An Extension of Some Results Due to Cox and Leland

We investigate an optimal portfolio allocation problem between a risky and a risk-free asset, as in [1]. They obtained explicit conditions for path-independence and optimality of allocation strategies when the price of the risky asset follows a geometric Brownian motion with constant asset characteristics. This paper analyzes and extends their results for dynamic investment strategies by allowing for non-constant returns and volatility. We adopt a continuous-time approach and appeal to well established results in stochastic calculus for doing so.


Introduction
Beginning with [2], diffusion processes have been the standard for modeling asset returns, despite empirical evidence that returns are not normally distributed.Dynamic asset allocations based on these processes have been prominent, for example see [3,4] and [5,6] provide a survey of this topic to the early 1990s.
Based on the work of [7] and [1,2] derived criteria for controls to optimize an investor's objectives.They restricted the case to a portfolio with only two assets, a risky one paying no dividends and a risk-free one with the price of the risky asset following a geometric Brownian motion process.
The restriction to a single risky asset involves no significant loss of generality since the setting can be taken as a mutual fund.[8] shows that if geometric Brownian motion models are adopted, the separation theorem of mutual funds can be applied: in a portfolio problem of allocating wealth across many risky assets, the problem can be reduced to that of choosing amongst combinations of a few funds formed from these assets.
However, [1] assume constancy of asset characteristics, which is restrictive.In addition, their use of the discretetime binomial model, converging in continuous-time by limiting the time intervals, is cumbersome and detracts from the economics of the issue.Nonetheless, their result of efficiency of path-independent strategies has been extensively cited in the literature, especially in the studies for hedge funds.[6] claim that, although the results presented by [1] were not well known at that time, path-independence of a strategy is often necessary for such a dynamic strategy to be optimal.In their study of hedge fund performance, when constructing a payoff function [9] stipulate that payoff must be a path-independent non-decreasing function of the index value, derived from [1].[10] extend the relevance of path-independence to the case when prices of risky assets follow an exponential Lévy process.On the other hand, path-independent strategies are not always attractive.[11] show that pathdependent strategies are suboptimal for risk-averse investors when the pricing model is a function of the risky asset price at terminal time.However, and not surprisingly, path-dependent strategies are preferred if the pricing model of the risky assets is itself path-dependent.
In this paper, we extend the results of [1] for more general asset return processes.We assume that the price of the riskless asset grows deterministically at a variable interest rate, and the price for the risky asset follows geometric Brownian motion, with both the drift and volatility being variable over both time and the stock price.Such a model mitigates some of the difficulties in explaining long-observed features of the implied volatility surface for option pricing.Hence it is possible to model derivatives more realistically.
Detailed references for such stochastic processes may be found in [12,13].Without loss of generality, we consider a world with a risky asset and a riskless asset, as in [1].We establish our results by application of a continuous-time approach and the use of partial differential equations (PDEs), rather than through stochastic calculus.We obtain explicit results for general dynamic strategies which allow for uncertainty as modeled in diffusion processes.These results extend those of [1].
Our results are concerned with maximizing some form of investor utility.In most former studies, when dealing with utility maximization, a particular form of utility function is specified.For example, a HARA utility is considered in [2]; an iso-elastic utility in [14]; and a CRRA power utility in [15].While the Hamilton-Jacobi-Bellman equation is a popular tool for utility maximization problems, [16] criticizes the use of an arbitrary "bequest function" as the boundary condition in [2]; the boundary behavior around zero terminal wealth may be inconsistent with his "bequest function".In our approach, the boundary condition is taken as an arbitrary utility function of terminal wealth, thereby avoiding this problem.[17] gives a more detailed review of expected utility maximization for strategies involving a risky and a riskless asset.Although he does not approach this problem in full generality, using the example of a power utility function, he shows how other cases can be solved with little effort.
In the working papers by [18,19], for a given utility function, the Feynman-Kac formula is used to find controls satisfying certain PDEs for utility maximization.We show that the Feynman-Kac formula can generally provide the solution to a control in terms of its terminal value.We also show that the terminal value satisfies some concave utility, without specifying its functional form.
The paper is organized as follows.First, for simplicity, we assume no cash flows, which corresponds to the pure "bequest" case of [7].This assumption is then later relaxed. Section 2 extends Proposition 1 of [1] for necessary and sufficient conditions for an investment strategy to be feasible, where the controls of the strategy are given as functions of time and the value of the risky asset. Section 3 develops necessary and sufficient conditions for an investment strategy to be path-independent, with controls defined as functions of time and the value of the portfolio (wealth).These results extend Proposition 2 of [1]. Section 4 establishes necessary and sufficient conditions for an investment strategy to optimize a concave utility, while imposing no constraints on portfolio allocations.This extends Proposition 3 of [1]. Sections 5 and 6 consider the case of non-negative allocations. Section 7 considers the situation when cash with drawals are admissible. Section 8 concludes.Then, when there are no cash withdrawals or injections (i.e. the strategy is self-financing),

Controls Based on
This implies that These equalities are consistent with the conditions of Proposition 1 of [1].Note that the expected return on the risky asset  does not appear in 5.
Differentiating Equation ( 5) with respect to s yields:  , W t.

