Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 40 (Optimal Investment)

Wealth and admissibility. The dollar holding $\theta$ earns the risky return, while $w-\theta$ earns the continuous-time bank account return. Removing consumption therefore givesHere $\theta$ is a dollar amount, not a number of shares; the number of shares is ${\theta }_t/S_t$. Both controls must use available information: take $\theta$ predictable and $c\ge 0$ progressively measurable, withalmost surely on every finite interval. Require a well-defined objective and an admissible trading strategy satisfying $w_t\ge 0$. At zero portfolio wealth this excludes continued risky gambling or positive consumption. The state-price budget constraint rules out doubling strategies. Throughout the diffusion calculations take $\sigma \ne 0$, positive discount $\rho$, positive decay $\lambda$, and finite value; degeneracies are discussed where they affect the conclusions.

Satisfaction and dynamic programming. Write the consumption satisfaction stock asThe product rule gives, almost everywhere in time,This state is a finite-variation process, so it has no quadratic covariation with portfolio wealth. Applying the Itô formula to the discounted value function over a short interval, then using dynamic programming, gives the interior HJB equationAssuming $V_{ww}<0$, completing the square givesThe last supremum is zero if $V_{\xi }-V_w\le 0$ and infinite otherwise. Thus the gradient constraint for unbounded consumption is $V_{\xi }\le V_w$, and $c=0$ wherever the inequality is strict. Since there is no direct penalty for a very large consumption rate, an active boundary can involve singular consumption control. In that relaxed interpretation the HJB equation isWith ordinary rate controls, this describes the supremum and its limiting transfer policy; it does not promise that an instantaneous transfer is attained by a finite rate.

Power reduction. An additive constant in the utility function only adds a control-independent constant divided by $\rho$ to the value, so normalize $U(\xi )={\xi }^{1-R}/(1-R)$. Put $p=1-R$ and $x=w/\xi$ for $\xi >0$. Scaling portfolio wealth, consumption satisfaction and the controls by the same positive number gives the wealth-to-satisfaction reductionConsequently the reduced HJB equation isIn the strict waiting region the first expression vanishes and consumption is zero. On a transfer region the second vanishes; integrating it gives $v(x)=K(1+x{)}^p$. This reflects preservation of $w+\xi$ during an instantaneous wealth-to-satisfaction transfer.

Why a waiting threshold is expected, and its qualification. The gradient constraint for unbounded consumption compares the benefit of increasing consumption satisfaction with the opportunity cost of spending financial portfolio wealth. When consumption satisfaction is already large relative to cash, waiting lets consumption satisfaction decay while financial portfolio wealth earns returns; consuming immediately can be wasteful. Homogeneity makes the comparison depend only on $x$. For fixed $s=w+\xi$, joint concavity of the value function makes $q_s(a)=V(s-a,a)$ concave in financial portfolio wealth $a$. It is nondecreasing because an immediate transfer can reproduce any smaller financial allocation. Consequently its derivative $V_w-V_{\xi }$ is nonnegative and nonincreasing in $a$: a strict waiting region, if present, starts at zero financial portfolio wealth and ends at a single transfer boundary. Homogeneity makes the corresponding boundary a ratio $x_{\ast }$. In the usual finite-boundary regime this gives $c=0$ for $x<x_{\ast }$, and transfers push a larger ratio down towards $x_{\ast }$. The condition is a comparison of marginal values, not the ordinary formula $c=(V_w{)}^{-1/R}$: current utility function here depends on consumption satisfaction, not on current consumption.

A positive threshold is not guaranteed by the printed hypotheses alone. A useful sufficient local test illustrates the intended argument. Let $a=\rho +\lambda p>0$. At zero portfolio wealth,Starting with a small extra portfolio wealth $\epsilon$, holding it in the continuous-time bank account until a fixed time $t$, then transferring it into consumption satisfaction, has right derivative in $\epsilon$ at zero equal toTherefore, if $\rho <r+\lambda R$, the portfolio wealth marginal value is strictly larger than the consumption satisfaction marginal value at zero. With the usual continuity of marginal values, there is a positive interval on which the gradient constraint for unbounded consumption is strict. The fixed-total-resource concavity argument then gives the threshold structure.

