Continuous-time Mean-variance Portfolio Selection Research Articles

Mean-variance portfolio selection with non-linear wealth dynamics and random coefficients

This paper studies the continuous time mean-variance portfolio selection problem with one kind of non-linear wealth dynamics. To deal the expectation constraint, an auxiliary stochastic control problem is firstly solved by two new generalized stochastic Riccati equations from which a candidate portfolio in feedback form is constructed, and the corresponding wealth process will never cross the vertex of the parabola. In order to verify the optimality of the candidate portfolio, the convex duality (requires the monotonicity of the cost function) is established to give another more direct expression of the terminal wealth level. The variance-optimal martingale measure and the link between the non-linear financial market and the classical linear market are also provided. Finally, we obtain the efficient frontier in closed form. From our results, people are more likely to invest their money in riskless asset compared with the classical linear market.

Reinforcement learning for continuous-time mean-variance portfolio selection in a regime-switching market

We propose a reinforcement learning (RL) approach to solve the continuous-time mean-variance portfolio selection problem in a regime-switching market, where the market regime is unobservable. To encourage exploration for learning, we formulate an exploratory stochastic control problem with an entropy-regularized mean-variance objective. We obtain semi-analytical representations of the optimal value function and optimal policy, which involve unknown solutions to two linear parabolic partial differential equations (PDEs). We utilize these representations to parametrize the value function and policy for learning with the unknown solutions to the PDEs approximated based on polynomials. We develop an actor-critic RL algorithm to learn the optimal policy through interactions with the market environment. The algorithm carries out filtering to obtain the belief probability of the market regime and performs policy evaluation and policy gradient updates alternately. Empirical results demonstrate the advantages of our RL algorithm in relatively long-term investment problems over the classical control approach and an RL algorithm developed for the continuous-time mean-variance problem without considering regime switches.

Continuous-Time Mean-Variance Portfolio Selection Under Non-Markovian Regime-Switching Model with Random Horizon

Continuous-Time Mean-Variance Portfolio Selection Under Non-Markovian Regime-Switching Model with Random Horizon

Reinforcement Learning for Continuous-Time Mean-Variance Portfolio Selection in a Regime-Switching Market

Reinforcement Learning for Continuous-Time Mean-Variance Portfolio Selection in a Regime-Switching Market

Mean-Variance Portfolio Selection with Dynamic Targets for Expected Terminal Wealth

In a market that consists of multiple stocks and one risk-free asset whose mean return rates and volatility are deterministic, we study a continuous-time mean-variance portfolio selection problem in which an agent is subject to a constraint that the expectation of the agent’s terminal wealth must exceed a target and minimize the variance of the agent’s terminal wealth. The agent can revise the expected terminal wealth target dynamically to adapt to the change of the agent’s current wealth, and we consider the following three targets: (i) the agent’s current wealth multiplied by a target expected gross return rate, (ii) the risk-free payoff of the agent’s current wealth plus a premium, and (iii) a weighted average of the risk-free payoff of the agent’s current wealth and a preset aspiration level. We derive the so-called equilibrium strategy in closed form for each of the three targets and find that the agent effectively minimizes the variance of the instantaneous change of the agent’s wealth subject to a certain constraint on the expectation of the instantaneous change of the agent’s wealth.

Continuous-time mean-variance portfolio selection with regime-switching financial market: Time-consistent solution

Abstract This paper studies optimal time-consistent strategies for the mean-variance portfolio selection problem. Especially, we assume that the price processes of risky stocks are described by regime-switching SDEs. We consider a Markov-modulated state-dependent risk aversion and we formulate the problem in the game theoretic framework. Then, by solving a flow of forward-backward stochastic differential equations, an explicit representation as well as uniqueness results of an equilibrium solution are obtained.

