The Black-Scholes model is a closed-form formula for pricing European options from 5 observable inputs. Black and Scholes (1973) derived it in "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, Vol. 81, No. 3, pp. 637–654. The formula w(x,t) = xN(d₁) - ce^(r(t*-t))N(d₂) excludes the expected return on the stock as an input. Empirical tests on a large body of call-option data showed option buyers consistently pay prices above the formula's predictions.
What the Study Found
Black and Scholes showed the formula requires exactly 5 inputs: stock price, exercise price, risk-free rate, variance rate, and time to expiration. The expected return on the stock does not appear in equation (13). The option delta, w₁(x,t) = N(d₁), gives the hedge ratio for maintaining a riskless position. The elasticity xw₁/w is always greater than 1, making the option always more volatile than the underlying stock. The gap between option buyer prices and formula values is larger for low-risk stocks than for high-risk stocks.
Methodology
The derivation assumes 7 ideal market conditions, including constant risk-free rate, log-normal stock prices, no dividends, and no transaction costs. A hedged portfolio of 1 share long and 1/w₁ options short has a return independent of the stock price. Setting that return equal to the risk-free rate yields w₂ = rw - rxw₁ - ½v²x²w₁₁. The unique solution satisfying w(x,t*) = max(x - c, 0) is equation (13).
Key Statistics
| Metric | Finding | Context |
|---|---|---|
| Call option formula | w(x,t) = xN(d₁) - ce^(r(t*-t))N(d₂) | Prices a European call from 5 observable inputs |
| Valuation PDE | w₂ = rw - rxw₁ - ½v²x²w₁₁ | No-arbitrage condition on the continuously adjusted hedged position |
| Option delta | w₁(x,t) = N(d₁) | Hedge ratio; shares of stock per short option to maintain a riskless position |
| Option beta | β_w = (xw₁/w) × β_x | Option beta equals stock beta multiplied by the price elasticity |
| Put-call parity | w(x,t) - u(x,t) = x - ce^(r(t-t*)) | Relation between European call and put on the same stock and exercise price |
| Put option formula | u(x,t) = -xN(-d₁) + ce^(-rt*)N(-d₂) | European put value derived from put-call parity |
| Option vs. stock volatility | xw₁/w always greater than 1 | Option is always more volatile than the underlying stock |
| Empirical deviation | Option buyers pay consistently above formula predictions | Gap is larger for low-risk stocks than for high-risk stocks |
Why This Matters
The formula extends to all corporate liabilities viewable as options on a firm's assets. Stockholders hold a call option to buy the firm from bondholders for the face value of debt. Black and Scholes showed that increasing debt raises the probability of default and reduces the market value of bonds. The same framework prices warrants by adjusting the exercise price for the firm's post-exercise equity structure.