Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 39 / 5

Let $Q$ be the given risk-neutral measure, and write $B_t=B_0{e}^{rt}$. Take $V$ to be classical on $t<T$ and continuous at its prescribed terminal value. By Itô formula and the pricing equation for a local volatility model,soThe discounted added asset is a local martingale, as is $S_t/B_t$, while $B_t/B_t=1$. Consequently $Q$ is an equivalent local martingale measure for all three assets.

Here is the arbitrage argument, including the usual admissible trading strategy condition. If $X$ is the discounted wealth of a self-financing strategy, it is a stochastic integral against the discounted asset prices, plus its initial value, and hence a local martingale. Assume $X_t\ge -c$ for a fixed $c$, as required for an admissible trading strategy. For a localizing sequence ${\tau }_n$, the stopped process $X^{{\tau }_n}$ is a martingale. Apply the conditional Fatou lemma to $X_{t\wedge {\tau }_n}+c\ge 0$: for $s\le t$, continuity and the stopped martingale identity give ${\mathbb{E}}_Q[X_t+c\mid {\mathcal{F}}_s]\le X_s+c$. Thus $X$ is a supermartingale. An arbitrage would have $X_0=0$, $X_T\ge 0$ and positive probability of $X_T>0$. Equivalence of $P,Q$ makes that probability positive under $Q$, contradicting ${\mathbb{E}}_Q[X_T]\le 0$. Therefore the augmented market has no admissible arbitrage.

Nonnegativity of $V$ gives the needed price process, but does not by itself make its discounted local martingale a true martingale. If $V$ is bounded, the latter follows by bounded localization, and its price is also $B_t{\mathbb{E}}_Q[g(S_T)/B_T\mid {\mathcal{F}}_t]$. The arbitrage proof above applies to the stated nonnegative classical solution without assuming that additional bound.

The delta replication from a local-volatility pricing equation uses the asset-unit holdingsTheir wealth is exactly $V(t,S_t)$. Moreover,where the final equality is the calculation in part (i). Thus the holdings are a self-financing strategy with terminal wealth $g(S_T)$, proving replication. Since $V\ge 0$, this replicating strategy has nonnegative wealth and is an admissible trading strategy. The equality of trading gains is unchanged by replacing the drift with its physical-measure value: the stock drift in Itô formula and in the stock holding changes by the same amount. If ${\pi }_t$ denotes wealth fractions, then, wherever $V>0$, its stock fraction is $S_t{V}_S/V$ and its bank fraction is $1-S_t{V}_S/V$.

The diffusion coefficient becomes $S\sigma (S)=\sqrt{S}$, so the stock is a driftless square-root diffusion, with zero taken as an absorbing boundary. Substitution of $V(t,S)=e^{A(t)S+B(t)}$ into the pricing equation for a local volatility model givesEquating the constant and linear terms and imposing the terminal payoff givesThus $1/A(t)=(1+t)/2$ and $B(t)=0$, and the price isThis is the exponential claim in a driftless square-root model. At the absorbing boundary its value is $1$, matching the payoff at zero.

The payoff in this part is unbounded, unlike the payoff assumed in part (i), so it is useful to verify that the proposed price is a finite conditional expectation. For $h>0$, starting from stock value $s$, defineThen $F_v+\frac{1}{2}x{F}_{xx}=0$, $F(h,x)=e^{-ux}$, and $0\le F\le 1$. By Itô formula, $F(v,S_v)$ is a bounded local martingale, hence a true martingale. It follows thatThis is also the Laplace transform of a sum of $N$ independent variables with exponential distribution of rate $2/h$, where $N$ has Poisson distribution of mean $2s/h$ and is independent of the summands: its transform is $\exp [(2s/h)((2/h)/(2/h+u)-1)]$. Uniqueness of the Laplace transform therefore gives the compound Poisson transition law of a driftless square-root diffusion. The exponential distribution jump has exponential moment $\mathbb{E}[e^Y]=(2/h)/(2/h-1)$ whenever $h<2$. Summing over the Poisson distribution yieldsFor the remaining horizon $h=1-t\le 1$, this is finite and, by the Markov property, is precisely ${\mathbb{E}}_Q[e^{S_1}\mid {\mathcal{F}}_t]=e^{2S_t/(1+t)}$. Thus the displayed process is indeed the risk-neutral pricing value and a true martingale. The replicating strategy from part (ii) has $h_t^S=[2/(1+t)]V(t,S_t)$ and $h_t^B=[1-2S_t/(1+t)]V(t,S_t)/B_t$.