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Behavioral Portfolio Selection in Continuous Time

Hanqing Jin, Xunyu Zhou

arXiv:0709.2830v1q-fin.PMmath.OCmath.PR

TL;DR

The paper addresses how behavioral portfolio models under cumulative prospect theory can become ill-posed when their components do not coordinate. It develops a distinct analytical approach for well-posed models in complete markets, finding simple binary-option-like optimal terminal wealth and discussing risky allocations.

  • Problem

    Behavioral portfolio models with S-shaped value functions and probability distortions can be ill-posed because their components may not coordinate properly.

  • Method

    The paper formulates a continuous-time cumulative prospect theory model and solves well-posed cases by reducing terminal-wealth optimization into gain and loss sub-problems.

  • Results

    Optimal terminal wealth resembles a portfolio of two binary options characterized by one number obtained from a simple two-dimensional programming problem.

  • Takeaways & Limitations

    The framework identifies ill-posed behavioral models and provides explicit optimal solutions whose structure resembles betting on good market states while accepting a fixed loss in bad states.

  • Takeaways & Limitations

    The paper does not claim to provide a satisfactory explanation of the equity premium puzzle, despite showing that investors may underweight stocks under certain conditions.

Abstract

from arXiv · show

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.

1 Introduction

The paper addresses the limited analytical treatment of continuous-time portfolio selection under cumulative prospect theory, where probability distortions and S-shaped preferences create modeling and optimization difficulties.

  • Behavioral motivation: Cumulative prospect theory captures reference-dependent gains and losses, non-uniform risk attitudes, loss aversion, and distorted probabilities.Its value function is concave for gains, convex for losses, and steeper for losses than gains.
  • Research gap: Continuous-time behavioral portfolio research was largely absent, while prior prospect-theory portfolio studies overwhelmingly considered single-period settings.The paper identifies only a specific continuous-time study without probability distortion as bearing on this setting.
  • Modeling challenge: Probability distortion replaces conventional expectation with Choquet integration and makes standard convexification techniques inapplicable.The distorted probability is a non-additive capacity rather than an ordinary probability measure.
  • Contributions: The paper establishes a general continuous-time CPT model, identifies ill-posed cases, develops a distinct solution approach, and derives explicit optimal terminal wealth.The study also examines how behavioral criteria influence equity allocations.
  • Modeling challenge: Well-posedness is central because behavioral models can allow arbitrarily high objective values when their components do not coordinate properly.The paper defines ill-posedness as an infinite supremum caused by an incorrectly set trade-off.
  • Main result: The resulting terminal wealth resembles two digital-option payoffs and supports a gambling policy that bets on good market states while accepting a fixed loss in bad states.The relevant parameters are obtained through a simple two-dimensional programming problem.

2 The Model

The model defines a complete continuous-time market and evaluates terminal wealth with cumulative prospect theory using separate gain and loss functions and probability distortions.

  • Market: The market contains a bank account and m stocks traded continuously, with asset prices modeled by progressively measurable Itô-process coefficients.The volatility matrix has full rank, and the discounted short rate is bounded below under the stated assumptions.
  • Terminal wealth: Completeness implies that any lower-bounded terminal claim satisfying E[ρξ] = x0 can be replicated by a tame admissible portfolio.The state-price density ρ is strictly positive and finite almost surely.
  • Market: Admissible portfolios generate self-financing wealth processes, and tame portfolios require discounted wealth to be almost surely bounded below.The wealth dynamics are given by dx(t) = [r(t)x(t) + B′(t)π(t)]dt + π(t)′σ(t)dW(t).
  • Behavioral criterion: CPT evaluates gains and losses relative to a terminal reference point of zero through separate value functions u+ and u− and distortions T+ and T−.The utility functions are increasing and concave, while the distortion functions are increasing maps from [0, 1] to [0, 1].
  • Behavioral criterion: With no probability distortion, the gain criterion reduces to expected utility; with distortion, it becomes a Choquet integral over a non-additive capacity.The framework extends to possibly continuous random variables and agrees with the discrete CPT definition.
  • Optimization: The portfolio problem reduces to maximizing V(X) over lower-bounded terminal wealth claims subject to E[ρX] = x0.After solving for optimal terminal wealth, the corresponding portfolio is obtained by replication.
  • Reference point: A general reference point can be incorporated through replication, but choosing a dynamically updating reference point is left for future study.The stated initial endowment is precisely the difference between initial wealth and discounted reference wealth.

