Hanqing Jin

Behavioral Portfolio Selection in Continuous Time

This paper formulates and studies a general continuous-time behavioral portfolio selection model under Kahneman and Tverskys (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 It\^o 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.

Time-Inconsistent Stochastic Linear{Quadratic Control

In this paper, we formulate a general time-inconsistent stochastic linear--quadratic (LQ) control problem. The time-inconsistency arises from the presence of a quadratic term of the expected state as well as a state-dependent term in the objective functional. We define an equilibrium, instead of optimal, solution within the class of open-loop controls, and derive a sufficient condition for equilibrium controls via a flow of forward--backward stochastic differential equations. When the state is one dimensional and the coefficients in the problem are all deterministic, we find an explicit equilibrium control. As an application, we then consider a mean-variance portfolio selection model in a complete financial market where the risk-free rate is a deterministic function of time but all the other market parameters are possibly stochastic processes. Applying the general sufficient condition, we obtain explicit equilibrium strategies when the risk premium is both deterministic and stochastic.

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 {\it 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, 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.

Buy Low and Sell High

In trading stocks investors naturally aspire to “buy low and sell high (BLSH)”. This paper formalizes the notion of BLSH by formulating stock buying/selling in terms of four optimal stopping problems involving the global maximum and minimum of the stock prices over a given investment horizon. Assuming that the stock price process follows a geometric Brownian motion, all the four problems are solved and buying/selling strategies completely characterized via a free-boundary PDE approach.

Time-Inconsistent Stochastic Linear–Quadratic Control: Characterization and Uniqueness of Equilibrium

In this paper, we continue our study on a general time-inconsistent stochastic linear--quadratic (LQ) control problem originally formulated in [6]. We derive a necessary and sufficient condition for equilibrium controls via a flow of forward--backward stochastic differential equations. When the state is one dimensional and the coefficients in the problem are all deterministic, we prove that the explicit equilibrium control constructed in \cite{HJZ} is indeed unique. Our proof is based on the derived equivalent condition for equilibria as well as a stochastic version of the Lebesgue differentiation theorem. Finally, we show that the equilibrium strategy is unique for a mean--variance portfolio selection model in a complete financial market where the risk-free rate is a deterministic function of time but all the other market parameters are possibly stochastic processes.

Numerical methods for portfolio selection with bounded constraints

This work develops an approximation procedure for portfolio selection with bounded constraints. Based on the Markov chain approximation techniques, numerical procedures are constructed for the utility optimization task. Under simple conditions, the convergence of the approximation sequences to the wealth process and the optimal utility function is established. Numerical examples are provided to illustrate the performance of the algorithms.

A Note on Semivariance

In a recent paper (Jin, Yan, and Zhou 2005), it is proved that efficient strategies of the continuous-time mean-semivariance portfolio selection model are in general never achieved save for a trivial case. In this note, we show that the mean-semivariance efficient strategies in a single period are always attained irrespective of the market condition or the security return distribution. Further, for the below-target semivariance model the attainability is established under the arbitrage-free condition. Finally, we extend the results to problems with general downside risk measures.

Illiquidity, position limits, and optimal investment for mutual funds

We study the optimal trading strategy of mutual funds that face both position limits and differential illiquidity. We provide explicit characterization of the optimal trading strategy and conduct an extensive analytical and numerical analysis of the optimal trading strategy. We show that the optimal trading boundaries are increasing in both the lower and the upper position limits. We find that position limits can affect current trading strategy even when they are not currently binding and other seemingly intuitive trading strategies can be costly. We also examine the optimal choice of position limits.

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.

Continuous-Time Markowitz's Problems in an Incomplete Market, with No-Shorting Portfolios

1 Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, Hong Kong, hqjin@se.cuhk.edu.hk 2 Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, Hong Kong, Tel.: 852-2609-8320, fax: 852-2603-5505, xyzhou@se.cuhk.edu.hk. Supported by the RGC Earmarked Grants CUHK 4175/03E, CUHK418605, and Croucher Senior Research Fellowship. We also thank an anonymous referee for helpful comments that have led to an improved version.