Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 211 / 3

A local martingale deflator makes both $YB$ and $YS$ local martingales. Since the filtration is generated by $W$, the martingale representation theorem and the finite-variation drift forced by $YB$ givefor a continuous adapted $\lambda$, where $\eta =-Y\lambda$. Applying the Itô product rule to $YS$ gives driftIt vanishes exactly when

SetThe boundedness of ${\xi }_T$ and positivity of the deflator make $M$ a nonnegative true martingale. By the Brownian martingale representation theorem, $d{M}_t={\eta }_{t}d{W}_t$. A self-financing wealth process with stock holding ${\pi }_t$ satisfiesThe product $YV$ then has diffusion coefficientChooseThen $Y_t{V}_t=M_t$, so $V_T={\xi }_T$ and the strategy replicates the claim. It is admissible because $V=M/Y$ is nonnegative.

For any other admissible replicating wealth $\tilde{V}$, the nonnegative local martingale $Y\tilde{V}$ is a supermartingale. HenceThe constructed strategy hasso this is the minimal replication cost.

For constant coefficients, the density process $Y_t{B}_t/B_0$ is a true exponential martingale and defines the risk-neutral measure $Q$. Under $Q$,The minimal value process is thereforeThe Markov property and the lognormal transition law make this a deterministic function of $(t,s)$, and

Apply Itô formula to $V(t,S_t)$. Its Brownian coefficient isThe self-financing portfolio's Brownian coefficient is ${\pi }_t\sigma S_t$. Since $S_t\sigma >0$, equality of the two value processes forcesThus the stock holding is the claim's option delta.