Continuous-time Portfolio Selection Research Articles

Optimal chonsei to monthly rent conversion choice given borrowing constraints

Chonsei, an exceptional but prevalent housing rental system in the Republic of Korea, functions as private loans between individuals, and its implicit interest rate is called the “chonsei to monthly rent conversion rate”. Although the chonsei to monthly rent conversion rate is much higher than the interest rate of savings accounts or mortgage loans, making chonsei a more expensive channel of financing, chonsei remains a dominant type of rent in Korea. This study provides a novel theoretical model in a continuous-time portfolio selection framework that allows a lessor, given borrowing constraints, to optimally choose between chonsei and monthly rent contracts to show a possible solution to this situation. We investigate the optimal choice between chonsei and the monthly rent of lessors using a rigorous theoretical model and find a closed-form solution to the model. We analyze how credit availability in the banking sector affects the choice problem, showing that the lessor may prefer the chonsei contract over the monthly rent contract, given the borrowing limit from a bank. In addition, we analyze the impact of other major factors, such as the market price of risk, the amount of chonsei deposit, the monthly rental rate, and the income rate, on the lessor’s optimal decision between chonsei and monthly rent.

BEHAVIORAL PORTFOLIO CHOICE UNDER HYPERBOLIC ABSOLUTE RISK AVERSION

This paper studies the optimal investment problem for a behavioral investor with probability distortion functions and an S-shaped utility function whose utility on gains satisfies the Inada condition at infinity, albeit not necessarily at zero, in a complete continuous-time financial market model. In particular, a piecewise utility function with hyperbolic absolute risk aversion (HARA) is applied. The considered behavioral framework, cumulative prospect theory (CPT), was originally introduced by [A. Tversky & D. Kahneman (1992) Advances in prospect theory: Cumulative representation of uncertainty, Journal of Risk and Uncertainty 5 (4), 297–323]. The utility model allows for increasing, constant or decreasing relative risk aversion. The continuous-time portfolio selection problem under the S-shaped HARA utility function in combination with probability distortion functions on gains and losses is solved theoretically for the first time, the optimal terminal wealth and its replicating wealth process and investment strategy are stated. In addition, conditions on the utility and the probability distortion functions for well-posedness and closed-form solutions are provided. A specific probability distortion function family is presented which fulfills all those requirements. This generalizes the work by [H. Jin & X. Y. Zhou (2008) Behavioral portfolio selection in continuous time, Mathematical Finance 18 (3), 385–426]. Finally, a numerical case study is carried out to illustrate the impact of the utility function and the probability distortion functions.

Behavioral Portfolio Choice under Hyperbolic Absolute Risk Aversion

This paper studies the optimal investment problem for a behavioral investor with probability distortion functions and an S-shaped utility function whose utility on gains satisfies the Inada condition at infinity, albeit not necessarily at zero, in a complete continuous-time financial market model. In particular, a piecewise utility function with hyperbolic absolute risk aversion (HARA) is applied. The considered behavioral framework, Cumulative Prospect Theory (CPT), was originally introduced by Tversky and Kahneman (1992). The utility model allows for increasing, constant or decreasing relative risk aversion. The continuous-time portfolio selection problem under the S-shaped HARA utility function in combination with probability distortion functions on gains and losses is solved theoretically for the first time, the optimal terminal wealth and its replicating wealth process and investment strategy are stated. In addition, conditions on the utility and the probability distortion functions for well-posedness and closed-form solutions are provided. A specific probability distortion function family is presented which fulfills all those requirements. This generalizes the work by Jin and Zhou (2008). Finally, a numerical case study is carried out to illustrate the impact of the utility function and the probability distortion functions.

Continuous-Time Portfolio Selection: A Cursory Survey

In this article we provide a short survey on continuous-time portfolio selection. We explain the pioneering contribution of Merton and the use of dynamic programming. Then, we discuss Bismut's application of the Pontryagin maximum principle to portfolio selection and the dual martingale approach. We also explain two models with potential applicability to practice: life-cycle models with explicit consideration of retirement and models with protection against decline in spending power.

Debiased expert forecasts in continuous-time asset allocation

Abstract Expert forecasts are an essential component of asset management and an important research topic. However, the effect of behavioral biases on expert forecasts is generally ignored. This paper examines the effect of biased expert forecasts on asset allocations. We find that biases have a significant impact on portfolios, explaining nearly 70% of excess risk-taking in our implementation. To address the effect of behavioral biases, we propose an integrated behavioral continuous-time portfolio selection model which we solve in closed form. The model applies general principles to identify and reduce the impact of five main behavioral biases. This paper concludes with a new personal fractional Kelly decomposition to account for the effect of opinions on the optimal asset allocation.

