Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 211 / 1 / b

Write $h=T-t$. Conditional on ${\mathcal{F}}_t^S$, independent increments of the Brownian motion givewhere $Y$ has the standard normal distribution and is independent of ${\mathcal{F}}_t^S$. Factoring $S_t$ out of the positive part identifies the normalized Black-Scholes call function:The European call option payoff is integrable, because $0\le (S_T-K{)}^{+}\le S_T$ and $\mathbb{E}{S}_T=S_0$. The tower property of conditional expectation therefore proves that $C$ is a true martingale. At $t=T$, the boundary value $F(0,m)=(1-m{)}^{+}$ gives $C_T=(S_T-K{)}^{+}$. The answer is the conditional payoff martingale