Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 39 / 4 / ii / Solution

Let $\Phi$ denote the standard normal distribution cumulative distribution function, and $\varphi$ its density. Put $\tau =T-t$. For $K>0$ and $t<T$, defineUnder the risk-neutral measure, conditional on ${\mathcal{F}}_t$,Thus $Q(S_T<K\mid {\mathcal{F}}_t)=\Phi (-d_2)$. The identity $e^{az}\varphi (z)=e^{a^2/2}\varphi (z-a)$ for the standard Gaussian distribution probability density function givesSplit the payout according to whether the stock exceeds the floor and discount. The risk-neutral pricing process isIts discounted value is ${\mathbb{E}}_Q[{\xi }_T/B_T\mid {\mathcal{F}}_t]$, a true martingale. Thus adding this asset preserves the equivalent martingale measure and gives no arbitrage. This is the guaranteed terminal stock floor in the Black-Scholes model. For $K\le 0$, positivity of the stock makes the claim simply $S_T$, and the price is ${\xi }_t=S_t$.