Past exam of the mathematical models of financial derivatives course of the University of Oxford / 2017 / 1

Apply Itô's formula to $f(u,s)=e^{-\alpha u}s$. Its second stock derivative is zero, and the two drift terms cancel, givingIntegrating and multiplying by $e^{\alpha T}$ solves the linear stochastic differential equation:The Itô integral has a deterministic integrand, so it is a centered Gaussian random variable independent of the current filtration ${\mathcal{F}}_t$. Its variance follows from the Itô isometry. With $\tau =T-t$,The second line is also the limit as $\alpha \to 0$. The normal law asserted here is conditional on ${\mathcal{F}}_t$; an arbitrary random initial value need not give an unconditional normal distribution.

Use a self-financing portfolio with ${\Delta }_t$ units of the traded asset and the remainder in the continuous-time bank account. If its value equals $V(S_t,t)$, its self-financing condition isMeanwhile Itô's formula givesMatching the Brownian motion coefficients gives the delta hedge ${\Delta }_t=V_S$. Matching drifts then cancels the physical drift $\alpha$, leaving the pricing equation for a linear asset with additive noise:The domain is the full real line: additive noise can take the asset through zero. The diffusion coefficient is $\sigma$, so the second-derivative term has no factor $S^2$. As usual, the replication argument assumes suitable differentiability and an admissible trading strategy.

The differential operator in part b is the infinitesimal generator of the auxiliary diffusion process $d{S}_u=r{S}_{u}du+\sigma d{W}_u$, with killing rate $r$. The Feynman-Kac formula therefore givesEquivalently, Itô's formula makes $e^{-r(u-t)}V(S_u,u)$ a local martingale under these dynamics; under appropriate growth and integrability assumptions it is a martingale, whose terminal conditional expectation yields the same formula. The expectation uses the drift-$r$ law, rather than the physical drift-$\alpha$ law in part a. This is risk-neutral pricing.

Apply part a with $\alpha =r$. For $\tau >0$, the transition density is the normal distribution with mean $m=S{e}^{r\tau }$ and positive variance $v={\sigma }^2(e^{2r\tau }-1)/(2r)$. Integrating the financial payoff against this probability density function givesThus $d_{\ast }=(x-S{e}^{r\tau })/\sqrt{v}$, as required. The variance is positive even if $r<0$, because numerator and denominator have the same sign. For $r=0$, replace $m$ by $S$ and $v$ by ${\sigma }^2\tau$; at expiry use the terminal value directly. The integral presumes an integrable financial payoff.