Mean-Variance Portfolio Selection with Tracking Error Penalization

This paper studies a variation of the continuous-time mean-variance portfolio selection where a tracking-error penalization is added to the mean-variance criterion. The tracking error term penalizes the distance between the allocation controls and a reference portfolio with same wealth and fixed weights. Such consideration is motivated as follows: (i) On the one hand, it is a way to robustify the mean-variance allocation in the case of misspecified parameters, by “fitting" it to a reference portfolio that can be agnostic to market parameters; (ii) On the other hand, it is a procedure to track a benchmark and improve the Sharpe ratio of the resulting portfolio by considering a mean-variance criterion in the objective function. This problem is formulated as a McKean–Vlasov control problem. We provide explicit solutions for the optimal portfolio strategy and asymptotic expansions of the portfolio strategy and efficient frontier for small values of the tracking error parameter. Finally, we compare the Sharpe ratios obtained by the standard mean-variance allocation and the penalized one for four different reference portfolios: equal-weights, minimum-variance, equal risk contributions and shrinking portfolio. This comparison is done on a simulated misspecified model, and on a backtest performed with historical data. Our results show that in most cases, the penalized portfolio outperforms in terms of Sharpe ratio both the standard mean-variance and the reference portfolio.

An Extended McKean — Vlasov Dynamic Programming Approach to Robust Equilibrium Controls under Ambiguous Covariance Matrix

This paper studies a general class of time-inconsistent stochastic control problems under ambiguous covariance matrix. The time-inconsistency is caused in various ways by a general objective functional and thus the associated control problem does not admit Bellman's principle of optimality. Moreover, we model the state by a McKean — Vlasov dynamics under a set of non-dominated probability measures induced by the ambiguous covariance matrix of the noises. We apply a game-theoretic concept of subgame perfect Nash equilibrium to develop a robust equilibrium control approach, which can yield robust time-consistent decisions. We characterize the robust equilibrium control and equilibrium value function by an extended optimality principle and then we further deduce a system of Bellman — Isaacs equations to determine the equilibrium solution on the Wasserstein space of probability measures. The proposed analytical framework is illustrated with its applications to robust continuous-time mean-variance portfolio selection problems with risk aversion coefficient being constant or state-dependent, under the ambiguity stemming from ambiguous volatilities of multiple assets or ambiguous correlation between two risky assets. The explicit equilibrium portfolio solutions are represented in terms of the probability law.

Provisions for bank deposit withdrawals and portfolio selection

The primary economic function of a bank is to redirect funds from savers to borrowers in an efficient manner, while increasing the value of the bank’s asset holdings in absolute terms. Within the regulatory framework of the Basel III accord, banks are required to maintain minimum liquidity to guard against withdrawals/liquidity risks. In this paper, we analyze a continuous-time mean-variance portfolio selection for a bank with stochastic withdrawal provisioning by relating the reserves as a proxy for the assets held by the bank. We then formulate an optimal investment portfolio selection for a banker by constructing a special Riccati equation as a continuous solution to the Hamilton–Jacobi–Bellman (HJB) equation under mean-variance paradigm. We obtain an explicit closed form solution for the optimal investment portfolio as well as the efficient frontier. The aforementioned modeling enables us to formulate a stochastic optimal control problem related to the minimization of the reserve, depository, and intrinsic risk that are associated with the reserve process.

Mean–Variance Portfolio Selection Under Volterra Heston Model

Motivated by empirical evidence for rough volatility models, this paper investigates continuous-time mean-variance (MV) portfolio selection under the Volterra Heston model. Due to the non-Markovian and non-semimartingale nature of the model, classic stochastic optimal control frameworks are not directly applicable to the associated optimization problem. By constructing an auxiliary stochastic process, we obtain the optimal investment strategy, which depends on the solution to a Riccati-Volterra equation. The MV efficient frontier is shown to maintain a quadratic curve. Numerical studies show that both roughness and volatility of volatility materially affect the optimal strategy.

Mean-Variance Portfolio Selection with Tracking Error Penalization

This paper studies a variation of the continuous-time mean-variance portfolio selection where a tracking-error penalization is added to the mean-variance criterion. The tracking error term penalizes the distance between the allocation controls and a reference portfolio with same wealth and fixed weights. Such consideration is motivated as follows: (i) On the one hand, it is a way to robustify the mean-variance allocation in case of misspecified parameters, by “fitting” it to a reference portfolio that can be agnostic to market parameters; (ii) On the other hand, it is a procedure to track a benchmark and improve the Sharpe ratio of the resulting portfolio by considering a mean-variance criterion in the objective function. This problem is formulated as a McKean-Vlasov control problem. We provide explicit solutions for the optimal portfolio strategy and asymptotic expansions of the portfolio strategy and efficient frontier for small values of the tracking error parameter. Finally, we compare the Sharpe ratios obtained by the standard mean-variance allocation and the penalized one for four different reference portfolios: equal-weights, minimum-variance, equal risk contributions and shrinking portfolio. This comparison is done on a simulated misspecified model, and on a backtest performed with historical data. Our results show that in most cases, the penalized portfolio outperforms in terms of Sharpe ratio both the standard mean-variance and the reference portfolio.