For a concrete counterexample to an unconditional positive threshold, take $R=1/2$, $\lambda =1$, $r=\mu =1/10$, $\sigma =1$ and $\rho =2$. There is zero market price of risk. The relaxed value isIndeed $F_{\xi }=F_w$, $F_{ww}<0$, and the optimized waiting residual, after division by ${\xi }^p$, isbecause $(r+\lambda )/a<1$ and the integrand is decreasing. The Itô formula gives an upper bound by $F$, while transferring all portfolio wealth into consumption satisfaction over intervals tending to zero attains that bound in the limit. Hence the transfer boundary is $x_{\ast }=0$ in this example. The positive-threshold explanation needs a parameter regime supporting a genuine waiting region. For $R>1$, even the zero-wealth value is finite only if $\rho >\lambda (R-1)$; otherwise decaying consumption satisfaction gives value $-\infty$.

The joint generator. With $\theta$ again measured in dollars, portfolio wealth obeysIts noise and the factor noise are driven by the same Brownian motion. Their quadratic covariation is $\sigma (x)\theta \alpha (x)dt$, so the diffusion generator has a cross derivative. Dynamic programming and the Itô formula giveAll coefficient functions in this formula are evaluated at $x$. The cross derivative is essential: it gives intertemporal hedging demand. Normalizing CRRA utility as $U(c)=c^p/p$, where $p=1-R$, the two optimizations give, at nonzero volatility,and henceHere $p(-1/R)=1-1/R$.

Homogeneity and the reduced equation. Scaling initial portfolio wealth and both controls preserves the portfolio wealth constraint and multiplies the objective by the positive number $b^p$. ThusThis expression is valid for both signs of $p$: $V$ is negative when $R>1$, but its portfolio wealth derivative is positive. Its derivatives areSubstitution yieldsThe resulting feedback isThe first portfolio term is myopic, and the second is intertemporal hedging demand.

Constant market price of risk. If $\mu -r=\sigma \kappa$ and volatility is nonzero, the portfolio term becomes $p(\kappa f+\alpha f^{\prime }{)}^2/(2Rf)$, so the magnitude of stock volatility disappears. Applying the power transformation of a complete-market investment equation $f=g^R$ cancels the squared-gradient terms and gives the further reductionFor ${\gamma }_M>0$ its economic solution is $g=1/{\gamma }_M$. ThusTo see why this solves the investment-consumption problem, optimize directly over Brownian portfolio exposures, writing $y=\sigma (x)\theta$. The portfolio wealth equation becomes $dw=ydW+(rw+\kappa y-c)dt$, which no longer contains $X$. With nonzero volatility the same admissible exposure processes are available for every factor state, so the attainable wealth-consumption pairs, and therefore the value, are exactly those of the Merton consumption-investment problem. This also excludes extraneous solutions of the linear equation without imposing artificial factor boundary data.

The printed boundedness assumptions do not ensure nonzero volatility or a finite value. The unsimplified HJB equation remains the correct control equation at a zero of $\sigma$. There the hedge term vanishes; if $\mu \ne r$, the riskless excess return gives arbitrage with unrestricted holdings. Under $\mu -r=\sigma \kappa$, a zero-volatility state offers only the bank exposure at that instant. For example $\sigma \equiv 0$, $\mu \equiv r$ satisfies this relation for any chosen $\kappa$, but its value uses ${\gamma }_0=[\rho -pr]/R$, not a fictitious nonzero risk premium. The constant-value formula using $\kappa$ presupposes access to the Brownian exposure, with sufficient integrability for the corresponding holdings. Additive utility constants again only shift $V$ by a constant divided by $\rho$.

