BEHAVIORAL PORTFOLIO SELECTION IN CONTINUOUS TIME

This paper formulates and studies a general continuous-time behavioral portfolio selection model under Kahneman and Tversky's (cumulative) prospect theory, featuring S-shaped utility (value) functions and probability distortions. Unlike the conventional expected utility maximization model, such a behavioral model could be easily mis-formulated (a.k.a. ill-posed) if its different components do not coordinate well with each other. Certain classes of an ill-posed model are identified. A systematic approach, which is fundamentally different from the ones employed for the utility model, is developed to solve a well-posed model, assuming a complete market and general Itô processes for asset prices. The optimal terminal wealth positions, derived in fairly explicit forms, possess surprisingly simple structure reminiscent of a gambling policy betting on a good state of the world while accepting a fixed, known loss in case of a bad one. An example with a two-piece CRRA utility is presented to illustrate the general results obtained, and is solved completely for all admissible parameters. The effect of the behavioral criterion on the risky allocations is finally discussed.     

OPTIONED PORTFOLIO SELECTION: MODELS AND ANALYSIS

The mean‐variance model of Markowitz and many of its extensions have been playing an instrumental role in guiding the practice of portfolio selection. In this paper we study a mean‐variance formulation for the portfolio selection problem involving options. In particular, the portfolio in question contains a stock index and some European style options on the index. A refined mean‐variance methodology is adopted in our approach to formulate this problem as multistage stochastic optimization. It turns out that there are two different solution techniques, both lead to explicit solutions of the problem: one is based on stochastic programming and optimality conditions, and the other one is based on stochastic control and dynamic programming. We introduce both techniques, because their strengths are very different so as to suit different possible extensions and refinements of the basic model. Attention is paid to the structure of the optimal payoff function, which is shown to possess rich properties. Further refinements of the model, such as the request that the payoff should be monotonic with respect to the index, are discussed. Throughout the paper, various numerical examples are used to illustrate the underlying concepts.     

ON OPTIMAL INVESTMENT FOR A BEHAVIORAL INVESTOR IN MULTIPERIOD INCOMPLETE MARKET MODELS

We study the optimal investment problem for a behavioral investor in an incomplete discrete‐time multiperiod financial market model. For the first time in the literature, we provide easily verifiable and interpretable conditions for well‐posedness. Under two different sets of assumptions, we also establish the existence of optimal strategies.     

ERRATUM TO “BEHAVIORAL PORTFOLIO SELECTION IN CONTINUOUS TIME”

We fill a gap in the proof of a (rather critical) lemma, Lemma B.1, in Jin and Zhou (2008: Math. Finance 18, 385–426). We also correct a couple of other minor errors in the same paper.     

MARKET SELECTION OF FINANCIAL TRADING STRATEGIES: GLOBAL STABILITY

In this paper we analyze the long-run dynamics of the market selection process among simple trading strategies in an incomplete asset market with endogenous prices. We identify a unique surviving financial trading strategy. Investors following this strategy asymptotically gather total market wealth. This result generalizes findings by Blume and Easlcy (1992) to any complete or incomplete asset market.     

TRACTABLE ROBUST EXPECTED UTILITY AND RISK MODELS FOR PORTFOLIO OPTIMIZATION

Expected utility models in portfolio optimization are based on the assumption of complete knowledge of the distribution of random returns. In this paper, we relax this assumption to the knowledge of only the mean, covariance, and support information. No additional restrictions on the type of distribution such as normality is made. The investor’s utility is modeled as a piecewise‐linear concave function. We derive exact and approximate optimal trading strategies for a robust (maximin) expected utility model, where the investor maximizes his worst‐case expected utility over a set of ambiguous distributions. The optimal portfolios are identified using a tractable conic programming approach. Extensions of the model to capture asymmetry using partitioned statistics information and box‐type uncertainty in the mean and covariance matrix are provided. Using the optimized certainty equivalent framework, we provide connections of our results with robust or ambiguous convex risk measures, in which the investor minimizes his worst‐case risk under distributional ambiguity. New closed‐form results for the worst‐case optimized certainty equivalent risk measures and optimal portfolios are provided for two‐ and three‐piece utility functions. For more complicated utility functions, computational experiments indicate that such robust approaches can provide good trading strategies in financial markets.     

