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Martingales and financial mathematics course of the University of Oxford

Past exam of the martingales and financial mathematics course of the University of Oxford

2010

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For <geometric Brownian motion>, $\log S_u=\log S_0+(\mu-\sigma^2/2)u+\sigma W_u$. Averaging in time gives
$$
\log A_t=\log S_0+\tfrac12(\mu-\sigma^2/2)t+\frac\sigma t I_t.
$$
Thus $\log A_t$ is normal with mean $\log S_0+(\mu-\sigma^2/2)t/2$ and variance $\sigma^2t/3$. The <moment-generating function of a normal distribution> gives
$$
\boxed{\mathbb EA_t=S_0\exp\left[\left(\frac\mu2-\frac{\sigma^2}{12}\right)t\right].}
$$

Solution

Past exam of the martingales and financial mathematics course of the University of Oxford / 2010 / B / 5 / b / Solution

Past exam of the martingales and financial mathematics course of the University of Oxford / 2010 / B / 5 / b / Solution

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Past exam of the martingales and financial mathematics course of the University of Oxford / 2010 / B / 5 / b / Solution

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