Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 35 / 4 / c / Solution

Let $K$ denote the strike, $\tau =T-t$, and suppose $S>0$, $K>0$, $\tau >0$ and $\sigma >0$. Under the Black-Scholes model and the risk-neutral measure, $\log S_T$ is normal withApplying risk-neutral valuation to the European call option payoff givesUse $\xi =\Phi ((u-m)/v)$, whose differential is precisely the normal density times $du$. The threshold is ${\xi }_K=\Phi ((\log K-m)/v)$, soEquivalently, rescale the tail interval to obtain $g_{\mathrm{tail}}(\eta )=e^{-r\tau }(1-{\xi }_K)\{\exp [m+v{\Phi }^{-1}({\xi }_K+(1-{\xi }_K)\eta )]-K\}$ on $0<\eta <1$.

For independent uniform inputs, the Monte Carlo estimator is ${\hat{C}}_N=N^{-1}{\sum }_{j=1}^{N}g(U_j)$. It is unbiased, converges by the strong law of large numbers, and has variance $\mathrm{Var}(g(U))/N$. The payoff has finite second moment because the lognormal stock has finite moments. Its estimated standard error is the sample standard deviation divided by $\sqrt{N}$.