Past exam of the mathematical models of financial derivatives course of the University of Oxford / 2021 / 1

The portfolio value is ${\Pi }_t={\varphi }_t{S}_t+{\psi }_t{B}_t$. The self-financing condition means that changes in holdings are financed by trading within the portfolio, with no external infusion or withdrawal. ThusThe holdings are predictable and satisfy the usual stochastic-integrability and admissibility conditions; without admissibility, doubling strategies would spoil the intended no-arbitrage interpretation.

For $V\in C^{1,2}$, Itô's formula givesMatch the Brownian coefficient with the self-financing portfolio by choosing the option delta $\varphi =V_s$ and bond value $\psi B=V-s{V}_s$. Equating drifts cancels the physical drift $\mu$ and givesThis is the pricing equation for a linear asset with additive noise. Its spatial domain is the whole real line, because the asset can be negative. In particular the diffusion term is ${\sigma }^2{V}_{ss}/2$, not ${\sigma }^2{s}^2{V}_{ss}/2$. The derivation is delta hedging and does not impose a boundary condition at $s=0$.

Set ${\theta }_t=(\mu -r)S_t/\sigma$. The Girsanov theorem uses the densityFor these nonexplosive linear stochastic differential equations, the linear drift change of measure ensures that $Z$ is a true martingale on each finite horizon. This matters: a stochastic exponential being a local martingale alone would not prove that its terminal expectation is one. Its strict positivity gives an equivalent risk-neutral measure $\mathbb{Q}$, under whichThe discounted asset is then a martingale. The Feynman-Kac formula, equivalently risk-neutral valuation of the replicable claim, givesHere the payoff must meet the usual integrability and replication assumptions; the bounded digital payoff below does.

The discounted stock over the remaining horizon satisfiesThe Itô integral on the right is a centered normal random variable with variance $a_{\tau }=(1-e^{-2r\tau })/(2r)$. Therefore $\mathbb{Q}(S_T\ge 0\mid S_t=s)=\Phi (d)$, where $d=s/(\sigma \sqrt{a_{\tau }})$. The digital option price and its option delta areThe formula also shows $V(t,0)=e^{-r\tau }/2$ for every $t<T$. At maturity use the specified step payoff, including its chosen value one at zero; the terminal discontinuity is not removed by the earlier-time smoothing.

Since $a_{\tau }$ does not depend on $\sigma$, differentiation of the digital option value gives its vega:Thus vega is negative for $s>0$, zero at $s=0$, and positive for $s<0$. Increasing volatility spreads the conditional normal distribution around its mean: when the current asset value is positive this reduces the chance of ending positive, while for a negative starting value it increases that chance. At zero the distribution stays symmetric and the probability remains one half. A digital payoff need not have the positive vega familiar from a European call option.