On this basis
Hence the left hand side of Equation ( 7) becomes For the right hand side of Equation ( 7): , Hence we have: viewed as a function of wealth, satisfies: Remark: This is the same as proposition 2 of [1] when 0.
W  

Controls That Are Compatible with a Concave Utility
Consider a control that maximizes an expected utility of terminal wealth at time : for some utility function and where is the physical measure under the process in 1.

  u W P
Then the Equation ( 5) is, regarded as a parabolic partial differential equation: The solution is given by the Feynman-Kac formula.The solution, expressed as a stochastic expectation, is: Here The expectation is taken with respect to the risk neutral process: Remark: It is known there are various conditions for the Feynman-Kac formula to hold, which are set out in the Appendix.A condition that q be bounded above zero is not onerous, as we are dealing with a risky asset.Some of these conditions may be relaxed significantly, and will be discussed in a further paper.
The probability density  , ; ,  s t S T


of s at time is governed by the Kolmogorov backward Equation The conditions for these results are also set out in the Appendix.
The critical implication of 9 is that and therefore where the expectation R E is taken with respect to the process:

Optimization
Thus for the utility function it suffices to find so as to maximize: Here the expectation P E and density  relate to the physical stock process: rather than to 10.This is subject to the initial condition: where the expectation is subject to the risk neutral process in 10.
 The resulting variation in is, to the second order:  with  being chosen to satisfy the initial condition 13.
The second order condition is so that a concave utility is required.The general solution for 0, is then given by 9.In the general case with r   not constant, we thus have the following extension of the existential results of Proposition 3 of [1]: ,0; , Remark: This is without qualification as to the existence of a solution to 8. In the case that   In the case that is given, the condition 0 S   is sufficient to determine However none of these conditions is mentioned in Proposition 3 of [1]. .
r   In this case, it is well known that 0 ln S s is normally distributed, with mean Hence 14 becomes: Differentiating with respect to we also have ,

Extension of Utility Characterization
It is of interest to consider whether the allocation to the risky asset is non-negative under more general conditions than indicated in Proposition 3 of [1]. .Then a strategy can be found to optimize a concave utility with the strategy is given by 14 and the Kac-Feynman formula 9. We also note the relation 12, which shows that if at time . T Differentiating 14 with respect to we have: The variant of Girsanov's theorem, as proved in the Appendix, confirms this result.
Remark: The conditions are sufficient, but by no means necessary.The Appendix shows that the density


of the risk neutral process for ln : with stochastic ,  is central to this issue.In particular, if the risk premium r   is non-stochastic, and  is concave in , Z then the result also holds.This situation may be investigated by noting that   , Z T  satisfies a parabolic PDE, which can in turn be investigated by the eigenfunctions of the operator

Constrained Strategies
The above discussion does not constrain the allocations to both the risky asset and the riskless asset to be nonnegative, which is often a requirement in practice.For this to apply, we have the additional constraints on terminal wealth: then Proposition 4 provides conditions for To provide that we need to have the terminal condition: If this holds, and 0, then Equation ( 12) implies: To ensure that 15 holds, consider the Lagrangian: The variation in  is, to the second order: .
The first order condition is thus: which can be written: and thus integrating over : The second order condition is as before: This leads to the following result: the strategy given by the Kac-Feynman formula 9, provides optimality over nonnegative allocations to the riskless asset, only if there is a solution of: for some   Remark: These are weaker conditions than provided in Proposition 3, as we are seeking optimality over a smaller class of allocations.Even weaker conditions may be found if the class of allocations is restricted to where both the risky and riskless assets are constrained to be non-negative.