Available portfolio wealth and the ruin boundary. Put $h=\bar{r}D$, the constant interest payment on the fixed loan. The loan principal is already included in available portfolio wealth; it is not a growing portfolio holding. ThereforeIn particular the interest outflow is $\bar{r}D$, not $(\bar{r}-r)D$. Writing $w=x+D$ would instead give net portfolio wealth drift $rx+(\mu -r)\theta -(\bar{r}-r)D-c$, which explains the distinction.

Let ${\tau }_0$ denote the ruin time, to avoid confusing it with a fixed terminal horizon. The objective stops at ${\tau }_0$; consequently the absorbing boundary is $V(0)=0$, without an obligation to keep financing the loan after ruin. Dynamic programming gives, for $w>0$,For increasing strictly concave value, put $z=V^{\prime }(w)>0$ and use inverse marginal utility $I$. The optimal controls and the optimized HJB equation arewhere $\tilde{U}(z)={\sup }_{c\ge 0}[U(c)-zc]$. For CRRA utility with $0<R<1$, write $p=1-R$ and $q=1-1/R<0$. Then

Dualization and the printed constant. Use the convex wealth-variable Legendre dualAt an interior maximizing portfolio wealth, $J^{\prime }=-w$, $J^{\prime \prime }=-1/V^{\prime \prime }>0$ and $V=J-z{J}^{\prime }$. The dual HJB equation is the linear Euler differential equationA trial power $z^t$ givesDirect substitution givesThe PDF prints an additional factor $1/2$ before ${\kappa }^2/(2R)$ in its definition of ${\gamma }_M$. That printed definition is inconsistent with its own identity for $Q(q)$. The expression above is the one used here; assume this corrected ${\gamma }_M>0$.

Solution when $\kappa \ne 0$. Assume positive discount $\rho$, nonzero $\sigma$, and $h>0$. Let $m>0$ and $n<0$ be the two roots of $Q$:Since $Q(q)<0$, $n<q<0<m$. Put $C=R/(p{\gamma }_M)$. For $r\ne 0$, the general interior solution isThe appropriate large-wealth condition is the Merton consumption-investment problem boundIndeed any original control consumes $c+h$ in the debt-free comparison model until ruin, and $U(c+h)\ge U(c)$. Dualizing this bound gives $0\le J(z)\le C{z}^q$. Because $n<q$, convexity and this upper bound force $B=0$ as $z\downarrow 0$.

At the other endpoint portfolio wealth reaches zero. If $z_{\ast }=V^{\prime }(0+)$, the dual ruin boundary with debt service requiresSolving these two equations givesFor $r>0$, $m>1$, and for $r<0$, $0<m<1$, so the quantity defining $z_{\ast }$ is positive in either case. These formulas determine the entire value. For each $w>0$, choose the unique $z\in (0,z_{\ast })$ satisfyingThenBoth terms in the bracket are positive, even when $r<0$ and $A<0$. Thus $J^{\prime \prime }>0$, $-J^{\prime }$ decreases from infinity to zero as $z$ increases, and the portfolio wealth inversion really is unique. The extended dual is continuously differentiable and convex.

There is no additional condition $J^{\prime \prime }(z_{\ast }^{-})=0$. In factAvailable portfolio wealth is killed at zero; the portfolio can have a nonzero limiting volatility immediately before ruin. Imposing a reflecting-boundary or zero-curvature condition would solve a different problem.

For $r=0$, the linear forcing resonates with the root $m=1$. Put $b=\rho +{\kappa }^2/2=Q^{\prime }(1)$. The dual and boundary constants instead areExtend by zero for $z\ge z_{\ast }$. Hereand the controls remain $c=z^{-1/R}$ and $\theta =\kappa z{J}^{\prime \prime }/\sigma$, with $J^{\prime \prime }=h/(bz)+Cq(q-1)z^{q-2}>0$.