Open-loop equilibrium strategy for mean-variance portfolio selection: A log-return model

<p style='text-indent:20px;'>This paper investigates a continuous-time mean-variance portfolio selection problem based on a log-return model. The financial market is composed of one risk-free asset and multiple risky assets whose prices are modelled by geometric Brownian motions. We derive a sufficient condition for open-loop equilibrium strategies via forward backward stochastic differential equations (FBSDEs). An equilibrium strategy is derived by solving the system. To illustrate our result, we consider a special case where the interest rate process is described by the Vasicek model. In this case, we also derive the closed-loop equilibrium strategy through the dynamic programming approach.

CONTINUOUS-TIME MEAN–VARIANCE OPTIMIZATION FOR DEFINED CONTRIBUTION PENSION FUNDS WITH REGIME-SWITCHING

Using mean–variance (MV) criterion, this paper investigates a continuous-time defined contribution (DC) pension fund investment problem. The framework is constructed under a Markovian regime-switching market consisting of one bank account and multiple risky assets. The prices of the risky assets are governed by geometric Brownian motion while the accumulative contribution evolves according to a Brownian motion with drift and their correlation is considered. The market state is modeled by a Markovian chain and the random regime-switching is assumed to be independent of the underlying Brownian motions. The incorporation of the stochastic accumulative contribution and the correlations between the contribution and the prices of risky assets makes our problem harder to tackle. Luckily, based on appropriate Riccati-type equations and using the techniques of Lagrange multiplier and stochastic linear quadratic control, we derive the explicit expressions of the optimal strategy and efficient frontier. Further, two special cases with no contribution and no regime-switching, respectively, are discussed and the corresponding results are consistent with those results of Zhou &amp; Yin [(2003) Markowitz’s mean-variance portfolio selection with regime switching: A continuous-time model, SIAM Journal on Control and Optimization 42 (4), 1466–1482] and Zhou &amp; Li [(2000) Continuous-time mean-variance portfolio selection: A stochastic LQ framework, Applied Mathematics and Optimization 42 (1), 19–33]. Finally, some numerical analyses based on real data from the American market are provided to illustrate the property of the optimal strategy and the effects of model parameters on the efficient frontier, which sheds light on our theoretical results.

Continuous-Time Mean-Variance Portfolio Selection: A Reinforcement Learning Framework

We approach the continuous-time mean-variance (MV) portfolio selection with reinforcement learning (RL). The problem is to achieve the best tradeoff between exploration and exploitation, and is formulated as an entropy-regularized, relaxed stochastic control problem. We prove that the optimal feedback policy for this problem must be Gaussian, with time-decaying variance. We then establish connections between the entropy-regularized MV and the classical MV, including the solvability equivalence and the convergence as exploration weighting parameter decays to zero. Finally, we prove a policy improvement theorem, based on which we devise an implementable RL algorithm. We find that our algorithm outperforms both an adaptive control based method and a deep neural networks based algorithm by a large margin in our simulations.

A simple solution to a continuous-time mean-variance portfolio selection via the mean-variance hedging

A simple solution to a continuous-time mean-variance portfolio selection via the mean-variance hedging

Continuous-time mean-variance portfolio selection with no-shorting constraints and regime-switching

The present article investigates a continuous-time mean-variance portfolio selection problem with regime-switching under the constraint of no-shorting. The literature along this line is essentially dominated by the Hamilton-Jacobi-Bellman (HJB) equation approach. However, in the presence of switching regimes, a system of HJB equations rather than a single equation need to be tackled concurrently, which might not be solvable in terms of classical solutions, or even not in the weaker viscosity sense as well. Instead, we first introduce a general result on the sign of geometric Brownian motion with jumps, then derive the efficient portfolio and frontier via the maximum principle approach; in particular, we observe, under a mild technical assumption on the initial conditions, that the no-shorting constraint will consistently be satisfied over the whole finite time horizon. Further numerical illustrations will be provided.