3 Ill-Posedness

The paper shows that behavioral portfolio optimization can be ill-posed when gain utility, state prices, or probability distortions permit unbounded value, motivating explicit well-posedness conditions.

  • General conditions: A model is ill-posed if its objective supremum is infinite, meaning the trade-off permits the objective value to be pushed arbitrarily high.The paper contrasts this with classical models, where global concavity usually supports well-posedness.
  • General conditions: Ill-posedness occurs when a feasible nonnegative claim has finite state-price cost but infinite prospective gain value.Theorem 3.1 establishes this directly, and the paper imposes an assumption to exclude the case.
  • Loss distortion: If u+(+∞) = +∞, the essential supremum of ρ is infinite, and T−(x) = x, then the optimization problem is ill-posed.The result highlights the role of undistorted losses when large gain payoffs can be financed through extreme state prices.
  • Loss distortion: A loss probability distortion is necessary for well-posedness when gain utility can grow without bound.The paper interprets the failure as borrowing to buy a huge payoff and betting on favorable market states.

4 Main Results

The paper decomposes the behavioral portfolio problem into auxiliary positive- and negative-part problems, identifies ill-posed cases, and solves well-posed models through a low-dimensional program. The resulting optimal terminal wealth is threshold-based and can be implemented as a combination of two binary options, producing a gambling-like payoff.

  • Problem decomposition: The original problem is split into positive and negative parts, then recombined through a mathematical program over an event and its associated initial price.The event is later represented by a state-price-density threshold, reducing the problem to two real decision variables.
  • Well-posedness: The model is ill-posed when its objective is unbounded, so the paper first identifies ill-posed cases before seeking optimal solutions.The authors define well-posedness by finiteness of the objective supremum and restrict optimization to well-posed cases.
  • Optimal wealth: The optimal terminal wealth changes sign according to whether the state pricing density is below or above a single threshold c*, obtained from the reduced optimization problem.The positive and negative regions coincide, up to a null set, with the states {ρ ≤ c*} and {ρ > c*}.
  • Optimal wealth: The optimal payoff is a combination of two binary options that is readily priced in the complete-market setting.The construction uses a contingent claim for favorable states and another claim for unfavorable states.
  • Economic interpretation: The strategy bets on good states by buying a claim above the reference wealth and selling or issuing a claim that creates a fixed loss in bad states.In a one-stock market, the good-state condition corresponds to the stock price exceeding a certain level.
  • Economic interpretation: The paper reports that this gambling-policy structure persists for the general behavioral model, extending an earlier special case without probability distortion.The earlier result used a two-piece power value function and no probability distortion.

5 Splitting

The splitting approach makes the original behavioral optimization equivalent to positive-part, negative-part, and event-selection subproblems. A rearrangement result then restricts the event to a state-price-density threshold and yields a binary-option characterization of optimal wealth.

  • Splitting: Any feasible wealth X induces an event A={X≥0} and positive-part price x+=E[ρX+], separating gains from losses.The complementary event represents the negative part and its corresponding price constraint.
  • Equivalence: The original problem is ill-posed if and only if the auxiliary event-selection problem is ill-posed.The proof maps unbounded objectives in either formulation into feasible solutions of the other.
  • Equivalence: When X* is optimal, its positive and negative parts solve the corresponding auxiliary problems for the induced event and price.The decomposition preserves optimality rather than merely providing an upper bound.
  • Threshold reduction: The event-selection problem is equivalent to one restricted to events of the form {ρ≤c}, replacing a difficult random-event decision with a scalar threshold.The relevant threshold lies in the admissible range of the state pricing density.
  • Threshold reduction: The final optimization uses the threshold c* and positive-part price x+ as decision variables, after which the positive and negative wealth components are solved conditionally.The resulting problem is a constrained optimization problem in R^2.
  • Final characterization: The resulting optimal wealth is characterized by two binary-option payoffs, with its gain region matching {ρ≤c*} up to a zero-probability set.The construction follows from the threshold reduction and the auxiliary optimality conditions.