Verification and transversality. The candidate is nonnegative, increasing, strictly concave and zero at ruin. Its HJB equation makes the discounted value plus accrued utility a local supermartingale for every admissible control, and a local martingale for the stated feedback. Localization at positive lower and finite upper portfolio wealth levels gives the finite-horizon comparison. The investment value transversality condition follows from the same debt-free bound: applying the Itô formula to $w^p$ and maximizing its risky term givesThe nonnegative consumption and debt-service drifts only decrease this bound. Thus the expected terminal candidate tends to zero. The feedback has at most linear growth, including a finite limit as portfolio wealth decreases to zero; stopping it at ruin gives an admissible policy. Letting localization levels and then the horizon tend to their limits proves that the candidate is the value, rather than just a formal dual solution. If $D=0$, the absorbing-debt boundary disappears and the ordinary Merton consumption-investment problem formula is recovered.

Zero market price of risk. With $\sigma \ne 0$ and $\kappa =0$, the dual equation is first order. If $\rho >r$, the preceding formulas remain valid with $m=\rho /(\rho -r)$ for $r\ne 0$, and the logarithmic formula with $b=\rho$ for $r=0$; there is no $B$ term and ${\theta }^{\ast }=0$. The same boundary and transversality argument verifies this deterministic consumption policy.

The remaining finite-value case has $r>0$ and $pr<\rho \le r$. Put ${\gamma }_0=(\rho -pr)/R$, $w_c=h/(rR)$ andThe correct convex dual and its corresponding value areThe two value branches have the same value and derivative at $w_c$. Above $w_c$, hold no stock and consume ${\gamma }_0(w-h/r)$; the surplus over $h/r$ grows at rate $r-{\gamma }_0\ge 0$. If $\rho =r$, the lower branch is attained by zero stock holding and constant consumption $ph/R$: portfolio wealth solves $\dot{w}=r(w-w_c)$ until ruin, and direct integration gives $z_{c}w$.

If $\rho <r$ and $0<w<w_c$, the lower branch is a supremum attained in a limit of increasingly rapid fair stock lotteries between zero and $w_c$, followed by the upper-branch policy on success. The success probability tends to $w/w_c$ and the fixed service cost during the lottery tends to zero. This is possible because unrestricted dollar holdings in the nonzero-volatility stock produce a fair Brownian motion exposure even when its excess drift is zero. The supporting linear branch has optimized waiting residual $(r-\rho )z_c(w-w_c)<0$; the fast lotteries, rather than a finite feedback optimizer, supply the missing control limit. The piecewise candidate is concave, has nonpositive waiting residual everywhere, and the preceding moment bound still supplies an upper-bound verification. This degenerate case can have a supremum without an ordinary maximizing strategy.

Pricing kernel and replication. In the nondegenerate Black-Scholes model, put $\kappa =(\mu -r)/\sigma$ and normalize the state-price density by ${\zeta }_0=1$. The process isThe density $e^{rt}{\zeta }_t$ changes probability to the risk-neutral measure. Under that measure $W_t+\kappa t$ is a Brownian motion and the stock drift is $r$. Thus an integrable contingent claim $H$ has time-$t$ priceIn the usual augmented natural Brownian filtration, the Brownian martingale representation theorem supplies a replicating strategy; this is the complete market assumption. For a nonnegative admissible trading strategy without intermediate consumption, the state-price budget constraint is $\mathbb{E}[{\zeta }_T{w}_T]\le w_0$, with equality for a fully invested replicated claim. In particular

Feasibility and the largest slope. First take the intended regime $r>0$, $T>0$, $w_0,S_0>0$, and the usual nonnegative portfolio wealth constraint. The terminal wealth floor isFor any feasible claim the state-price budget constraint impliesTherefore $\alpha \le w_0/S_0$. Conversely, when $0\le \alpha \le w_0/S_0$, hold $\alpha$ shares and put the remaining $w_0-\alpha S_0$ in the continuous-time bank account. Its terminal portfolio wealth isHenceAt equality, $\xi =\bar{\alpha }{S}_T$ has cost exactly $w_0$. Positivity of the state-price density forces $w_T=\bar{\alpha }{S}_T$ almost surely: any strict improvement would cost more. Invest all initial portfolio wealth in $w_0/S_0$ shares and hold them until $T$.

