Black-Scholes is the foundation of options pricing. Every quant trading firm expects you to understand it deeply — not just plug numbers in. This guide covers the formula, what each piece means, the intuition behind it, its assumptions, and the exact questions interviewers ask.

The Formula

European Call Option Price

C = S · N(d₁) − K · e^(−rT) · N(d₂)

d₁ = [ ln(S/K) + (r + σ²/2) · T ] / (σ · √T)

d₂ = d₁ − σ · √T

For a European put: P = K · e^(−rT) · N(−d₂) − S · N(−d₁). You can also derive the put price directly from the call price using put-call parity: P = C − S + K · e^(−rT).

What Each Variable Means

Symbol	Meaning	Typical interview value
S	Current stock price (spot)	$100
K	Strike price of the option	$100 (at-the-money)
r	Continuously compounded risk-free interest rate	5% per year
T	Time to expiry in years	1 year, or 0.25 (3 months)
σ	Annualized volatility of the stock	20% (σ = 0.20)
N(·)	Standard normal CDF — probability that a N(0,1) variable is ≤ the argument	N(0) = 0.5

The Intuition: Cost of Delta-Hedging

The Black-Scholes price is the cost of a continuously rebalanced delta hedge. Here is the clearest way to read the formula:

S · N(d₁) is the expected receipt of stock if the option finishes in the money, adjusted for the fact that you hold N(d₁) shares (the delta) to replicate the option payoff. N(d₁) is the option delta — the fraction of a share you must hold at every moment to stay hedged.

K · e^(−rT) · N(d₂) is the present value of the expected payment at expiry. N(d₂) is the risk-neutral probability that the option expires in the money (i.e., that S_T > K at expiry under the risk-neutral measure).

So the option price = (expected stock received) − (expected cash paid), both present-valued. This is why options become more expensive when volatility rises: higher σ increases the probability of large upside moves while the downside remains capped at zero. The option buyer captures asymmetry.

Key Assumptions

Interviewers do not just want you to recite the formula — they want you to know where it breaks down:

Constant volatility. In practice, volatility smiles and skews show that σ is not constant across strikes and expiries. This is why practitioners use implied volatility surfaces and local/stochastic vol models.
Log-normal stock prices. Returns are assumed normally distributed. Real returns have fat tails and negative skew — crashes happen more often than the model predicts.
Continuous trading with no transaction costs. Delta-hedging must be done continuously, which is impossible. Discrete rebalancing introduces hedging error proportional to γ (gamma).
No dividends. The basic formula ignores discrete cash dividends. The standard adjustment is to replace S with S − PV(dividends) or use the Merton continuous-dividend extension.
No arbitrage / frictionless markets. In practice, bid-ask spreads, borrowing constraints, and short-sell restrictions all matter.

Common Interview Questions

What happens to a call price when volatility increases?

It increases. Vega = ∂C/∂σ > 0 always for both calls and puts. Higher volatility raises the probability of large moves in any direction, but the option holder benefits from upside while being insulated from downside (their loss is capped at the premium paid). More volatility = more valuable optionality.

What happens to a call price as T → 0?

C → max(S − K, 0) — the intrinsic value. With no time left, the option is worth exactly what it pays at expiry. Time value decays to zero, which is why theta is negative for long options.

What happens as σ → ∞?

C → S. As volatility becomes infinite, the stock could go arbitrarily high, so the call (which captures all of that upside) approaches the value of the stock itself. N(d₁) → 1 and the discounted strike term becomes negligible.

Is Black-Scholes actually used in practice?

Yes, but not naively. Practitioners use it as a quoting convention (implied volatility) rather than a literal pricing model. They observe market prices, invert Black-Scholes to get the implied vol, and then trade based on the vol surface. The real models in use at trading desks are local vol (Dupire), stochastic vol (Heston, SABR), and jump-diffusion models — all of which reduce to Black-Scholes in limiting cases.

What is put-call parity?

C − P = S − K · e^(−rT). This is a no-arbitrage relationship that holds independently of Black-Scholes: a long call and short put with the same strike and expiry is equivalent to a forward contract on the stock. If this relationship is violated, there is a riskless arbitrage.

Black-Scholes Formula Explained