Options Pricing for Quant Interviews: Black-Scholes, Greeks, and Put-Call Parity

Options Pricing Basics for Quant Interviews: Black-Scholes, Greeks, Put-Call Parity

Options pricing comes up in nearly every quant interview at firms that trade options — Optiver, SIG, Citadel Securities, Jane Street, Akuna, IMC, every options-market-making prop shop, and quant researchers at hedge funds dealing with derivatives. You don’t need to derive Black-Scholes from scratch in an interview, but you do need to understand the model, its inputs, the Greeks (especially delta and gamma), put-call parity, and how to reason about option pricing intuitively when prices change. This guide covers what’s actually tested, with the level of mathematical rigor interviewers expect — not more, not less.

Foundations: What an Option Is

A call option gives the holder the right (not obligation) to buy the underlying at strike price K on or before expiration T. A put option gives the right to sell at K. European options can only be exercised at expiration; American options can be exercised any time before expiration.

The payoff at expiration:

• Call: max(S_T – K, 0)
• Put: max(K – S_T, 0)

where S_T is the stock price at expiration. This is the only thing the option is worth at expiry. Before expiration, the option’s price reflects expected future payoff plus time value.

Put-Call Parity

The most important options identity. For European options on a non-dividend-paying stock:

C – P = S – K·e^-rT

Where C = call price, P = put price, S = current stock price, K = strike, r = risk-free rate, T = time to expiry.

Why? Consider two portfolios:

• Portfolio A: long one call, short one put. Payoff at expiry = (S_T – K) regardless of whether S_T > K or S_T < K. This is a synthetic forward.
• Portfolio B: long the stock, short K dollars cash (borrow K, will repay K·e^rT at expiry). Payoff = S_T – K.

Both portfolios have identical payoffs, so by no-arbitrage, they must cost the same today. C – P = S – K·e^-rT.

Why interviewers love this: demonstrates no-arbitrage reasoning, lets you compute either C or P given the other, exposes intuition about synthetic positions.

Common interview question: “Stock at $100, 1-year European call at $105 strike costs $8, risk-free rate 5%. What should the put cost?” Apply formula: P = C – S + K·e^-rT = 8 – 100 + 105·e^-0.05 ≈ 8 – 100 + 99.88 ≈ $7.88.

Black-Scholes Model

The standard pricing model for European options. The closed-form solution for a call:

C = S·N(d₁) – K·e^-rT·N(d₂)

where:

• d₁ = [ln(S/K) + (r + σ²/2)T] / (σ√T)
• d₂ = d₁ – σ√T
• N(·) is the cumulative normal distribution
• σ is the annualized volatility of the underlying

The formula for a put follows from put-call parity.

What you need to know:

• The five inputs: S, K, T, r, σ (and possibly dividends)
• How each input affects the price (higher σ → higher option price; higher T → higher option price; higher S → higher call price, lower put price; etc.)
• Key assumptions: log-normal price distribution, constant volatility, constant risk-free rate, continuous trading, no transaction costs, no dividends
• Major limitations: real markets violate constant-volatility (volatility smile) and log-normality (fat tails)

What you don’t need: deriving the formula from PDE methods or risk-neutral measure changes. Interviewers care that you understand what the model says and what its inputs mean.

The Greeks

The “Greeks” are partial derivatives of option price with respect to inputs. They tell you how the option price changes when something moves.

Delta (Δ)

∂C/∂S, the rate of change of option price with respect to stock price. For a call, delta ranges 0 to 1; for a put, -1 to 0. At-the-money options have delta near 0.5 (call) or -0.5 (put). Deep in-the-money options have delta near 1 (call) or -1 (put). Deep out-of-the-money options have delta near 0.

Interpretation: if delta = 0.4 and the stock moves up $1, the option moves up roughly $0.40. Useful for hedging.

Common interview question: “If you’re long 10 calls with delta 0.5 each, how do you delta-hedge?” Sell 5 shares (10 × 0.5 = 5 delta exposure to neutralize).

Gamma (Γ)

∂²C/∂S², the rate of change of delta with respect to stock price. Highest at-the-money near expiration — delta changes fastest there.

Interpretation: high gamma means delta-hedging needs to be rebalanced frequently as the stock moves. Long-option positions have positive gamma; short-option positions have negative gamma.

Vega (V)

∂C/∂σ, sensitivity to volatility. Highest for at-the-money options with longer time to expiry. Higher implied volatility → higher option prices for both calls and puts.

Interpretation: if vega = 0.20, a 1-percentage-point increase in implied vol increases option price by $0.20.