Behavioral Benchmarked Investment Management with Expert Forecasts

We propose a continuous-time portfolio selection model that explains the active-passive continuum. Our model illuminates the pivotal role of expert opinions and factors in the asset allocation process. In the model, investors aim to outperform a benchmark. As securities and benchmark's drift depends on unobservable factors, portfolio selection becomes a partial observation risk-sensitive control problem. We find that the optimal investment policy combines Kelly, benchmark-tracking, and intertemporal hedging portfolios. The Kelly portfolio sums two alpha-generating strategies. Our simulation reveals the importance of debiasing expert opinions and shows that factor choice is crucial at every stage of the investment process.

Continuous-Time Portfolio Selection: A Cursory Survey

In this article we provide a short survey on continuous-time portfolio selection. We explain the pioneering contribution of Merton and the use of dynamic programming. Then, we discuss Bismut's application of the Pontryagin maximum principle to portfolio selection and the dual martingale approach. We also explain two models with potential applicability to practice: life-cycle models with explicit consideration of retirement and models with protection against decline in spending power.

Penalty Method for Portfolio Selection with Capital Gains Tax

Many finance problems can be formulated as a singular stochastic control problem, where the associated Hamilton-Jacobi-Bellman (HJB) equation takes the form of variational inequality and its penalty approximation equation is linked to a regular control problem. The penalty method, as a finite difference scheme for the penalty equation, has been widely used to numerically solve singular control problems, and its convergence analysis in literature relies on the uniqueness of solution to the original HJB equation problem. We consider a singular stochastic control problem arising from continuous-time portfolio selection with capital gains tax, where the associated HJB equation problem admits infinitely many solutions. We show that the penalty method still works and converges to the value function which is the minimal (viscosity) solution of the HJB equation problem. Numerical results are presented to demonstrate the efficiency of the penalty method and to better understand optimal investment strategy in the presence of capital gains tax. Our approach sheds light on the robustness of the penalty method for general singular stochastic control problems.

Debiased Expert Opinions in Continuous Time Asset Allocation: Supplementary Material

This paper examines the effect of biased expert opinions on asset allocation. Expert opinions, such as brokerage research and analyst views, are an essential component of the asset management sector and an important research topic. However, the effect of behavioral biases on expert opinions is generally ignored. We find that biases have a significant impact on portfolios, explaining nearly half of the effect of expert opinions in our implementation. To address the impact of behavioral biases on expert opinions, we propose an integrated behavioral continuous time portfolio selection model which we solve in closed form. This model recommends general principles to identify and reduce the impact of five main behavioral biases. The model concludes with a new personal fractional Kelly decomposition to account for the effect of opinions on the optimal asset allocation.

A Reinterpretation of the Optimal Demand for Risky Assets in Fund Separation Theorems

In a continuous-time portfolio selection model with N risky assets and K state variables driving their risk and return parameters, we derive simple expressions for the allocation to each asset in the K + 1 risky funds of the (K + 2)-fund separation theorem. We show that the allocation to any given risky asset in each fund can be written in terms of the parameters of a regression of the excess returns of this asset on those of the N − 1 remaining assets. We also use these parameters to provide quantitative measures of the increase in Sharpe ratio of the speculative demand, or in the maximum correlation of each hedging demand with respect to the corresponding risk factor, associated with the introduction of a new asset in the investment universe. Finally, we show that in a multiperiod setting, an asset is “spanned” by others if and only if it improves neither the maximum Sharpe ratio of the speculative demand nor the maximum correlations of the hedging demands with the risk factors. This paper was accepted by Lauren Cohen, finance.

Closed-Form Optimal Strategies of Continuous-Time Options with Stochastic Differential Equations

A continuous-time portfolio selection with options based on risk aversion utility function in financial market is studied. The different price between sale and purchase of options is introduced in this paper. The optimal investment-consumption problem is formulated as a continuous-time mathematical model with stochastic differential equations. The prices processes follow jump-diffusion processes (Weiner process and Poisson process). Then the corresponding Hamilton-Jacobi-Bellman (HJB) equation of the problem is represented and its solution is obtained in different conditions. The above results are applied to a special case under a Hyperbolic Absolute Risk Aversion (HARA) utility function. The optimal investment-consumption strategies about HARA utility function are also derived. Finally, an example and some discussions illustrating these results are also presented.

Optimal consumption–investment strategy under the Vasicek model: HARA utility and Legendre transform

This paper studies the optimal consumption–investment strategy with multiple risky assets and stochastic interest rates, in which interest rate is supposed to be driven by the Vasicek model. The objective of the individuals is to seek an optimal consumption–investment strategy to maximize the expected discount utility of intermediate consumption and terminal wealth in the finite horizon. In the utility theory, Hyperbolic Absolute Risk Aversion (HARA) utility consists of CRRA utility, CARA utility and Logarithmic utility as special cases. In addition, HARA utility is seldom studied in continuous-time portfolio selection theory due to its sophisticated expression. In this paper, we choose HARA utility as the risky preference of the individuals. Due to the complexity of the structure of the solution to the original Hamilton–Jacobi–Bellman (HJB) equation, we use Legendre transform to change the original non-linear HJB equation into its linear dual one, whose solution is easy to conjecture in the case of HARA utility. By calculations and deductions, we obtain the closed-form solution to the optimal consumption–investment strategy in a complete market. Moreover, some special cases are also discussed in detail. Finally, a numerical example is given to illustrate our results.

