Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 40 (Optimal Investment)

Take $S_t$ to be the ex-dividend stock price: the date-$t$ output has already been paid when trading occurs. Receiving ${\theta }_t{d}_t$ units of fruit and selling ${\theta }_t-{\theta }_{t+1}$ shares gives the discrete dividend budget equationThe entering holding ${\theta }_t$ is chosen with date-$(t-1)$ information, while $C_t$ and the next holding ${\theta }_{t+1}$ may use date-$t$ information. The output is the consumption good and price numéraire; its stochastic law is exogenous to the individual's price-taking choice.

For the interior differentiable case, attach an adapted Lagrange multiplier ${\Lambda }_t$ to each budget constraint. On a finite horizon, with the terminal holding fixed or its continuation value included, the optimization Lagrangian isAn adapted variation in $C_t$ gives ${\Lambda }_t={\beta }^t{U}^{\prime }(C_t)$. An arbitrary date-$t$ measurable variation in ${\theta }_{t+1}$ affects two adjacent terms. Its coefficient must have zero conditional mean, soThese are the Euler equations of marginal utility pricing in a dividend economy. If the utility function is merely concave, a supporting slope replaces $U^{\prime }$; binding consumption or trading restrictions give the corresponding inequalities rather than unconstrained equalities.

With known initial information and ${\Lambda }_0>0$, normalize ${\zeta }_t={\Lambda }_t/{\Lambda }_0$. This is a state-price density for the traded asset. In particular,The ratio is the stochastic discount factor: payment in a state with greater marginal utility has a higher marginal value. For a priced contingent payoff $H$ at $t+1$, its price is ${\mathbb{E}}_t[({\zeta }_{t+1}/{\zeta }_t)H]$; pricing arbitrary claims uniquely also requires market completeness. The deflated cumulative gains ${\zeta }_t{S}_t+{\sum }_{s=1}^t{\zeta }_s{d}_s$ form a martingale whenever the displayed terms are integrable.

Iterating the Euler equation givesUnder the dividend-price transversality condition and the needed integrability, this becomes the fundamental priceThus an infinite-horizon price is not automatically just a dividend sum: its terminal term must vanish. Telescoping the individual budget constraints similarly givesA no-Ponzi lower restriction on this final term gives the consumption budget inequality; its vanishing gives equality. Together with the supporting-line inequality for a concave utility function, the Euler equations and these terminal conditions supply the usual sufficiency argument for an admissible candidate. In an incomplete market, each optimal agent's marginal utility can supply a different state-price density pricing the same traded tree.

For a competitive equilibrium with one productive asset, the primitives are the probability law and information filtration of total output $d$, each agent's utility function and discount factor, the initial ownership shares ${\theta }_0^j$ summing to one, and the permitted consumption and trading strategies. Prices are not primitives of the equilibrium problem. The derived objects are the ex-dividend price $S$, each agent's consumption $C^j$ and holdings ${\theta }^j$, and their marginal-utility pricing processes. At a common price, every agent solves the individual problem, and resource and share clearing requireThe individual discrete dividend budget equations, optimality Euler equations and terminal conditions from part (i), together with these clearing equations, define equilibrium even when the market is incomplete. Summing the individual budget constraints is consistent with total fruit consumption.

For the given constant absolute risk aversion utility, $U_j^{\prime }(c)=e^{-{\gamma }_{j}c}$. Hence each interior optimum satisfiesThe same $S$ must satisfy all these equations, but their conditional expectation identities do not by themselves identify the different agents' pricing densities pointwise.

To illustrate explicit exponential-utility risk sharing, first consider a complete or otherwise implementable interior allocation. With positive constant welfare weights $a_j$, a pointwise resource multiplier ${\nu }_t$ satisfiesWriteSolving the marginal equations and summing the consumptions givesThe output risk is shared in proportion to risk tolerance ${\gamma }_j^{-1}$; differences in patience produce the time-dependent transfer term. The common normalized state-price density is ${\zeta }_t={\nu }_t/{\nu }_0$. The weights, up to a common scale, are determined by the initial wealth budgetsand, for a fundamental price,If consumption is required to be nonnegative and the interior formula produces a negative allocation, replace it byFor $d_t>0$ this determines the resource multiplier uniquely. At zero total output all constrained consumptions are zero and the corresponding marginal inequalities apply.

