Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 2 / 27K (Stochastic Financial Models)

A standard Brownian motion has $B_0=0$, continuous paths almost surely, and independent increments with $B_t-B_s\sim N(0,t-s)$ for $0\le s<t$. With a specified filtration, the increment is also independent of the information available at time $s$.

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

Expand the payoff as $(S_T-2K+K^2/S_T)1_{\{S_T>K\}}$. The same risk-neutral measure applies. For any real $p$, completing the square gives the truncated lognormal momentUsing $p=1,0,-1$ and discounting by $e^{-rT}$ givesThe inverse-price moment explains both the extra volatility factor and the shifted normal argument in the last term.