Solution

Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 32 / 5 / Solution

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

{title2=$F(v,m)$}

For $v,m\geq0$, define $F(v,m)=\mathbb E[(e^{-v/2+\sqrt vY}-m)^+]$, where $Y$ has the <standard normal distribution>. For $v,m>0$, completing the square in the <standard normal density> gives
$$
F(v,m)=\Phi(d_1)-m\Phi(d_2),\qquad d_1=\frac{-\log m+v/2}{\sqrt v},\quad d_2=d_1-\sqrt v,
$$
where $\Phi$ is the <standard normal distribution function>. The boundary values are $F(0,m)=(1-m)^+$ and $F(v,0)=1$. Multiplying by the current <stock> price produces a zero-interest <Black-Scholes formula> with total variance $v$ and relative strike $m$.

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 Incoming links: Normalized Black-Scholes call function