Local Discontinuous Galerkin Method for Portfolio Optimization with Transaction Costs

We study the continuous time portfolio selection problem over a finite horizon for an investor who maximizes the expected utility of terminal wealth and faces transaction costs. The portfolio consists of a risk-free asset, and a risky asset whose price is modeled as a geometric Brownian motion. The problem can be formulated as a stochastic singular control or an impulse control problem depending on whether the transaction costs are of proportional or fixed type. Due to the intractability of the problem, modelers resort to numerical methods to obtain approximations of solutions to the problem. In this paper we propose a stable and high-order computational scheme to solve this problem, which is capable of handling any form of transaction costs. Specifically, we implement the Local Discontinuous Galerkin (LDG) Finite Element Method (FEM) to solve the resulting convection-diffusion Partial Differential Equation (PDE), and obtain error estimates for the LDG method. Moreover, we prove the convergence of the scheme. Our numerical experiments show the order of accuracy of the LDG method and illustrate the optimal policies under various kinds of transaction costs.

Continuous-time portfolio selection under ambiguity

In a financial market, the appreciation rates of stocks are statistically difficult to estimate, and typically only some confidence intervals in which the rates reside can be estimated. In this paper we study continuous-time portfolio selection under ambiguity, in the sense that the appreciation rates are only known to be in a certain convex closed set and the portfolios are allowed to be based on only the historical stocks prices. We formulate the problem in both the expected utility and the mean--variance frameworks, and derive robust portfolios explicitly for both models.

Continuous-Time Mean-Variance Portfolio Selection with Inflation in an Incomplete Market

This paper concerns a continuous-time portfolio selection problem with inflation in an incomplete market. By using the approach of more general stochastic linear quadratic control technique (SLQ), we obtain the optimal strategy and efficient frontier to this problem. Furthermore, a numerical example is also provided.

A Perturbation Approach to Continuous-Time Portfolio Selection Under Stochastic Investment Opportunities

This paper studies portfolio selection in continuous-time models with stochastic investment opportunities. We consider asset allocation problems where preferences are specified as power utility derived from terminal wealth as well as consumption-savings problems with recursive utility Epstein-Zin preferences. The paper approximates the associated dynamic programming problem by perturbing the coefficients of the stochastic dynamics. We represent the Hamilton-Jacobi-Bellman equation as a series of partial differential equations that can be solved iteratively in closed-form through computer algebra software, at any desired accuracy.

Optimal portfolio selection in a Lévy market with uncontrolled cash flow and only risky assets

This article considers an investor who has an exogenous cash flow evolving according to a Lévy process and invests in a financial market consisting of only risky assets, whose prices are governed by exponential Lévy processes. Two continuous-time portfolio selection problems are studied for the investor. One is a benchmark problem, and the other is a mean-variance problem. The first problem is solved by adopting the stochastic dynamic programming approach, and the obtained results are extended to the second problem by employing the duality theory. Closed-form solutions of these two problems are derived. Some existing results are found to be special cases of our results.

Continuous-Time Portfolio Selection and Option Pricing under Risk-Minimization Criterion in an Incomplete Market

We study option pricing with risk-minimization criterion in an incomplete market where the dynamics of the risky underlying asset are governed by a jump diffusion equation. We obtain the Radon-Nikodym derivative in the minimal martingale measure and a partial integrodifferential equation (PIDE) of European call option. In a special case, we get the exact solution for European call option by Fourier transformation methods. Finally, we employ the pricing kernel to calculate the optimal portfolio selection by martingale methods.

Continuous-time safety-first portfolio selection with jump-diffusion processes

This article is concerned with continuous-time portfolio selection based on a safety-first criterion under discontinuous price processes (jump-diffusion processes). The solution of the corresponding Hamilton–Jacobi–Bellman equation of the problem is demonstrated. The analytical solutions are presented when there does not exist any riskless asset. Moreover, the problem is also discussed while there exists one riskless asset.

Penalty methods for continuous-time portfolio selection with proportional transaction costs

This paper is concerned with numerical solutions to a singular stochastic control problem arising from continuous-time portfolio selection with proportional transaction costs. The associated value function is governed by a variational inequality with gradient constraints. We propose a penalty method to deal with the gradient constraints and employ a finite difference discretization. Convergence analysis is presented. We also show that the standard penalty method can be applied in the case of a single risky asset where the problem can be reduced to a standard variational inequality. Numerical results are given to demonstrate the efficiency of the methods and to examine the behavior of the optimal trading strategy.