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The Black-Scholes formula for a European call option is: C = S · N(d1) − K · e^(−rT) · N(d2). This single equation changed finance. Here is what every piece means and how to use it.
The formula has two terms. S · N(d1) is the expected stock position, adjusted for the probability-weighted amount you receive if the option is exercised. K · e^(−rT) · N(d2) is the present value of the strike price, weighted by the probability of exercise. The call price is the difference: what you expect to receive minus what you expect to pay.
d1 = [ln(S/K) + (r + σ²/2) · T] / (σ · √T). d2 = d1 − σ · √T. The term ln(S/K) captures moneyness. The term (r + σ²/2) · T adjusts for drift and the convexity of the lognormal distribution. Dividing by σ√T normalises everything into standard deviation units so that N() can be applied.
S = $49, K = $50, r = 5%, σ = 20%, T = 20 weeks (0.3846 years). Step 1: σ√T = 0.20 × √0.3846 = 0.1241. Step 2: ln(49/50) = −0.0202. Step 3: (0.05 + 0.02) × 0.3846 = 0.0269. Step 4: d1 = (−0.0202 + 0.0269) / 0.1241 = 0.054. Step 5: d2 = 0.054 − 0.1241 = −0.070. Step 6: N(0.054) = 0.522, N(−0.070) = 0.472. Step 7: C = 49 × 0.522 − 50 × e^(−0.019) × 0.472 = 25.58 − 23.14 = $2.44.
The formula assumes the stock follows geometric Brownian motion with constant volatility, continuous trading is possible at zero cost, the risk-free rate is constant, and there are no dividends. Every assumption is wrong. The market knows it. But BSM remains the universal pricing language because it provides a one-to-one mapping between option price and implied volatility — a translation device that enables comparison across all options.
The first term S times N(d1) represents the expected stock payout weighted by the hedge ratio. The second term K times e to the minus rT times N(d2) is the present value of the strike weighted by the probability of exercise. The call price is the difference between what you expect to receive (stock) and what you expect to pay (strike).
N(d1) is the delta of the option, the hedge ratio. N(d2) is the risk-neutral probability that the option expires in the money. They differ by sigma times the square root of T. For ATM options, the difference is small. For long-dated or high-volatility options, the gap can be significant.
First compute sigma times the square root of T. Then d1 equals the natural log of S over K, plus r plus half sigma squared times T, all divided by sigma root T. d2 is simply d1 minus sigma root T. Then apply the cumulative normal distribution function N() to both values to get the probabilities used in the pricing formula.
FinLingo covers Black-Scholes in Level 3 — 8 units from binomial intuition to the full formula. Price options in real time in The Lab. Level 1 is free.
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