Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 2 / 27K / ii

In the Black-Scholes model, absence of arbitrage and completeness give the price as the discounted expectation under the risk-neutral measure. Under that measure,The physical drift $\mu$ therefore does not enter the option price. PutFor the standard normal distribution function $\Phi$, $\mathbb{P}(S_T>K)=\Phi (d_2)$. Completing the square in the Gaussian density also gives $\mathbb{E}[S_T{1}_{\{S_T>K\}}]=S_0{e}^{rT}\Phi (d_1)$. Expanding the call payoff into these two truncated moments yields