Black-Scholes-Merton and Option Sensitivities

An options market maker quotes a 1-month at-the-money call on an equity ETF. A customer buys at her offer, leaving her short calls. By the close, she has bought ETF shares, added a second option position and rebalanced as the market drifted up. Each hedge changes a different sensitivity in her risk system. The Greeks describe those sensitivities and help size the next adjustment.

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.
• Constant risk-free rate r.
• No arbitrage; perfect short-sale availability.

Common mistakes

• Confusing N(d₁) and N(d₂). N(d₁) is the non-dividend-paying call delta and weights the stock term in the price. N(d₂) is the risk-neutral probability of the call 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 $e^{-qT}$ 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. Equality holds for non-dividend stock under the standard nonnegative-rate assumptions. With dividends, early exercise just before the ex-dividend date can add value, so the American call can be worth more.

Bottom line

• BSM call price: $C = S_0 N(d_1) - K e^{-rT} N(d_2)$; put: $P = K e^{-rT} N(-d_2) - S_0 N(-d_1)$. With dividends q, replace $S_0$ by $S_0 e^{-qT}$ and use (r - q) in d_1's drift.
• $d_1, d_2$: $d_1 = [\ln(S_0/K) + (r - q + \sigma^2/2) T] / (\sigma \sqrt{T})$; $d_2 = d_1 - \sigma \sqrt{T}$. $N(d_2)$ is the risk-neutral probability of finishing in the money.
• Delta $\Delta = N(d_1)$ for calls (between 0 and 1); $\Delta = N(d_1) - 1$ for puts (between -1 and 0). It is the share-equivalent to hedge one option.
• Gamma $\Gamma = \phi(d_1) / (S_0 \sigma \sqrt{T})$. 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 $\sigma\sqrt{T}$. 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 7,029 words, 47 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

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