Rational Option Pricing Bounds

Definition

Rational option pricing bounds are restrictions on option and warrant prices that follow solely from the assumption that investors prefer more wealth to less (no dominant securities) — without specifying any distributional model for the underlying asset. Merton (1973, Sections 2–4) derives thirteen such theorems; they hold regardless of the stochastic process governing the stock price or the form of the pricing model.

Key Ideas

Lower bound (Theorem 1): $f(S,\tau;E) \geq \max[0,\, S - EP(\tau)]$ ; the floor uses the present value of the exercise price $EP(\tau)$ , not $E$ itself.
No early exercise of American calls (Theorem 2): An American call on a non-dividend-paying stock is never optimally exercised before expiration; it equals the European call.
Perpetual warrant equals stock (Theorem 3): $F(S,\infty;E) = S$ .
Convexity in exercise price (Theorem 4): Warrant price is convex in $E$ .
Diversification hurts option holders (Theorem 7): A warrant on a diversified portfolio is worth less than or equal to a portfolio of warrants on the individual constituents; option holders benefit from concentration of risk.
Monotonicity in variance (Theorem 8): If stock $A$ is riskier than stock $B$ in the Rothschild-Stiglitz sense (mean-preserving spread), then $f_A \geq f_B$ .
European put-call parity (Theorem 12): $g(S,\tau;E) = f(S,\tau;E) - S + EP(\tau)$ ; follows by pure replication.
American put early exercise (Theorem 13): An American put is almost always optimally exercised before expiration; European put-call parity does not extend to American puts.

How It Works

Lower Bound (Theorem 1)

At expiration, the call pays $\max(S^*-E,0)$ and the portfolio [long stock, short discount bond with face $E$ ] pays $S^*-E$ . The call dominates the portfolio in the state $S^* < E$ ; hence $f \geq S - EP(\tau)$ .

No Early Exercise of American Calls (Theorem 2)

When $P(\tau) < 1$ (positive interest rates), the lower bound gives $f \geq S - EP(\tau) > S - E$ . Since an exercised call yields only $S-E$ , the market value always exceeds the exercise value. Early exercise destroys the interest value of the deferred payment $E$ .

Put-Call Parity (Theorem 12)

Portfolio [long call, long bond with face $E$ , short stock] has payoff:

$S^* > E$ : $(S^*-E) + E - S^* = 0$
$S^* \leq E$ : $0 + E - S^* = E - S^*$

This replicates the European put exactly. Hence $g = f - S + EP(\tau)$ .

Why American Puts Are Different (Theorem 13)

For a deep in-the-money put ( $S \to 0$ ), immediate exercise yields $E$ today. The European put cannot be worth more than $E$ regardless of time to expiration. If $r > 0$ , receiving $E$ now is strictly preferable to receiving $E$ at expiration discounted at $P(\tau)$ . Hence early exercise is optimal for sufficiently in-the-money American puts. The argument does not apply to calls because the payoff $S-E$ is bounded only from below.

Dividend Adjustments

When the stock pays dividends, Theorem 2 may fail. Merton shows that early exercise becomes optimal if the dividend rate $d$ exceeds $rE$ (the interest income from investing the exercise price). The threshold condition $E > d/r$ ensures no early exercise for constant dividend $d$ .

Why It Matters

These restrictions are the logical foundation for all option pricing: any model (Black-Scholes, stochastic volatility, jump-diffusion) must satisfy them.
Theorem 2 (no early exercise of calls) is universally applied in practice: American calls on non-dividend-paying stocks are priced as European calls.
Theorem 8 (monotonicity in variance) establishes that implied volatility is the market's key free parameter; the direction of the option-variance relationship is model-free.
Theorem 12 (put-call parity) is the first arbitrage identity of any options market; deviations from it represent the most basic mispricings.

Open Questions

Theorem 2 fails when dividends are positive; no closed-form exists for the optimal early-exercise boundary of a finite-maturity American call with dividends.
The Rothschild-Stiglitz ordering (Theorem 8) is a partial order — many stock pairs are incomparable, limiting the bound's practical content.
For finite-maturity American puts, the early-exercise boundary must be solved numerically; only the perpetual case has a closed form: $G(S) = \frac{E}{1+\gamma}\!\left[\frac{(1+\gamma)S}{\gamma E}\right]^{-\gamma}$ , $\gamma=2r/\sigma^2$ .