6 Positive Part Problem

The positive-part problem is recast as a Choquet maximization problem and solved under a monotonicity condition linking the distribution of the state-price density to probability distortion. This condition yields explicit optimal solutions and clarifies how distortion affects solvability.

  • Formulation: The positive-part problem is a special Choquet maximization problem solved conditionally on A={ω:ρ≤c}.The conditional formulation uses the probability measure P(·|A) and a normalized distortion.
  • Boundary cases: If x+=0, the optimal solution is X*=0 with value v+(c,x+)=0; infeasible cases have value −∞.These are the boundary cases in the positive-part problem.
  • Optimal solution: For positive feasible wealth, the optimizer has the form X*(λ)=(u′)^−1(λρ) on {ρ≤c}, where λ>0 uniquely satisfies E[ρX*(λ)]=x+.The solution is characterized by the budget constraint and the state-price density.
  • Dependence on c: The value v+ is strictly increasing in c whenever the newly included states have positive probability.Any c̄>c with P{c<ρ≤c̄}>0 strictly improves the positive-part value.
  • Monotonicity condition: Assumption 4.1 requires monotonicity of F−1(z)/T+′(z), equivalently that T+′(F(x))/x is non-increasing.The condition is economically interpreted as preventing distortion from increasing relative risk seeking by more than 1.
  • Distortion examples: For reversed S-shaped distortions, the condition is characterized by j(x)≤0 below c0 and 0≤j(x)≤1 above c0.A constructed lognormal example produces reversed S-shaped distortions satisfying this condition, with T+′ diverging near probabilities 0 and 1.

7 Negative Part Problem

The negative-part problem is formulated as a Choquet minimization problem on the complement of the gain region, with boundedness and conditional-budget constraints. General minimization results characterize feasibility, optimality, and boundary cases.

  • Formulation: The negative-part problem is a Choquet minimization problem with feasible solutions required to be almost surely bounded above.Under a mild condition, optimal solutions to the more general minimization problem automatically satisfy this bound.
  • Conditional reduction: On A^C, the problem uses the conditional measure P(·|A^C), normalized distortion T−(xP(A^C))/T−(P(A^C)), and budget E_A^C[ρY]=(x+−x0)/P(A^C).This conditional formulation reduces the negative-part problem to the general problem treated in Appendix D.
  • Boundary cases: If c=ρ̄ and x+=x0, the optimal negative-part solution is X*=0 with v−(c,x+)=0.This is the zero-loss boundary case.
  • Boundary cases: If c=ρ̄ and x+≠x0, the negative-part problem has no feasible solution and v−(c,x+)=+∞.The terminal allocation cannot satisfy the required constraints in this boundary configuration.
  • Optimality: For interior c, existence of an optimal negative-part solution is characterized by a separate minimization problem under strict concavity of u− at 0.The theorem transfers the general Appendix D characterization to the portfolio problem.

8 Proof of Main Results

The proof decomposes the behavioral optimization into positive and negative subproblems, establishes their equivalence to auxiliary programs, and reconstructs the optimal terminal wealth. The auxiliary problems have identical supremum values, supporting the main theorem.

