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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

2008

{c}

Applying <Itô's formula> to $\log S$ gives the <geometric Brownian motion>
$$
\boxed{S_t=s\exp\{(\mu-\tfrac12\sigma^2)t+\sigma W_t\}.}
$$
The Gaussian exponential moment $\mathbb E e^{aW_t}=e^{a^2t/2}$ then yields
$$
\mathbb E S_t^2=s^2e^{(2\mu-\sigma^2)t}\mathbb E e^{2\sigma W_t}
=\boxed{s^2e^{(2\mu+\sigma^2)t}.}
$$
This expectation is under the physical measure and therefore uses drift $\mu$.

Solution

Past exam of the martingales and financial mathematics course of the University of Oxford / 2008 / B / 5 / a / Solution

Past exam of the martingales and financial mathematics course of the University of Oxford / 2008 / B / 5 / a / Solution

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Past exam of the martingales and financial mathematics course of the University of Oxford / 2008 / B / 5 / a / Solution

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