A single traded tree need not span the contingent transfers in the general risk-sharing formula. A genuine equilibrium example within the literal one-tree market is obtained by taking all ${\beta }_j=\beta$ and initial shares ${\theta }_0^j=\Gamma /{\gamma }_j$. ThenEvery agent simply keeps the initial shares, all budget constraints and clearing conditions hold, and their discounted marginal utilities coincide. The fundamental price above satisfies every Euler equation. For example, nonnegative bounded output ensures the required price and utility integrability and the terminal terms vanish; the supporting-line argument from part (i) then verifies individual optimality. For heterogeneous discount factors or arbitrary initial ownership, the displayed coupled Euler-budget-clearing system remains the correct one-tree formulation; using the unconstrained Pareto allocation as a traded equilibrium requires an additional financing check.

Use ${\theta }_t$ for the dollar amount invested in the stock; the number of shares is ${\theta }_t/S_t$. The bank account contains $w_t-{\theta }_t$ dollars. With no intermediate consumption, the self-financing portfolio equation isIf instead $\theta$ denotes share holdings, replace each dollar ${\theta }_t$ here by $S_t{\theta }_t$. The equation says that trading only reallocates existing portfolio wealth; gains come from cash interest and stock returns.

For the standard unconstrained complete-diffusion formulation, assume the completed natural Brownian filtration, initial wealth $w_0>0$, and nonnegative admissible wealth. Define the market price of risk ${\kappa }_t=({\mu }_t-r_t)/{\sigma }_t$. The boundedness assumptions make $\kappa$ bounded. The state-price density normalized at zero isIts money-market-adjusted process $Z_t=e^{{\int }_0^t{r}_{s}ds}{\zeta }_t$ is a true martingale by the Novikov condition. Under $d\mathbb{Q}=Z_{T}d\mathbb{P}$, the Girsanov theorem gives $W_t^{\mathbb{Q}}=W_t+{\int }_0^t{\kappa }_{s}ds$ as a Brownian motion, and the discounted stock has zero drift.

The Itô product rule applied to the wealth equation givesFor nonnegative admissible wealth this is a nonnegative local martingale, hence a supermartingale. Therefore every affordable terminal payoff $X=w_T$ satisfies the state-price budget constraint $\mathbb{E}[{\zeta }_{T}X]\le w_0$. Nonzero volatility and the Brownian martingale representation theorem allow replication of nonnegative claims with finite state-price cost.

For a differentiable strictly concave utility function with the usual conditions ensuring an interior integrable optimizer, put $I=(U^{\prime }{)}^{-1}$, the inverse marginal utility. Pointwise maximization of $U(x)-y{\zeta }_{T}x$ givesThe second equation determines the positive constant $y$. To prove optimality, use the concave supporting-tangent inequality:Taking expectations and using the budget bound proves that the candidate dominates every admissible affordable payoff, whenever the utility expectations are well defined. This is the complete-market terminal utility optimizer.

The actual wealth and dollar portfolio are as explicit as general adapted coefficients permit. LetThe Brownian martingale representation theorem, with localization if only $L^1$ integrability is initially known, gives $h$. The dollar portfolio from a deflated wealth martingale isThe bank holds $w_t^{\ast }-{\theta }_t^{\ast }$ dollars. Equivalently, with $B_t=e^{{\int }_0^t{r}_{s}ds}$, $w_t^{\ast }=B_t{\mathbb{E}}^{\mathbb{Q}}[w_T^{\ast }/B_T\mid {\mathcal{F}}_t]$. These formulas need no unjustified differentiability of a value function with random coefficients.

