Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 35 / 3

A process $M_n$ is a $\mathbb{P}$-martingale relative to the filtration $({\mathcal{F}}_n)$ when it is adapted, each $M_n$ is integrable, andHere adapted means $M_n$ is ${\mathcal{F}}_n$-measurable, and integrable means ${\mathbb{E}}_{\mathbb{P}}|M_n|<\infty$. Thus its present value is the conditional mean of its next value given all present information.

By the tower property of conditional expectation and the defining martingale identity,Iteration givesThe integrability in the definition ensures these expectations exist.

A previsible strategy selects $({\varphi }_n,{\psi }_n)$ using only ${\mathcal{F}}_{n-1}$, before the price changes from time $n-1$ to time $n$ occur. Thus these holdings cannot exploit future information.

The self-financing portfolio condition values both the old and newly rebalanced holdings at the same time-$n$ prices:Their equality says that rebalancing uses only the proceeds of selling existing assets: there is no external deposit or withdrawal. In particular, the changes in holdings have zero net cost at the current prices. The holdings for the following period may use the newly available information in ${\mathcal{F}}_n$.

At time $n-1$, the self-financing portfolio can be valued using the holdings about to be carried over the next period:Subtracting this from $V_n={\varphi }_n{S}_n+{\psi }_n{B}_n$ gives the gains identityConversely, suppose the gains identity and the valuation $V_n={\varphi }_n{S}_n+{\psi }_n{B}_n$ hold. Rearranging the identity gives $V_{n-1}={\varphi }_n{S}_{n-1}+{\psi }_n{B}_{n-1}$. Comparing with the valuation using $({\varphi }_{n-1},{\psi }_{n-1})$ shows that rebalancing at $n-1$ costs nothing. Applying this at each rebalancing time recovers the self-financing condition.

Assume the bank account stays positive and write ${\tilde{S}}_n=S_n/B_n$ and ${\tilde{V}}_n=V_n/B_n$. Dividing the time-$n$ rebalancing equality by $B_n$ givesThe discounted bank account has constant price one. This is exactly the self-financing equality in the discounted market. Equivalently, subtract the corresponding valuation at $n-1$ to obtain the discounted wealth equation in discrete timeThis algebra uses only positivity of $B_n$, and does not require it to be deterministic.

The usual replication argument needs two hypotheses implicit in the pricing conclusion: the strategy must replicate the payoff, $V_N=X$, and its discounted gains must be integrable. Bounded predictable risky holdings suffice for the latter. Under these conditions, the equivalent martingale measure $\mathbb{Q}$ and part (c) giveThus discounted wealth is a martingale, and repeated conditioning givesFor a normalized initial bank account, $B_0=1$, deterministic terminal bank value, and trivial initial information, this reduces to the intended risk-neutral pricing formulaIf $B_N$ is random it must remain inside the expectation. If a claim is not replicable, the existence of an equivalent martingale measure alone does not make its arbitrage-free price unique.

The discounted wealth martingale integrability condition cannot simply be dropped. For a concrete counterexample, set $B_n=1$, $S_0=3/2$, $S_1=1+U$, and $S_2=1+U+\epsilon /2$, where $U$ is uniform on $(0,1)$ and $\epsilon$ an independent fair sign. These positive bounded prices form a martingale in the natural filtration. Start with zero holdings; at time one buy $1/U$ shares financed by borrowing $S_1/U$. This is previsible and self-financing, but its terminal wealth is $\epsilon /(2U)$, whose absolute expectation is infinite. It is therefore not a martingale. The stated pricing proof applies to the standard integrable replication class, rather than all unconstrained previsible strategies.