Merton (1973) — Theory of Rational Option Pricing

Summary

Merton (1973) is a two-part paper published in the inaugural issue of the Bell Journal of Economics and Management Science alongside Merton's own Intertemporal Capital Asset Pricing Model (ICAPM). Part I derives model-free rational restrictions on option and warrant prices using only the preference assumption that investors prefer more wealth to less: lower bounds, early-exercise conditions, convexity in strike, monotonicity in riskiness, and put-call parity. Part II provides an alternative derivation of the Black-Scholes formula under strictly weaker assumptions — stochastic interest rates are explicitly accommodated, CAPM is not invoked — and extends the framework to dividends, the American perpetual put, down-and-out barrier options, and callable warrants.

Key Claims

Theorem 1 (lower bound): $f(S,\tau;E) \geq \max[0, S - EP(\tau)]$ where $P(\tau)$ is the discount bond price; the relevant floor uses the present value of the exercise price.
Theorem 2 (no early exercise of American calls): On a non-dividend-paying stock an American call is never optimally exercised early; it equals the European call. Early exercise yields $S-E < S-EP(\tau) \leq f$ .
Theorem 8 (riskiness): A warrant on a riskier stock (Rothschild-Stiglitz mean-preserving spread) is worth at least as much; option value is nondecreasing in variance.
Theorem 12 (European put-call parity): $g(S,\tau;E) = f(S,\tau;E) - S + EP(\tau)$ ; follows by pure replication, no distributional assumption required.
Theorem 13 (American put early exercise): An American put is almost always optimally exercised before expiration; European put-call parity does not extend to American puts.
Alternative B-S derivation (eq. 38): Under geometric Brownian motion (GBM) for the stock and any Itô process for the bond, the European call satisfies $H(S,P,\tau;E) = EP(\tau)\cdot y\!\left[S/EP(\tau),\;\int_0^\tau V^2(s)\,ds\right]$ where $V^2(s)=\sigma^2+\delta^2-2\rho\sigma\delta$ ; reduces to the standard Black-Scholes formula when $\delta=0$ .
American perpetual put (Section 8): $G(S,\infty;E) = \frac{E}{1+\gamma}\!\left[\frac{(1+\gamma)S}{\gamma E}\right]^{\!-\gamma}$ , $\gamma=2r/\sigma^2$ ; optimal exercise threshold $C^*=\gamma E/(1+\gamma)$ .
Down-and-out call (Section 9): Barrier option formula via Brownian first-passage distribution; first closed-form barrier option formula in the academic finance literature.

Concepts Introduced or Extended

Entities Mentioned

Quotes

"In the derivation of the Black-Scholes formula, it was assumed implicitly that the option buyer and seller can trade in shares continuously. But, in fact, to derive the formula it is only necessary that the option buyer or seller be able to hedge."

"Theorem 8 states that the rational warrant price is a nondecreasing function of the riskiness of the stock... put another way, two warrants identically specified in every way except that one is written on a 'riskier' stock will be priced such that the warrant on the riskier stock is at least as valuable."

My Take

The lasting contribution of Part I is a complete, model-free restriction system: any option pricing model must satisfy Theorems 1–13, and violations are arbitrage opportunities. This foundation is logically prior to Black-Scholes and remains its cleanest statement. Part II's key insight is that the B-S derivation requires neither CAPM nor constant interest rates — only no-sure-thing-profits among correlated securities and Itô's lemma — making the formula more robust than the original paper implied. The extensions (barrier options, callable warrants, American put perpetual) are methodologically dense but historically first: Section 9 contains the first closed-form barrier option in the literature, predating practitioners' awareness by roughly a decade.