  • Decomposition: The proof solves the positive and negative parts separately before combining them through the ultimate optimization problem.The auxiliary problems provide the components needed for Theorem 4.1.
  • Auxiliary bounds: For any feasible pair, Lemma 8.1 supplies the key inequality linking the positive and negative objective terms.The displayed inequality bounds the negative contribution using the positive-part parameters.
  • Equivalent programs: Problems (21) and (14) have the same supremum values.The proof establishes both inequalities, including the case where the supremum is infinite.
  • Reconstruction: An optimizer of the main problem induces optimizers for the positive and negative subproblems on the regions {ρ≤c*} and {ρ>c*}.The terminal wealth is reconstructed by combining the region-specific solutions associated with c* and x+*.
  • Main result: Theorem 4.1 follows by transferring optimality between the auxiliary programs and the original terminal-wealth problem.The proof also handles degenerate cases such as x+*=0 and c*=ρ̄.

9 An Example with Two-Piece CRRA Utility Functions

The two-piece CRRA example shows that the behavioral model can be well-posed, unattainable, or ill-posed depending on the coordination of loss aversion, probability distortion, and the market state-price density. The gain and loss initial-wealth cases have different optimal-portfolio structures.

  • Setup: The example assumes lognormal ρ and two-piece CRRA functions u+(x)=x^α and u−(x)=k−x^α, with 0<α<1 and k−>0.The probability distortions satisfy the stated assumptions, including Assumption 4.1 for T+.
  • Initial gain: For x0≥0, inf_c>0 k(c)≥1 yields a replicating optimal portfolio, whereas inf_c>0 k(c)<1 makes the problem ill-posed.The scalar comparison k(c) determines whether the gain-side objective is bounded.
  • Initial loss: For x0<0, inf_c>0 k(c)>1 yields well-posedness and conditional existence of an optimizer, while inf_c>0 k(c)<1 is ill-posed.When the infimum equals 1, the supremum is 0 but is not attained.
  • Optimal portfolios: When an interior minimizer c*>0 exists in the loss case, the optimal portfolio replicates the corresponding explicit terminal wealth; when c*=0 is uniquely minimizing, the optimal solution is also explicit.The terminal-wealth formulas depend on k(c*) and α.
  • Behavioral coordination: The threshold inf_c>0 k(c) captures coordination among utility functions, probability distortions, and the market represented by ρ.The example uses α=0.88 and k−=2.25 from Tversky and Kahneman’s parameterization.
  • Economic interpretation: The optimal strategy differs by initial wealth: positive wealth supports buying a claim above the reference point, while negative wealth leads to a different allocation structure.The reference point can represent a future liability that must be fulfilled.

10 How Behavioral Criterion Affects Risky Allocation

The paper’s two-piece CRRA example derives closed-form optimal portfolios under behavioral criteria and compares risky allocations with the conventional utility model.

  • The example uses power utilities u+(x)=x^α and u−(x)=k−x^α with time-invariant market parameters.
  • Under x0≥0 and inf_c>0 k(c)≥1, the optimal wealth-portfolio pair is available in closed form.
  • The optimal portfolio replicates a claim composed of state-contingent components associated with the distorted-market state variable.
  • For c0=1, the optimal portfolio and its risky-asset ratio simplify further.
  • The behavioral investor underweights risky assets relative to conventional utility allocations under some conditions, and overweights them under others.

11 Concluding Remarks

The concluding section presents the paper as an initiating continuous-time CPT portfolio model, emphasizing coordinated well-posedness, a distinct solution method, simple binary-option-like payoffs, and behavioral effects on risky allocation.

  • The paper introduces a general continuous-time portfolio-selection model under cumulative prospect theory with S-shaped utilities and probability distortions.
  • Behavioral models can be ill-posed, so well-posedness requires coordination among the market, utility function, and probability distortions.
  • The paper develops a solution approach fundamentally different from methods used in conventional dynamic asset-allocation models.
  • The optimal terminal payoff is related to binary options characterized by a single number, while the strategy bets on good market states.
  • In the specific two-piece CRRA case, the behavioral criterion changes risky allocation and can underweight stocks under certain conditions.
  • The analysis assumes a small investor and market properties including no arbitrage and market completeness; incomplete-market behavioral models remain open.

