Past exam of the mathematical models of financial derivatives course of the University of Oxford / 2017 / 3 / a / Solution

A European call option has nonnegative financial payoff, so its price is nonnegative under no-arbitrage. A second arbitrage comparison buys one call and a zero-coupon bond paying $K$, and shorts one share. Its terminal value is $(S_T-K{)}^{+}+K-S_T=(K-S_T{)}^{+}\ge 0$, so its initial cost cannot be negative. ConsequentlyThe holder of an American call option can choose to wait until expiry, so $C_t^a\ge C_t^e$. Since $r>0$ and $t<T$, an in-the-money call has $C_t^a\ge S_t-K{e}^{-r(T-t)}>S_t-K$: immediate exercise sacrifices value. An out-of-the-money exercise pays zero and cannot improve the holder's position. This proves no early exercise of a call without dividends: waiting is always at least as good, and is strictly better for positive immediate exercise payoff. In a degenerate market with zero continuation value an out-of-the-money holder can be indifferent; the Black-Scholes model with positive volatility gives strictly positive continuation value before expiry.