Theta (Θ)

∂C/∂T (or more often, the negative, as time decreases). The “time decay” of an option. Options lose value as expiration approaches, especially at-the-money options near expiry.

Interpretation: if theta = -0.05, the option loses $0.05 in value per day from time decay alone (holding other inputs constant).

Rho (ρ)

∂C/∂r, sensitivity to interest rates. Generally small for short-dated options; matters more for long-dated.

Common Interview Questions

“Stock at $50, 6-month at-the-money call. Volatility 30%, risk-free 0%. Roughly what’s the call worth?”

Without computing the full Black-Scholes: at-the-money calls trade roughly at 0.4 × S × σ × √T (a useful approximation). Plug in: 0.4 × 50 × 0.30 × √0.5 ≈ 0.4 × 50 × 0.30 × 0.707 ≈ 4.24, so roughly $4–$5. Black-Scholes confirms ≈ $4.20.

“Stock goes up by $1. Option went up by $0.40. What can you infer about delta?”

Delta ≈ 0.40, suggesting the option is somewhere between out-of-the-money and at-the-money.

“What happens to a call’s value if interest rates rise?”

Goes up. The intuition: holding a call is cheaper than holding stock outright; rising rates make the deferred-payment benefit of the call (you don’t pay K until expiry) more valuable. Rho is positive for calls and negative for puts.

“How does volatility affect option prices?”

Higher implied vol → higher prices for both calls and puts. Why? Higher vol means more probability mass in the tails, which means more probability of extreme positive payoffs for both options. Even though there’s also more probability of extreme negative payoffs, the option’s downside is capped at zero (you don’t exercise if it’s not profitable), so the asymmetric payoff means you benefit from upside without symmetric downside.

“You’re long a straddle. What’s your risk exposure?”

A straddle = long call + long put at same strike. Net delta near zero (calls and puts offset), so directionally neutral. But long gamma (you benefit from large moves either direction), long vega (you benefit from rising vol), short theta (you bleed time value). You make money if the stock moves a lot or implied vol rises; you lose money if the stock stays still and vol falls.

“Why are equity put options often more expensive than calls of equivalent moneyness?”

Volatility skew. Markets typically price down-side puts at higher implied vols than upside calls because crashes are more violent and faster than rallies (the “leverage effect” where falling stock prices increase realized volatility). This is a market reality, not a Black-Scholes prediction (BS assumes constant vol).

Market-Making Frame

For options market makers, every option has a fair value (computed from a model + market-implied vols) and a market price. If the market price is below fair value, you buy. If above, you sell. The Greeks describe your risk exposure after the trade. Most market-making profits come from collecting the bid-offer spread on flow, not from being right about direction; the Greeks framework lets you stay directionally hedged while collecting spread.

Common interview frame: “I’m a customer; I want to buy 100 calls at strike 105 expiring in 30 days. Stock is at 100, vol is 25%. What’s your offer?” Strong candidates: compute Black-Scholes value, add a market-maker’s spread (widening for higher uncertainty, lower liquidity, harder-to-hedge positions), articulate hedging plan after the trade.

Interview Tactical Tips

• Know put-call parity cold. Re-derivable in 30 seconds; lets you sanity-check almost any option-pricing question.
• Approximation over computation. The 0.4·S·σ·√T approximation for at-the-money calls is enough for most interview reasoning.
• Talk through Greeks visually. “Delta is the slope of the option price vs stock price line. Gamma is the curvature. Theta is time pulling the curve down.”
• Don’t memorize the full Black-Scholes formula. Memorizing C = S·N(d₁) – K·e^-rT·N(d₂) is fine but you won’t need to compute N(d₁) by hand. Knowing the structure is enough.
• For deeper rounds: understand risk-neutral measure intuitively (option prices are expectations of payoff under a measure where stock grows at risk-free rate, discounted at risk-free rate), but you don’t need to do the measure-change math live.

Practice Strategy

Weeks 1–2: work through put-call parity examples, Greeks intuition, volatility-pricing intuition. Hull’s Options, Futures, and Other Derivatives chapters 1–6 (the introductory derivatives sections) cover everything you need at interview depth.

Weeks 3–4: options-specific brainteasers from Joshi’s Quant Job Interview Questions and Answers. Mock 5–10 problems explaining your reasoning out loud.

Weeks 5+: if targeting options market makers (Optiver, SIG, Akuna, IMC), spend time on more nuanced topics: volatility surfaces, dividend handling, American option early-exercise considerations.

Frequently Asked Questions

See also: Breaking Into Quant Finance and Wall Street: 2026 Guide • Probability Brainteasers for Quant Interviews • Jane Street Interview Guide