Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 4 / 29K (Stochastic Financial Models)

In a Black-Scholes model, the risk-free asset and risky asset satisfywhere $B$ is a Brownian motion, $\sigma >0$, and the coefficients are constant. With $S_0^0=1$,so the risky asset is a geometric Brownian motion.

The model has a unique equivalent martingale measure $\mathbb{Q}$ under whichand the discounted stock is a martingale. The time-zero value of an integrable time-$T$ contingent claim $C$ is therefore

For a European call option, the pointwise inequality $(S_T-K{)}^{+}\ge S_T-K$ and risk-neutral pricing give

Under $\mathbb{Q}$,DefineThe standard normal cumulative distribution function $\Phi$ then givesSubstitution into the discounted expected payoff yields the Black-Scholes formula

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.