Allowance for Cash Flows
The previous relations can be extended to accommodate portfolios with cash withdrawals.Let us now consider the situation when an investor is allowed to withdraw from their investment, at a rate .Such as before, we discuss the cash withdrawn from the portfolio in two cases, a function of price of the risky asset and time, , or a function of total wealth and time, K W t .
Total wealth then obeys the generalised relation:

Controls That Are Functions of the Value of the Risky Asset and Time
In analogy with section 2 consider the case where the controls, , G H and , are all functions of K   , s t , where the process of s is the same as in 1.
Allowing for cash withdrawals, 2 generalizes to This implies that and the same condition as in 4, which is consistent with Proposition and hence 5 generalizes to: It may be shown similarly that 7 generalizes to: Now we can formalize the above results as: Proposition 6: Necessary and sufficient conditions for the differentiable functions , and to be the controls of a self-financing investment strategy are that: for all s and t .Notice that Proposition 1 of [1] is a special case of this generalized form with a constant diffusion for price of the risky asset, that is 0 s   .

Controls That Are Functions of the Value of the Portfolio Wealth and Time
In analogy with section 3, consider the situation when the controls, and K , are functions of .G

  , W t
As is a control, we also have: Then Equation ( 7) can be shown to generalize to: viewed as a function of wealth, satisfies:

Compatibility with Investor Objectives
We now consider if the processes of controls and are compatible with rational investor objectives.
where, without loss of generality, we write This is again a parabolic PDE in , and the Feynman-Kac formula can be applied to find a solution as: but now the discount factor includes: As before, the wealth W is completely determined by the terminal wealth   S  , along with the control   , .k s t In the case with cash withdrawals are admissible, we consider not only the utility from terminal wealth for an investor, but also the utility from consumption financed by the cash withdrawals . Therefore, the problem of choosing optimal portfolio and consumption rules for an investor over a period of is to maximize an aggregate utility of the following form: is the utility from consumption, with .The initial and terminal times are specified at and t , as is the initial condition that some initial stock price The expectation P E is specified as before in Section 4 for the physical stock process.
The Lagrangian is then given by: This optimization problem is exactly of continuous stochastic control [20,VII.10].Define an optimal expected value function given the stock price s at time : The process terminates at time , at which time the utility of terminal wealth is assessed, with the boundary condition The fundamental PDE for the control is: However, we follow an alternative, but simpler, approach.Given consider a small variation in say localized at time for some constant ,  which induces a variation

T d t
And thus:  The second order condition in  is: for some constant > 0.
 Remark: As K c is a decreasing function in , K this implies that cash withdrawals should increase in the wealth achieved, but should decrease where such wealth is less likely to be achieved.This corresponds to the conditions contained in Proposition 4 of [1].

Conclusions
In this paper, we address two related issues, based on the work by [1].
First, we examine the characteristics of optimal portfolio controls.Rather than assuming constant expected returns and volatility, we consider the more realistic situation with the expected return and volatility of risky assets are non-constant, or even stochastic.
Second, we consider whether a given investment strategy is consistent with expected utility maximization.We apply several techniques of the calculus of variations to show that, under mild conditions, optimal portfolio controls are compatible with some concave utility function.Unlike most papers in the literature, we do not specify a particular form of utility function.
It would be interesting to extend these results to more general asset models, for example where the risky asset follows a jump diffusion process, or where volatility of the return on the risky asset is itself a stochastic process.    will be taken assuming all other variables are held constant.We summarize various conditions on and q for results to hold.Most of these are cited from [12] and the references therein.

 
is globally bounded from zero across   This holds under the following conditions [12, Theorem 5.15]:  C1.  C2.

Feynman-Kac Formula
Consider the Equation ( 5), regarded as a parabolic partial differential Equation: where are functions of , , p q r   , , s t as in 5.This is subject to the boundary condition In this section, we allow to be a function of both r  

, . s t
The solution is given by the Feynman-Kac formula, expressed as a stochastic expectation:

Girsanov's Theorem
Both the physical process 1 and risk-neutral process 10 for .
s can be simplified under Itô's lemma by making the transformation ln z s  Thus in logarithmic terms the physical process can be described by:  

Z Z
This can be shown directly from 31.However we take an approach that illustrates a relationship with Girsanov's theorem, and allows a generalization to the case where the parameters are stochastic.

  
Make the transformation  .
We then have from 29: (so that the risk premium r   is not stochastic).Then: It may be concluded from comparing this equation with 30 that    , ; , , ; ,  x t X T z t X R T     , so that changing variables: This last condition clearly holds in the case of non-stochastic parameters as in 31.However it is a condition on the risk neutral process only, and may hold in other cases where the stock volatility  is stochastic.
and also the Kolmogorov forward equation of Proposition 3 of[1].

2 ,
  are non-stochastic (i.e.independent of s ) and

Proposition 5 :
Given a concave utility  ,


, to the first order in d :

8
Hence we have the following result.Proposition Given concave utility functions for terminal wealth   u W and for consumption   c K , and a terminal wealth with density the optimal cash flow control is given by

Stock Price s
.