Section 25.4

No-Arbitrage Bounds

Monotonicity, convexity (butterflies), calendar and American-versus-European constraints, bounds on spreads.

Toolkit

No-arbitrage bounds come from comparing option portfolios to dominated or dominating payoffs: if portfolio $A$ ’s payoff is always $\ge$ portfolio $B$ ’s, then $\mathrm{Price}(A)\ge \mathrm{Price}(B)$ today, else trade the cheap one against the rich one and lock in a riskless profit (discounting any time- $0$ cash flow).

Key static constraints for calls/puts at strikes $K_1<K_2<K_3$ (same expiry) and maturities $T_1<T_2$ :

Monotonicity: $C(K)$ decreasing, $P(K)$ increasing in $K$ .
Vertical spread bound: $0\le C(K_1)-C(K_2)\le (K_2-K_1)e^{-rT}$ , similarly for puts.
Convexity (butterfly): $C(K_2)\le \lambda C(K_1)+(1-\lambda)C(K_3)$ , where $\lambda=\frac{K_3-K_2}{K_3-K_1}$ ; butterfly payoff $\ge 0$ so its price $\ge 0$ .
Calendar: on non-dividend stock, $C_{T_2}(K)\ge C_{T_1}(K)$ (American always $\ge$ European, same strike/expiry).
American $\ge$ European for calls and puts; equality for non-dividend calls.
Box spread: $[C(K_1)-C(K_2)]+[P(K_2)-P(K_1)] = (K_2-K_1)e^{-rT}$ .

Problems