DISUTILITY, OPTIMAL RETIREMENT, AND PORTFOLIO SELECTION

We study the optimal retirement and consumption/investment choice of an infinitely-lived economic agent with a time-separable von Neumann–Morgenstern utility. A particular aspect of our problem is that the agent has a retirement option. Before retirement the agent receives labor income but suffers a utility loss from labor. By retiring, he avoids the utility loss but gives up labor income. We show that the agent retires optimally if his wealth exceeds a certain critical level. We also show that the agent consumes less and invests more in risky assets when he has an option to retire than he would in the absence of such an option.
An explicit solution can be provided by solving a free boundary value problem. In particular, the critical wealth level and the optimal consumption and portfolio policy are provided in explicit forms.     

OPTIMAL CONSUMPTION AND PORTFOLIO SELECTION WITH INCOMPLETE MARKETS AND UPPER AND LOWER BOUND CONSTRAINTS

We study an optimal consumption and portfolio selection problem for an investor by a martingale approach. We assume that time is a discrete and finite horizon, the sample space is finite and the number of securities is smaller than that of the possible securities price vector transitions. the investor is prohibited from investing stocks more (less, respectively) than given upper (lower) bounds at any time, and he maximizes an expected time additive utility function for the consumption process. First we give a set of budget feasibility conditions so that a consumption process is attainable by an admissible portfolio process. Also we state the existence of the unique primal optimal solutions. Next we formulate a dual control problem and establish the duality between primal and dual control problems. Also we show the existence of dual optimal solutions. Finally we consider the computational aspect of dual approach through a simple numerical example.     

A CONVEX STOCHASTIC OPTIMIZATION PROBLEM ARISING FROM PORTFOLIO SELECTION

A continuous-time financial portfolio selection model with expected utility maximization typically boils down to solving a (static) convex stochastic optimization problem in terms of the terminal wealth, with a budget constraint. In literature the latter is solved by assuming a priori that the problem is well-posed (i.e., the supremum value is finite) and a Lagrange multiplier exists (and as a consequence the optimal solution is attainable). In this paper it is first shown that, via various counter-examples, neither of these two assumptions needs to hold, and an optimal solution does not necessarily exist. These anomalies in turn have important interpretations in and impacts on the portfolio selection modeling and solutions. Relations among the non-existence of the Lagrange multiplier, the ill-posedness of the problem, and the non-attainability of an optimal solution are then investigated. Finally, explicit and easily verifiable conditions are derived which lead to finding the unique optimal solution.     

OPTIMAL PORTFOLIO, CONSUMPTION-LEISURE AND RETIREMENT CHOICE PROBLEM WITH CES UTILITY

We study optimal portfolio, consumption-leisure and retirement choice of an infinitely lived economic agent whose instantaneous preference is characterized by a constant elasticity of substitution (CES) function of consumption and leisure. We integrate in one model the optimal consumption-leisure-work choice, the optimal portfolio selection, and the optimal stopping problem in which the agent chooses her retirement time. The economic agent derives utility from both consumption and leisure, and is able to adjust her supply of labor flexibly above a certain minimum work-hour, and also has a retirement option. We solve the problem analytically by considering a variational inequality arising from the dual functions of the optimal stopping problem. The optimal retirement time is characterized as the first time when her wealth exceeds a certain critical level. We provide the critical wealth level for retirement and characterize the optimal consumption-leisure and portfolio policies before and after retirement in closed forms. We also derive properties of the optimal policies. In particular, we show that consumption in general jumps around retirement.     

PORTFOLIO CHOICE VIA QUANTILES

A portfolio choice model in continuous time is formulated for both complete and incomplete markets, where the quantile function of the terminal cash flow, instead of the cash flow itself, is taken as the decision variable. This formulation covers a wide body of existing and new models with law‐invariant preference measures, including expected utility maximization, mean–variance, goal reaching, Yaari's dual model, Lopes' SP/A model, behavioral model under prospect theory, as well as those explicitly involving VaR and CVaR in objectives and/or constraints. A solution scheme to this quantile model is proposed, and then demonstrated by solving analytically the goal‐reaching model and Yaari's dual model. A general property derived for the quantile model is that the optimal terminal payment is anticomonotonic with the pricing kernel (or with the minimal pricing kernel in the case of an incomplete market if the investment opportunity set is deterministic). As a consequence, the mutual fund theorem still holds in a market where rational and irrational agents co‐exist.     