Optimal payoff below the feasibility limit. For $\alpha <\bar{\alpha }$ the floor is strictly positive, and its price is strictly less than $w_0$. Assume the utility function is increasing, differentiable and strictly concave, satisfies the Inada conditions, and has the integrability needed for the finite-budget optimization. These are the usual hypotheses implicit in using inverse marginal utility. For each positive multiplier $\lambda$, maximizeseparately in every state. Its derivative decreases through zero at $I(\lambda {\zeta }_T)$, so the floored marginal utility optimizer isThe multiplier is characterized byUnder the stated integrability hypotheses the left side is continuous and decreasing, tends to the floor cost as $\lambda \uparrow \infty$, and tends to infinity as $\lambda \downarrow 0$. It is strictly decreasing wherever it exceeds the floor cost: on the event where the inverse-marginal-utility payoff exceeds the floor, a larger multiplier strictly reduces that payoff. Therefore the budget determines a unique finite multiplier. For CRRA utility the inverse marginal utility is $I(y)=y^{-1/R}$; lognormal moments provide the needed integrability.

For completeness, pointwise maximality gives, for any feasible competing terminal portfolio wealth $Y$,Taking expectations and using $\mathbb{E}[{\zeta }_{T}Y]\le w_0=\mathbb{E}[{\zeta }_T{w}_T^{\ast }]$ proves optimality. Strict concavity gives uniqueness of the terminal claim. Its price processis nonnegative and starts from $w_0$; claim replication therefore turns the payoff optimizer into an admissible portfolio.

What the missing interest-rate hypothesis changes. The PDF does not explicitly assume $r>0$ or give the utility and admissibility hypotheses above. These omissions matter. At $r=0$, under nonnegative admissibility, the largest feasible slope remains $w_0/S_0$: for larger slopes the positive-part floor costs strictly more than $w_0$, since $S_T$ has support $(0,\infty )$. But every $\alpha \le w_0/S_0$ already gives floor cost exactly $w_0$, so the only feasible terminal claim is $\xi$. There is no spare budget for an inverse-marginal-utility improvement. For example $\alpha =0$, $\kappa \ne 0$ and CRRA utility make $\mathbb{E}[{\zeta }_T\max \{w_0,I(\lambda {\zeta }_T)\}]>w_0$ for every finite $\lambda$. The prescribed positive finite multiplier then does not exist.

For negative $r$, nonnegative admissibility requires the effective floor ${\xi }_{+}=\max \{\xi ,0\}$. Define its costThis is continuous and convex. For $\alpha \le w_0/S_0$ it equals $w_0{e}^{-rT}+\alpha S_0(1-e^{-rT})$, exceeding $w_0$ below the endpoint. At ${\alpha }_0=w_0/S_0$, $G({\alpha }_0)=w_0$ and $G^{\prime }({\alpha }_0)=S_0(1-e^{-rT})<0$. Above that endpoint the lognormal stock gives strict convexity, and $G(\alpha )\to \infty$. Thus there is a unique second root ${\alpha }_1>{\alpha }_0$ of $G({\alpha }_1)=w_0$, the feasible slopes are $[{\alpha }_0,{\alpha }_1]$, and the largest is ${\alpha }_1$. Equivalently, for $\alpha >{\alpha }_0$ the floor price is $\alpha$ times the European call option price with strike $S_0-w_0/\alpha$. At the upper endpoint replicate ${\xi }_{+}$; in the interval with strict budget slack the same floored marginal utility optimizer applies with ${\xi }_{+}$.

If instead portfolio wealth may be negative and utility is defined on all real portfolio wealth, the floor itself has its affine replication cost: at $r=0$ every slope is feasible, and at negative $r$ every $\alpha \ge w_0/S_0$ is feasible. There is then no largest finite slope. The intended stock-only endpoint and strict-slack optimizer use positive interest and standard nonnegative admissibility.