Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 4 / 29K / v

At time $t$, immediate exercise is worth $(S_t-K{)}^{+}$, while waiting is worth the European call option price with remaining maturity $\tau =T-t$. The conditional version of part (iii) givesUnder the standard Black-Scholes assumption $r\ge 0$, this is at least $S_t-K$, while a call value is nonnegative. Hence $C(t,S_t)\ge (S_t-K{)}^{+}$, and the investor should wait until $T$; with $\sigma >0$ and $\tau >0$, waiting has strictly positive time value.

If negative interest rates are allowed, the unconditional instruction “wait” need not be correct. The exact decision is to exercise whenand otherwise to continue.