Black-Scholes-Merton and Option Sensitivities | Free FRM Part I Lesson

An options market maker quotes a 1-month at-the-money call on the S&P 500 at $32.10. The customer hits the bid; the market maker is now short one call. By the close, she has bought 540 shares of an S&P ETF, sold 28 nearby strikes against a wing position, and rebalanced twice as the market drifted up 0.6%. None of those trades are speculative. They're all driven by Greeks reading off her risk system. The Greeks are not advanced theory; they are the operational language of an options book.

Black-Scholes-Merton is the continuous-time limit of the binomial tree from the prior lesson. As the number of steps grows without bound, and $u$ , $d$ , and $p$ are calibrated to match drift and variance, the binomial price converges to BSM. The tree is your sandbox for building intuition; BSM is the closed-form answer.

Black-Scholes-Merton prices European options on a non-dividend-paying stock under a set of strong assumptions:

Stock price follows geometric Brownian motion with constant drift μ and constant volatility σ.
Continuous trading, no transaction costs, no taxes.

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Common mistakes

Confusing N(d₁) and N(d₂). N(d₁) is delta and the conditional expected stock weight. N(d₂) is the risk-neutral probability of finishing ITM. They're different numbers; mixing them in the BSM formula gives a nonsense price.
Forgetting dividends in delta and other Greeks. Dividend-paying stock requires $e^{- q T}$ factors throughout. A 5% dividend yield over 1 year shaves about 5% off raw delta. Trap: a question says "stock pays 3% dividend" but the candidate uses non-dividend formulas; off by several percent on every Greek.
Assuming American calls equal European calls always. True for non-dividend stock. False for dividend-paying stock. The American call can be early-exercised just before ex-dividend. Trap: a question describes an in-the-money American call on a dividend-paying stock and asks if it equals the European; the answer is no.

Bottom line

BSM call price: $C = S_{0} N (d_{1}) - K e^{- r T} N (d_{2})$ ; put: $P = K e^{- r T} N (- d_{2}) - S_{0} N (- d_{1})$ . With dividends q, replace $S_{0}$ by $S_{0} e^{- q T}$ and use (r - q) in d_1's drift.
$d_{1}, d_{2}$ : $d_{1} = [ln (S_{0} / K) + (r - q + σ^{2} /2) T] / (σ T)$ ; $d_{2} = d_{1} - σ T$ . $N (d_{2})$ is the risk-neutral probability of finishing in the money.
Delta $Δ = N (d_{1})$ for calls (between 0 and 1); $Δ = N (d_{1}) - 1$ for puts (between -1 and 0). It is the share-equivalent to hedge one option.
Gamma $Γ = ϕ (d_{1}) / (S_{0} σ T)$ . Same for calls and puts. Largest at the money and near expiry. Drives delta-hedging rebalancing frequency.

Exam shortcut

When a BSM problem appears, write down d₁ first using the formula. d₂ is just d₁ minus $σ T$ . Then look up N(d₁) and N(d₂). These two numbers, plus S, K, and the discount factor, build the call price. Put price comes from put-call parity if you have the call.

The full lesson (about 3,437 words, 23 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

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