The printed assumption that $U$ is only increasing and concave does not guarantee that the interior formula exists, or even that the optimal value is finite. In general replace the inverse formula by an affordable selectionwhere $\mathcal{D}$ is the wealth domain; boundary choices or supporting slopes handle nondifferentiability. Existence, finite cost and replication still have to hold. A concrete obstruction is the unbounded linear terminal-wealth utility: take $U(x)=x$, $r=0$, $\mu =\sigma =1$, and the Brownian filtration. Then ${\zeta }_T=e^{-W_T-T/2}$. For $A_n=\{{\zeta }_T<1/n\}$, the bounded nonnegative claimis replicable, costs $w_0$, and has $\mathbb{E}{X}_n>n{w}_0$. Thus there is no finite optimal expected utility under the literal utility hypotheses alone. Also, bounded coefficients do not ensure market completeness in an arbitrarily enlarged filtration containing untraded randomness; the full replication formula uses the Brownian market information assumption stated above.

Assume, as in the Bayesian model, that the unknown drift $\alpha$ is independent of the driving Brownian motion, with known positive prior precision ${\tau }_0$. Recover the observed signal from the positive stock price:For a fixed time $t$, the centered observed path $Y_s-(s/t)Y_t$ is a Brownian bridge independent of $(\alpha ,Y_t)$: it is independent of the prior and has zero covariance with $W_t$ in their joint Gaussian law. Thus the whole observed history has the same posterior information about $\alpha$ as its endpoint. This proves Brownian endpoint sufficiency for a constant drift rather than assuming it.

The likelihood of that endpoint, as a function of $\alpha$, is proportional to $\exp (\alpha Y_t-{\alpha }^{2}t/2)$. Multiplying by the normal distribution prior and completing the square gives the Gaussian Brownian drift filterDefine the observed innovation process ${\hat{W}}_t=Y_t-{\int }_0^t{\hat{\alpha }}_{s}ds$. It is a continuous observed-filtration martingale: for $s<u$, independence of future noise and the tower property of conditional expectation give $\mathbb{E}[\alpha -{\hat{\alpha }}_u\mid {\mathcal{F}}_s^S]=0$, so the conditional mean of its increment vanishes. Its quadratic variation is $t$, since subtracting the finite-variation drift from $Y$ does not change quadratic variation. The Lévy characterization of Brownian motion makes $\hat{W}$ an observed-filtration Brownian motion. Differentiating the posterior mean givesConversely, the last equation determines $\hat{\alpha }$ from $\hat{W}$ with deterministic coefficients and known initial value, and $Y=\hat{W}+\int \hat{\alpha }dt$. Thus the observed and innovation filtrations coincide, supplying the natural Brownian information needed for replication. Consequently the observed stock dynamics are $d{S}_t/S_t=\sigma d{\hat{W}}_t+\sigma {\hat{\alpha }}_{t}dt$. Its observed market price of risk is ${\kappa }_t={\hat{\alpha }}_t-r/\sigma$, and its normalized state-price density solvesThe posterior mean is unbounded, so bounded-coefficient Novikov condition reasoning from Question 2 is not available automatically. A direct finite-horizon argument supplies true pricing. Relative to driftless Wiener measure for $Y$, the Bayesian mixture has observation densityA law with observation drift $k=r/\sigma$ instead has density $e^{k{Y}_t-k^{2}t/2}$. ThereforeThis is the finite-horizon pricing density for Gaussian drift learning. The ratio changes the physical observed law into a genuine probability law, hence has mean one. Indeed, differentiation of the explicit likelihood, using $d{Y}_t=d{\hat{W}}_t+{\hat{\alpha }}_{t}dt$, givesThe Itô formula therefore gives $d{D}_t=-D_t{\kappa }_{t}d{\hat{W}}_t$, agreeing with the stochastic exponential above. Under this pricing law the stock drift is $\sigma k=r$.

For logarithmic utility, terminal marginal utility is $1/w_T$. The state-price budget constraint and the marginal condition imply $w_T^{\ast }=1/(y{\zeta }_T)$ and $1/y=w_0$. Since ${\zeta }_T{w}_T^{\ast }=w_0$ is constant, its conditional-expectation pricing process is constant as well. Thus the entire optimal wealth process isIndeed, for any other nonnegative admissible terminal wealth $X$, the elementary inequality $\log (X/w_T^{\ast })\le X/w_T^{\ast }-1$ givesThis directly verifies optimality. Expected candidate log wealth is finite because $\mathbb{E}{\hat{\alpha }}_t^2={\hat{\alpha }}_0^2+{\tau }_0^{-1}-({\tau }_0+t{)}^{-1}$ and its time integral is finite on every fixed horizon. Applying the Itô formula to $1/\zeta$ givesThe observed self-financing portfolio with risky dollar fraction ${\pi }_t$ has diffusion coefficient ${\pi }_t\sigma$. Matching the coefficients proves the log-optimal investment with Gaussian drift learning ruleThe remaining dollars are held in the bank account; the fraction may be negative or exceed one because the question imposes neither short-sale nor borrowing restrictions.

