Black-Scholes (1973) The Pricing of Options and Corporate Liabilities

Summary

Black and Scholes (1973) derive the no-arbitrage price of a European call option by constructing a continuously rebalanced delta hedge — long one share, short $1/w_1$ options — whose instantaneous covariance with the market is zero, so no-arbitrage forces it to earn the riskless rate $r$ . This yields a second-order partial differential equation (PDE) that is solved analytically via a change of variables reducing it to the heat equation, producing a closed-form formula. A parallel capital asset pricing model (CAPM)-based derivation, requiring only that $\beta_{\text{option}} = (xw_1/w)\cdot\beta_{\text{stock}}$ , recovers the same PDE without any hedging argument. The paper also reinterprets corporate equity and debt as contingent claims on firm assets.

Key Claims

Continuous delta rebalancing eliminates all systematic risk from the hedged position, forcing it to earn $r$ . This yields the fundamental Black-Scholes PDE: $\tfrac{1}{2}v^2x^2w_{11} + rxw_1 + w_2 - rw = 0$ (eq. 7), where $v$ is the instantaneous standard deviation of stock returns, $x$ is the current stock price, and $w$ is the option value.
With boundary condition $w(x,t^*) = \max(x-c,0)$ , a substitution $y = \ln x + (r - \tfrac{1}{2}v^2)(t^*-t)$ , $\tau = v^2(t^*-t)$ converts (7) to the standard heat equation, which yields the closed-form solution: $w(x,t) = xN(d_1) - ce^{-r(t^*-t)}N(d_2)$ , with $d_1 = [\ln(x/c) + (r+v^2/2)(t^*-t)]/(v\sqrt{t^*-t})$ and $d_2 = d_1 - v\sqrt{t^*-t}$ .
CAPM alternative derivation: write the option's expected return as $\alpha_w - r = (xw_1/w)(\alpha - r)$ via the instantaneous $\beta$ relationship. Substituting $\alpha_w = (w_2 + \alpha xw_1 + \tfrac{1}{2}v^2x^2w_{11})/w$ into this equation and simplifying, the $\alpha$ and $r$ terms cancel to recover equation (7) exactly — preference parameters drop out independently of the hedging argument.
European put: $u(x,t) = ce^{-r(t^*-t)}N(-d_2) - xN(-d_1)$ from put-call parity.
Corporate liabilities: equity = call on firm assets with strike = face value of debt; bond value = $x - w(x,t)$ . Coupon bonds are compound options (each coupon is an option on the right to pay future coupons). This framework anticipates Merton (1974) structural credit models.
Empirical tests (over-the-counter call data): option buyers paid above the formula on average. Low-variance stocks show the largest relative deviations. Variance rate is systematically underestimated by the market relative to what the formula requires.

Concepts Introduced or Extended

Entities Mentioned

Quotes

"If the hedge is maintained continuously, then the approximations mentioned above become exact, and the return on the hedged position must be completely certain."

"One of the striking features of the warrant pricing formula is that it does not depend on investors' attitudes toward risk."

"In general, since over-the-counter options are written at-the-money when issued, our evidence indicates that buyers of options are paying more than our formula suggests they should."

My Take

The delta-hedge argument is elegant precisely because it does not require any model of equilibrium risk premia. What Black and Scholes recognized — which Samuelson, Merton, and Boness had not — is that the preference parameters drop out entirely because of the perfect instantaneous hedge, not because of any assumption about the market price of risk. The CAPM derivation makes this even cleaner: the excess-return terms cancel algebraically when $\beta_{\text{option}}$ is expressed in terms of $\beta_{\text{stock}}$ , making preference-independence a consequence of the elasticity relationship rather than a distributional assumption. The corporate liabilities extension is prescient: treating equity as a call on firm assets with the bond as the residual reframes all corporate securities as derivatives, laying the groundwork for structural credit risk models.