Past exam of the mathematical models of financial derivatives course of the University of Oxford / 2015 / 2

Let $\tau =T-t$. The product rule and Itô's formula givewhere the Black-Scholes equation cancels the drift. Under the usual pricing growth and integrability conditions this local martingale is a martingale. Its terminal value is $f(S_T)$, since the exponential factor becomes one at $T$. Taking a conditional expectation givesThe dynamics already use the risk-neutral measure. This is the Feynman-Kac formula for the given pricing problem; a differential equation alone, without the relevant growth condition, would not justify upgrading a local martingale or uniqueness.

Apply Itô's formula to the logarithm of the positive geometric Brownian motion. The quadratic-variation correction givesIntegrating yieldsThe increment is a Gaussian random variable with variance $T-t$, independent of current information. Writing it as $W_{\tau }$ means a fresh future Brownian motion ${\tilde{W}}_v=W_{t+v}-W_t$, evaluated at $\tau$; it need not equal the original process at time $T-t$.

Take the terminal financial payoff $f(S)=S$. The function $V(S,t)=S$ satisfies the Black-Scholes equation: its time derivative and second stock derivative vanish, while $rS{V}_S-rV=rS-rS=0$. It also has the correct terminal value. Part a therefore givesThus the discounted stock is a martingale. The explicit lognormal distribution in part b supplies the necessary finite expectation, so here this is a true martingale identity.

Under the risk-neutral measure, set $a=\sigma \sqrt{\tau }$ and $m=\log (S/K)+(r-{\sigma }^2/2)\tau$. Then $\log (S_T/K)=m+aZ$ with $Z$ a standard normal random variable, and $d_{-}=m/a$. The log-call option is the discounted Gaussian positive-part expectation:since ${\varphi }^{\prime }(z)=-z\varphi (z)$. ThereforeHere $\varphi$ and $\Phi =N$ are the standard normal density and standard normal distribution function. At expiry the continuous terminal limit is $(\log (S/K){)}^{+}$.