Resolving Time Inconsistency in Financial Decision Problems With Non-Expectation Operator: From Internal Conflict to Internal Harmony by Strategy of Self-Coordination

When a stochastic decision problem is time inconsistent, the decision maker would always be troubled by his conflicting decisions “optimally” derived from his time-varying preferences at different time instants. The long-run self (LR) of the decision maker pursues the long-term optimality and thus would like to take the pre-committed strategy. On the other hand, the short-run selves (SRs) of the decision maker at different time instants lack foresight, bow to short-term temptations and would like to take the so-called time consistent strategy. As these two extreme strategies have been criticized for their incomplete viewpoints in dealing with time inconsistency, the emerging literature of self-control game models recognizes an importance to strike a balance between the LR’s and SRs’ interests. While most risk measures in financial decisions are of a non-expectation nature, the existing results of self-control game models are confined to situations where the decision maker’s preferences involve only expectation operators. We take a step forward to develop in this study an operable unified two-tier dual-self game model with commitment by punishment in continuous time setting, which can cope with general time inconsistent stochastic financial decision problems with both expectation and non-expectation operators in the objective function. By attaching to both the preferences of LR and SRs certain punishment terms which quantitatively evaluate the internal conflict among different selves, our game model can align the interests of the LR and SRs to any desired degree. The equilibrium strategy, termed strategy of self-coordination in this paper, achieves some degree of internal harmony among various selves. We demonstrate in the paper successful applications of our dual-self game model to two long standing challenges in the literature, the continuous-time investment and consumption problem with quasi-hyperbolic discounting and the continuous-time mean-variance portfolio selection problem.

Dynamic Mean-Risk Asset Allocation and Myopic Strategies: A Universal Portfolio Rule

In a market that consists of multiple stocks and one risk-free asset whose mean return rates and volatility are deterministic, we study a continuous-time mean-variance portfolio selection problem in which an agent is subject to a constraint that the expectation of her terminal wealth must exceed a target and minimizes the variance of her terminal wealth. The agent can revise her expected terminal wealth target dynamically to adapt to the change of her current wealth, and we consider the following three targets: (i) the agent's current wealth multiplied by a target expected gross return rate, (ii) the risk-free payoff of the agent's current wealth plus a premium, and (iii) a weighted average of the risk-free payoff of the agent's current wealth and a pre-set aspiration level. We derive the so-called equilibrium strategy in closed form for each of the three targets and find that the agent effectively minimizes the variance of the instantaneous change of her wealth subject to a certain constraint on the expectation of the instantaneous change of her wealth.

Explicit solutions for continuous time mean–variance portfolio selection with nonlinear wealth equations

This paper concerns the continuous time mean–variance portfolio selection problem with a special nonlinear wealth equation. This nonlinear wealth equation has a nonsmooth coefficient and the dual method developed in Ji (2010) does not work. We invoke the HJB equation of this problem and give an explicit viscosity solution of the HJB equation. Furthermore, via this explicit viscosity solution, we obtain explicitly the efficient portfolio strategy and efficient frontier for this problem. Finally, we show that our nonlinear wealth equation can cover three important cases.

Big Data Challenges of High-Dimensional Continuous-Time Mean-Variance Portfolio Selection and a Remedy.

Investors interested in the global financial market must analyze financial securities internationally. Making an optimal global investment decision involves processing a huge amount of data for a high-dimensional portfolio. This article investigates the big data challenges of two mean-variance optimal portfolios: continuous-time precommitment and constant-rebalancing strategies. We show that both optimized portfolios implemented with the traditional sample estimates converge to the worst performing portfolio when the portfolio size becomes large. The crux of the problem is the estimation error accumulated from the huge dimension of stock data. We then propose a linear programming optimal (LPO) portfolio framework, which applies a constrained ℓ1 minimization to the theoretical optimal control to mitigate the risk associated with the dimensionality issue. The resulting portfolio becomes a sparse portfolio that selects stocks with a data-driven procedure and hence offers a stable mean-variance portfolio in practice. When the number of observations becomes large, the LPO portfolio converges to the oracle optimal portfolio, which is free of estimation error, even though the number of stocks grows faster than the number of observations. Our numerical and empirical studies demonstrate the superiority of the proposed approach.