B Two Auxiliary Optimization Problems

This appendix reduces two auxiliary distribution-constrained optimization problems to monotonic rearrangements, establishing comonotonic or anti-comonotonic optimal solutions and uniqueness under atomlessness.

  • The auxiliary problems are introduced to simplify the behavioral portfolio-selection model, although they are highly non-convex.
  • Given a strictly positive atomless Y and target distribution G, the constructions G−1(F(Y)) and G−1(1−F(Y)) solve the corresponding problems.
  • For a non-decreasing transformation, matching X with h(Y) maximizes E[XY], with equality characterized by X∈[h(Y−),h(Y+)] almost surely.
  • For a non-increasing transformation, matching X with h(Y) minimizes E[XY], with the analogous equality condition.
  • When the relevant objective values are finite, these solutions are unique almost surely.
  • The first solution is comonotonic with Y, while the second is anti-comonotonic with Y.

C A Choquet Maximization Problem

This section transforms a Choquet maximization problem into a distribution-function problem, identifies conditions for well-posedness and explicit solutions, and records unresolved cases and positivity properties.

  • The original Choquet optimization is non-convex, so the paper changes variables and transforms it into a convex problem.
  • Any optimal solution must be anti-comonotonic with ξ and can therefore be represented as G−1(Z), where Z=1−Fξ(ξ).
  • The distribution-function problem is equivalent to the original problem, and optimal solutions correspond through X∗=(G∗)−1(Z).
  • After relaxing monotonicity pointwise, the candidate solution is determined by the inverse marginal utility and the ratio λFξ−1(z)/T′(z).
  • If Fξ−1(z)/T′(z) is non-decreasing, the candidate is non-decreasing and solves the transformed problem; otherwise, an explicit optimal solution remains open.
  • Under the stated monotonicity and relative-risk-aversion conditions, well-posedness and existence of a unique optimum are equivalent for every a>0.
  • When the relevant value is finite, the Lagrange multiplier is selected to satisfy the budget constraint; infinite value implies ill-posedness.
  • With positive initial budget, any optimal solution is positive almost surely.

D A Choquet Minimization Problem

The appendix reduces the Choquet minimization problem to distributional and quantile formulations, then characterizes optimal solutions through corner-point structure and an equivalent scalar problem. It also gives a replication result for truncated power claims in a complete-market setting.

  • Distributional reduction: Problem (51) is transformed into a distributional problem whose optimizer is represented by X∗ = (G∗)^−1(Z), with Z = Fξ(ξ).Conversely, every optimizer of Problem (51) induces an optimal distribution function G∗ and has this quantile representation almost surely.
  • Corner-point analysis: Because the reformulated objective is concave, its solution has a corner-point structure rather than the structure associated with the utility-minimization problem.The proof uses step-function approximations and convex-combination arguments to identify the relevant corner solutions.
  • Optimal solution structure: Optimal solutions to Problem (53), when they exist under strict concavity at 0, have a threshold form: zero below b and constant q(b) above it.Feasibility determines the constant as q(b), reducing the search to an optimal threshold b ∈ [0,1).
  • Boundedness: Any optimal solution X∗ to (51) is uniformly bounded from above when the threshold-form optimizer is uniformly bounded.The construction extends g to t = 1 by setting g(1) = q(b), after which Proposition D.1 transfers boundedness to X∗.
  • Equivalent formulations: Problems (53) and (55) have the same infimum values, and Theorem D.1 extends this equivalence to Problems (51) and (55).Under strict concavity at 0, Problem (51) admits an optimizer exactly when the associated problem admits an optimizer, with the optimal solution represented through c∗.
  • Claim replication: Theorem E.1 provides a wealth-portfolio pair replicating the truncated power claim ρα1ρ∈(c1,c2), with a separate argument covering c2 = +∞.The claim has a binary- or digital-option-like payoff structure, although ρ is not an underlying stock.
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