know for a quant trading interview?

Delta, gamma, theta, and vega cover nearly every question at options market makers, with rho occasionally appearing for rates-sensitive roles. You need signs, definitions, where each Greek peaks (gamma and theta are largest at-the-money near expiry, vega grows with the square root of time), and the gamma-theta trade-off. Formula recall matters less than being able to explain how each Greek behaves as spot, time, and volatility change.

Do interviewers ask you to derive the Black-Scholes formula?

Full derivations are rare in trading interviews; they are more common in quant researcher rounds at firms that emphasize stochastic calculus. Traders are instead tested on Black-Scholes intuition: what N(d1) and N(d2) mean, how the price responds to each input, and quick approximations like the ATM call value of roughly 0.4 times spot times vol times root-T. Knowing the replication argument at a high level is expected either way.

Is delta the same as the probability an option expires in the money?

No, and this is one of the most common trick questions. Delta of a call is N(d1), while the risk-neutral probability of finishing in the money is N(d2), which is strictly smaller. The gap between them grows with volatility and time to expiry, so for long-dated or high-vol options the two numbers can differ substantially.

What is the most common put-call parity interview question?

The standard format gives you a call price, put price, spot, and strike, and asks whether an arbitrage exists. You check C minus P against S minus the discounted strike; any mismatch is captured by trading the conversion or reversal (buy the cheap side, sell the rich side, hedge with stock). Follow-ups usually test whether you know the clean equality only holds for European options without dividends.

Practice the real thing

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