Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 39 / 3

In the one-period model, initial holdings are a deterministic vector $H\in {\mathbb{R}}^n$, chosen with trivial initial information, and the terminal payoff of the portfolio is $H^T{P}_1$. A contingent claim ${\xi }_1$ is replicable if some such vector has $H^T{P}_1={\xi }_1$ almost surely; its initial claim replication cost is $H^T{P}_0$. A complete market replicates every claim in the specified terminal-information class; for the arguments below, it suffices that every bounded measurable claim is replicable. The $n$ assets count all assets of the model, including a cash asset if one is traded.

These are one-period definitions. If arbitrary terminal information were already available initially and holdings could depend on it, the finite-atomic conclusion in the next part would not follow; the deterministic-holdings convention is essential.

The span of the $n$ asset payoff functions has dimension at most $n$. If there were $n+1$ disjoint measurable events $A_1,\dots ,A_{n+1}$ of positive probability, their indicator functions would be linearly independent as random variables modulo almost-sure equality. Indeed, restricting a zero linear combination to $A_j$ forces its $j$th coefficient to vanish. Market completeness would put all these independent functions inside a span of dimension at most $n$, a contradiction.

This implies the stronger meaningful partition conclusion: terminal information has at most (n) positive-probability atoms. Start with the whole sample space and split any event which is not a probability atom of a measure into two positive-probability measurable subsets. Each split increases the number of disjoint positive events, so no more than $n-1$ splits are possible. The resulting partition has $m\le n$ components, and each is a probability atom of a measure, since otherwise another split would be possible. Null sets can be included in a component without altering any random variable modulo null sets.

On such an atom every measurable real-valued random variable is constant almost surely: if its distribution on that atom were not concentrated at one value, an appropriate level set would split the atom. Hence every terminal claim is described by its $m$ values. This proves the finite branching bound in a complete market for one period, rather than the vacuous weaker observation that the whole space itself is one event.

For each terminal event $A$, market completeness supplies a portfolio $H_A$ replicating its indicator function. The two pricing identities giveThe constant payoff is also replicable, so these positive pricing variables are integrable under the stated finite pricing expectations. Equivalently, the preceding finite-atom result makes them finite-valued modulo null sets. Taking $A=\{X>Y\}$, the equality says that the nonnegative variable $(X-Y)1_A$ has expectation zero; thus $X\le Y$ almost surely. Reversing the roles givesThis is uniqueness of a one-period pricing density in a complete market. It uses replication of every event, not only matching a few marginal asset expectations.

Let $H$ replicate ${\xi }_1$, so ${\xi }_1=H^T{P}_1$. The positive-definite Gram matrix $Q=\mathbb{E}[P_1{P}_1^T]$ is invertible. Multiplication by $P_1$ and taking expected values giveAll quantities are integrable: the finite-atom representation from completeness gives finite terminal values, and the matrix $Q$ has finite entries. Positive definiteness also guarantees uniqueness of holdings, since $h^T{P}_1=0$ would imply $h^{T}Qh=0$ and hence $h=0$.

Thus the replication cost isConsequently the one-period Gram-matrix replication formula isThe symmetry of $Q^{-1}$ makes the displayed orientations consistent. Positivity of this $Z$ would require an additional no-arbitrage condition; positive definiteness of the payoff Gram matrix alone proves the representation and uniqueness, not positivity of prices across states.