Use column vectors for dollar holdings ${\theta }_t$ and put $b=\mu -r1$, $h={\sigma }^{-1}b$ and $V=\sigma {\sigma }^T$. The normalized state-price density isThe vector $h$ is the market price of risk. The density $e^{rt}{\zeta }_t$ defines an equivalent risk-neutral measure, under which $W_t^{\mathbb{Q}}=W_t+ht$ is Brownian motion and every stock has drift $r$. A terminal claim $H$ with finite state-price cost is priced at ${\zeta }_t^{-1}\mathbb{E}[{\zeta }_{T}H\mid {\mathcal{F}}_t]$. Nonsingularity of $\sigma$ and the natural Brownian market information make this a complete market; in particular the optimal nonnegative payoff below can be replicated.

Let $q=n^{-1}1$ and $v={\sigma }^{T}q$. Apply the Itô formula to each log price, whose return variance is $V_{ii}$:Summing and dividing by $n$ proves the geometric stock index formulaThus $J_t=J_0\exp (v^T{W}_t+a_{J}t)$, where $a_J=q^T\mu -\mathrm{tr}(V)/(2n)$. The factor $1/n$ applies to the whole bracket. The geometric average need not itself be a self-financing portfolio; its instantaneous drift is $a_J+|v{|}^2/2$.

If ${\theta }_t^i$ is the dollar amount invested in stock $i$, the bank account holds $w_t-1^T{\theta }_t$. The self-financing portfolio equation isEquivalently, ${\pi }_t={\theta }_t/w_t$ gives $d{w}_t/w_t=(r+{\pi }_t^{T}b)dt+{\pi }_t^T\sigma d{W}_t$. For share-holding notation replace ${\theta }_t^i$ by $S_t^i$ times the number of shares.

Static expected utility maximization uses the terminal budget $\mathbb{E}[{\zeta }_T{w}_T]\le w_0$. Since $J_T$ is an exogenous positive benchmark, differentiating $U(x/J_T)$ in $x$ gives $J_T^{-1}{U}^{\prime }(x/J_T)$. An interior Lagrange multiplier $\lambda >0$ for the budget therefore givesFor CRRA utility, write $p=1/R$. Since $U^{\prime }(y)=y^{-R}$, solving this marginal equation givesEvery positive lognormal moment is finite here, so the budget determines the constant uniquely. PutThenThis is also a sufficiency condition: concavity gives $U(X/J_T)-U(w_T^{\ast }/J_T)\le \lambda {\zeta }_T(X-w_T^{\ast })$, and taking expectations and using the budget inequality proves global optimality.

To find the whole optimal process, set $c=a_J-r-|h{|}^2/2$ andSince ${\zeta }_t{J}_t=J_0\exp ((v-h{)}^T{W}_t+ct)$, the moment-generating function of a normal distribution givesPricing the terminal payoff yields the explicit benchmark-relative power-utility portfolio wealthIts proportional diffusion coefficient is $\ell =ph+(1-p)v$. Since the wealth diffusion coefficient for dollar fractions is ${\sigma }^T\pi$, replication requiresUsing ${\sigma }^{-T}h=V^{-1}b$ gives the requested fixed proportions:The drift matches the self-financing portfolio equation as well. Indeed, $\ell =h+(1-p)(v-h)$, so the drift of $\log w^{\ast }$ from the explicit formula isConsequently an equivalent form of the solution isThe bank account fraction is $1-1^T{\pi }^{\ast }$, with negative values representing borrowing. The first stock term is ordinary power-utility risk exposure, while the second hedges the random benchmark. At the logarithmic limit $R=1$ the benchmark term disappears, consistently with $\log (w_T/J_T)=\log w_T-\log J_T$.