DYNAMIC PORTFOLIO OPTIMIZATION WITH A DEFAULTABLE SECURITY AND REGIME‐SWITCHING

We consider a portfolio optimization problem in a defaultable market with finitely‐many economical regimes, where the investor can dynamically allocate her wealth among a defaultable bond, a stock, and a money market account. The market coefficients are assumed to depend on the market regime in place, which is modeled by a finite state continuous time Markov process. By separating the utility maximization problem into a predefault and postdefault component, we deduce two coupled Hamilton–Jacobi–Bellman equations for the post‐ and predefault optimal value functions, and show a novel verification theorem for their solutions. We obtain explicit constructions of value functions and investment strategies for investors with logarithmic and Constant Relative Risk Aversion utilities, and provide a precise characterization of the directionality of the bond investment strategies in terms of corporate returns, forward rates, and expected recovery at default. We illustrate the dependence of the optimal strategies on time, losses given default, and risk aversion level of the investor through a detailed economic and numerical analysis.     

HEDGING AND PORTFOLIO OPTIMIZATION UNDER TRANSACTION COSTS: A MARTINGALE APPROACH12

We derive a formula for the minimal initial wealth needed to hedge an arbitrary contingent claim in a continuous-time model with proportional transaction costs; the expression obtained can be interpreted as the supremum of expected discounted values of the claim, over all (pairs of) probability measures under which the “wealth process” is a supermartingale. Next, we prove the existence of an optimal solution to the portfolio optimization problem of maximizing utility from terminal wealth in the same model, we also characterize this solution via a transformation to a hedging problem: the optimal portfolio is the one that hedges the inverse of marginal utility evaluated at the shadow state-price density solving the corresponding dual problem, if such exists. We can then use the optimal shadow state-price density for pricing contingent claims in this market. the mathematical tools are those of continuous-time martingales, convex analysis, functional analysis, and duality theory.     

PORTFOLIO MANAGEMENT WITH TRANSACTION COSTS: AN ASYMPTOTIC ANALYSIS OF THE MORTON AND PLISKA MODEL

We examine the Morton and Pliska (1993) model for the optimal management of a portfolio when there are transaction costs proportional to a fixed fraction of the portfolio value. We analyze this model in the realistic case of small transaction costs by conducting a perturbation analysis about the no-transaction-cost solution. Although the full problem is a free-boundary diffusion problem in as many dimensions as there are assets in the portfolio, we find explicit solutions for the optimal trading policy in this limit. This makes the solution for a realistically large number of assets a practical possibility.     

STABILITY OF THE UTILITY MAXIMIZATION PROBLEM WITH RANDOM ENDOWMENT IN INCOMPLETE MARKETS

We perform a stability analysis for the utility maximization problem in a general semimartingale model where both liquid and illiquid assets (random endowments) are present. Small misspecifications of preferences (as modeled via expected utility), as well as views of the world or the market model (as modeled via subjective probabilities) are considered. Simple sufficient conditions are given for the problem to be well posed, in the sense that the optimal wealth and the marginal utility‐based prices are continuous functionals of preferences and probabilistic views.     

A NOTE ON THE QUANTILE FORMULATION

Many investment models in discrete or continuous‐time settings boil down to maximizing an objective of the quantile function of the decision variable. This quantile optimization problem is known as the quantile formulation of the original investment problem. Under certain monotonicity assumptions, several schemes to solve such quantile optimization problems have been proposed in the literature. In this paper, we propose a change‐of‐variable and relaxation method to solve the quantile optimization problems without using the calculus of variations or making any monotonicity assumptions. The method is demonstrated through a portfolio choice problem under rank‐dependent utility theory (RDUT). We show that this problem is equivalent to a classical Merton's portfolio choice problem under expected utility theory with the same utility function but a different pricing kernel explicitly determined by the given pricing kernel and probability weighting function. With this result, the feasibility, well‐posedness, attainability, and uniqueness issues for the portfolio choice problem under RDUT are solved. It is also shown that solving functional optimization problems may reduce to solving probabilistic optimization problems. The method is applicable to general models with law‐invariant preference measures including portfolio choice models under cumulative prospect theory (CPT) or RDUT, Yaari's dual model, Lopes' SP/A model, and optimal stopping models under CPT or RDUT.