Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 40 / 6

Normalize $B_0=1$ and put ${\tilde{S}}_t=e^{-rt}{S}_t$. Its dynamics are $d{\tilde{S}}_t={\tilde{S}}_t\sigma (t,S_t)d{W}_t$. Bounded volatility makes this stochastic exponential a true martingale on each finite horizon, by the Novikov condition. It also gives a finite second moment: stopping the Itô formula for $S^2$ and applying the Gronwall inequality yields $\mathbb{E}{S}_T^2\le S_0^2{e}^{(2r+L^2)T}$ when $\sigma \le L$.

The discounted payoff $\xi =e^{-rT}(S_T-K{)}^{+}$ is thus square-integrable. Define the nonnegative martingaleThe Brownian martingale representation theorem says that every square-integrable martingale in the Brownian filtration has a representation $M_t=M_0+{\int }_0^t{h}_{u}d{W}_u$ with predictable $h$ and $\mathbb{E}{\int }_0^T{h}_u^{2}du<\infty$. Since $\tilde{S}>0$ and $\sigma >0$, chooseThen discounted gains satisfy $d{M}_t={\pi }_{t}d{\tilde{S}}_t$. Consequently $X_t=B_t{M}_t={\varphi }_t{B}_t+{\pi }_t{S}_t$ is self-financing, nonnegative, and hence an admissible trading strategy. At maturity $X_T=(S_T-K{)}^{+}$, proving claim replication at cost $C(T,K)$.

For minimality, discounted wealth of any admissible self-financing portfolio is a local martingale bounded below, hence a supermartingale by localization and the conditional Fatou lemma. Thus any such replication with initial wealth $x$ obeysTogether with the constructed portfolio, this provesThis is Brownian representation replication in a local volatility market. The given drift $r$ means that the original probability measure already serves as the risk-neutral measure.

Differentiating the European call option price in strike gives, for $T>0$,The strike derivative uses dominated convergence; the second uses the continuous density. Also,Differentiate the supplied time-integral identity and the discount factor. Continuity of the density supplies the diffusion-term derivative. For the tail first moment, continuity in time follows from continuous stock paths, locally uniformly bounded second moments, and the absence of an atom at $K$. ThereforeHence the Dupire equation isIts initial condition is $C(0,K)=(S_0-K{)}^{+}$; natural strike boundaries are $C(T,0)=S_0$ and $C(T,K)\to 0$ as $K\to \infty$. These are consistent with the discounted stock martingale and integrable tails. Where $C_{KK}>0$, the same identity gives local volatility recovery from call prices:The equation evolves in maturity and strike, unlike the backward option-value equation in calendar time and spot.

The Brownian martingale representation theorem argument in part (a) also replicates the bounded put payoff. The discounted stock is a true martingale, so the elementary terminal payoff identity yields put-call paritySet $H(T,K)=-S_0+K{e}^{-rT}$. Then $H_T=-rK{e}^{-rT}$, $H_K=e^{-rT}$, and $H_{KK}=0$. ThusLinearity of the Dupire equation and $P=C+H$ giveThe initial payoff is $P(0,K)=(K-S_0{)}^{+}$, and $P(T,0)=0$. Therefore calls and puts obey the same maturity-strike differential equation, with their respective initial and boundary data.