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Year 3 of the mathematics course of the University of Oxford

Mathematical models of financial derivatives course of the University of Oxford

Past exam of the mathematical models of financial derivatives course of the University of Oxford

2023

Apply the <Itô product rule> with deterministic <integrating factor> $e^{\alpha u}$:
$$
d(e^{\alpha u}x_u)=\alpha\widetilde\mu e^{\alpha u}du+\sigma e^{\alpha u}dZ_u.
$$
Integrate from $t$ to $T$ and multiply by $e^{-\alpha T}$ to obtain
$$
\boxed{x_T=x_t e^{-\alpha(T-t)}+\widetilde\mu(1-e^{-\alpha(T-t)})+\sigma e^{-\alpha T}\int_t^T e^{\alpha u}\,dZ_u}.
$$
This is the explicit solution behind the <Ornstein-Uhlenbeck transition distribution>.

Solution

Past exam of the mathematical models of financial derivatives course of the University of Oxford / 2023 / 3 / b / Solution

Past exam of the mathematical models of financial derivatives course of the University of Oxford / 2023 / 3 / b / Solution

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Past exam of the mathematical models of financial derivatives course of the University of Oxford / 2023 / 3 / b / Solution

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