Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 211 / 6

Apply the Itô formula to ${\xi }_t=V(t,S_t)$. The backward equation cancels its drift, leavingBounded $a$ and $V_S$ make this Itô integral square-integrable on the finite horizon; in addition $V$ itself is bounded. Hence $\xi$ is a true martingale, not just a local martingale, and ${\xi }_T=g(S_T)$. Taking conditional expectations givesThe stock is understood to be adapted to the stated Brownian filtration, with its initial value fixed there. This is the Feynman-Kac formula in the zero-potential, zero-drift case.

Differentiate the backward equation for $V$ with respect to the stock state, and put $w=V_S$. The product derivative of $a^2/2$ is $a{a}^{\prime }$, givingThe assumed smoothness and bounded derivatives put $w$ in the uniqueness class for the equation defining $U$. Therefore $U=V_S$, and ${\pi }_t=V_S(t,S_t)$ is the option delta. Substituting in the Itô integral from part (a) and using $d{S}_t=a(S_t)d{W}_t$ gives the self-financing representationThe boundedness hypotheses justify both stochastic integrals. This derivation does not divide by $a$ and remains valid even at states with zero diffusion amplitude.

The positive density process is the stochastic exponentialSince $a^{\prime }$ is bounded, the Novikov condition holds; $Z$ is a true martingale with $\mathbb{E}{Z}_T=1$ and $Z_T>0$. Thus it defines the stated equivalent probability measure. The positive sign in the exponent means that the Girsanov theorem givesUnder this change of measure, applying the Itô formula to ${\pi }_t=U(t,S_t)$ produces precisely the drift in its backward equation, which vanishes:Bounded $a$ and $U_S$ make $\pi$ a true martingale under $\hat{\mathbb{P}}$. Its terminal value is $g^{\prime }(S_T)$. ConsequentlyThe sign can also be checked before changing measure: under the original measure $d\pi =-a{a}^{\prime }{U}_{S}dt+a{U}_{S}dW$, while the quadratic covariation term in $d(Z\pi )$ is $Za{a}^{\prime }{U}_{S}dt$, exactly cancelling its drift.

Take the original measure as the risk-neutral measure for this zero-interest model, or interpret $S$ as a discounted stock price. ${\xi }_t$ is the arbitrage-free value of the European contingent claim with payoff $g(S_T)$, and ${\pi }_t$ is its option delta and replicating stock holding. Part (b) supplies the self-financing portfolio:With the cash asset identically one, its value is ${\xi }_t$ and its gains satisfy $d{\xi }_t={\pi }_{t}d{S}_t$, so it exactly replicates $g(S_T)$. This delta hedge is admissible since the assumed nonnegative $V$ gives nonnegative wealth. In this continuous-path setting the holdings can be taken predictable.

The auxiliary change of measure in part (c) expresses the option delta as an expectation of terminal payoff sensitivity. It changes the stock drift to $a{a}^{\prime }$ and is generally not the original risk-neutral measure for the zero-interest stock market. Its role is to value sensitivity under the tilted state dynamics, while the actual replicating strategy trades under the original dynamics. No general claim of market completeness is needed, including when $a$ can vanish.