Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 41 / 2 / iii / Solution

In a complete market, terminal-wealth optimization can first be solved over terminal payoffs and then implemented by replication. Let initial wealth be $x_0>0$, and take an increasing differentiable strictly concave utility function $U$ on $(0,\infty )$ satisfying the Inada conditions. The state-price budget constraint for a terminal wealth $X$ is ${\mathbb{E}}_P[{\zeta }_{n}X]\le x_0$. Monotonicity spends the whole budget at the optimum.

Let $I=(U^{\prime }{)}^{-1}$ be the inverse marginal utility. Choose $y>0$ to satisfy ${\mathbb{E}}_P[{\zeta }_{n}I(y{\zeta }_n)]=x_0$. In a finite tree with positive state probabilities, this left side is continuous and strictly decreasing from infinity to zero, so $y$ exists uniquely. The complete-market terminal utility optimizer isTo prove optimality, concavity gives $U(X)\le U(X^{\ast })+U^{\prime }(X^{\ast })(X-X^{\ast })=U(X^{\ast })+y{\zeta }_n(X-X^{\ast })$. Taking expectations and using the budget proves $\mathbb{E}U(X)\le \mathbb{E}U(X^{\ast })$. Strict concavity gives uniqueness of the optimal terminal wealth. The finite-market setting avoids additional infinite-state existence and integrability issues.

Replicate $X^{\ast }$ using the backward recursion of part (i). Its wealth process is $V_k=B_k{\mathbb{E}}_Q[X^{\ast }/B_n\mid {\mathcal{F}}_k]$, and the two-successor difference quotient gives its stock holdings. This self-financing strategy starts with $x_0$ by the budget equality and remains positive at every node, since it is the discounted conditional expectation of a strictly positive terminal wealth.

For logarithmic utility, $U(x)=\log x$, $I(z)=1/z$, the multiplier is $y=1/x_0$ and $X^{\ast }=x_0/{\zeta }_n=x_0{B}_n/L_n$. For CRRA utility with risk aversion $\gamma >0$, $\gamma \ne 1$, the normalized payoff isConstraints such as prohibiting short sales change the attainable terminal-payoff set and require a constrained optimization; the unconstrained complete-market formula uses the stated assumptions.