Optimal mean-variance portfolio selection under regime-switching-induced stock price shocks (2507.19824v1)

Abstract: In this paper, we investigate mean-variance (MV) portfolio selection problems with jumps in a regime-switching financial model. The novelty of our approach lies in allowing not only the market parameters -- such as the interest rate, appreciation rate, volatility, and jump intensity -- to depend on the market regime, but also in permitting stock prices to experience jumps when the market regime switches, in addition to the usual micro-level jumps. This modeling choice is motivated by empirical observations that stock prices often exhibit sharp declines when the market shifts from a bullish'' to abearish'' regime, and vice versa. By employing the completion-of-squares technique, we derive the optimal portfolio strategy and the efficient frontier, both of which are characterized by three systems of multi-dimensional ordinary differential equations (ODEs). Among these, two systems are linear, while the first one is an $\ell$-dimensional, fully coupled, and highly nonlinear Riccati equation. In the absence of regime-switching-induced stock price shocks, these systems reduce to simple linear ODEs. Thus, the introduction of regime-switching-induced stock price shocks adds significant complexity and challenges to our model. Additionally, we explore the MV problem under a no-shorting constraint. In this case, the corresponding Riccati equation becomes a $2\ell$-dimensional, fully coupled, nonlinear ODE, for which we establish solvability. The solution is then used to explicitly express the optimal portfolio and the efficient frontier.

• The paper derives optimal portfolio strategies and efficient frontiers under regime-switching, addressing stock price jumps.
• It formulates a complex stochastic control problem using Riccati equations and ODEs for both unconstrained and no-shorting scenarios.
• The study establishes feasibility conditions essential for mean-variance optimization in dynamic markets with abrupt regime shifts.

Optimal Portfolio Selection with Regime-Switching and Stock Price Shocks

This paper addresses the problem of optimal mean-variance (MV) portfolio selection in a financial market characterized by regime-switching and jumps in stock prices induced by these regime shifts. The model extends classical MV frameworks by incorporating a continuous-time Markov chain to represent market regimes, with stock prices experiencing jumps concurrent with regime transitions, alongside standard micro-level jumps. The paper derives explicit optimal portfolio strategies and efficient frontiers, and it examines both unconstrained and no-shorting constrained investment scenarios.

Model Formulation and Feasibility

The financial market model comprises one risk-free asset and multiple tradable stocks. The stock prices follow jump-diffusion dynamics modulated by a continuous-time Markov chain, $\alpha$, representing the market regime. A key feature is the inclusion of jumps in stock prices that occur when the Markov chain transitions between regimes. This is modeled using terms of the form $S_{k,t-} (1 + \gamma_{k,t}^{i,j})$, where $\gamma_{k,t}^{i,j}$ represents the jump in the $k$-th stock price when the regime switches from $i$ to $j$. This addition captures the empirical observation that market regime shifts often correlate with abrupt stock price movements. The investor aims to minimize the variance of the terminal wealth, $X_T$, subject to a target expected return, $z$. The problem is formulated as a standard MV optimization, and its feasibility is addressed by deriving conditions under which the set of admissible portfolios is non-empty.

A crucial step is establishing the feasibility of the MV problem, i.e., ensuring the existence of a portfolio that satisfies the expected return constraint. The paper derives conditions for feasibility, which depend on the market parameters and the regime-switching dynamics. These conditions, given by equations (2.7) and (2.8), provide practical criteria for verifying whether the MV problem is well-posed.

Unconstrained MV Portfolio Selection

In the absence of trading constraints, the optimal portfolio strategy and efficient frontier are derived using a completion-of-squares technique. The problem is transformed into a stochastic linear-quadratic (LQ) control problem, and the optimal control is obtained in feedback form. The optimal portfolio is characterized by three systems of multi-dimensional ODEs: a fully coupled, highly nonlinear Riccati equation (3.2), and two linear ODEs (3.3) and (3.4). The nonlinearity of the Riccati equation arises from the regime-switching-induced stock price shocks, making the problem significantly more complex than models without such shocks. In particular, the paper highlights that in the absence of regime-switching-induced stock price shocks ($\gamma^{i,j} = 0$), the Riccati equation degenerates into a linear ODE. This underscores the complexity introduced by the regime-switching-induced jumps.

Theorem 3.1 establishes the existence and uniqueness of solutions to the Riccati equation and the associated linear ODEs. This result is critical for ensuring that the optimal portfolio strategy is well-defined. Theorem 3.2 provides the optimal control in feedback form, given by equation (3.5), and the corresponding optimal value function, given by equation (3.6). These results offer explicit expressions for the optimal investment strategy and the resulting portfolio value.

Theorem 3.3 characterizes the optimal portfolio and efficient frontier for the original MV problem. The optimal portfolio, given by equation (3.8), is expressed in terms of the solutions to the ODE systems. The efficient frontier, given by equation (3.9), provides a relationship between the expected return and the variance of the optimal portfolio.

MV Portfolio Selection with No-Shorting Constraint

The paper extends its analysis to the case where short-selling of stocks is prohibited. The no-shorting constraint introduces additional complexity, leading to a $2\ell$-dimensional, fully coupled, nonlinear ODE (4.3). The paper establishes the solvability of this Riccati equation and derives the optimal portfolio and efficient frontier.

The paper defines mappings $H_{+,t}^i$ and $H_{-,t}^i$ in equations (4.1) and (4.2), which are essential for characterizing the Riccati equation under the no-shorting constraint. These mappings capture the impact of the non-negativity constraint on the portfolio selection process. Theorem 4.1 establishes the existence of a unique positive solution to the Riccati equation (4.3) under the no-shorting constraint. This result is crucial for ensuring that the optimal portfolio strategy is well-defined in this constrained setting.

Theorem 4.2 provides the optimal feedback control for the LQ problem under the no-shorting constraint, given by equation (4.4). This control is expressed in terms of the solutions to the Riccati equation and the initial wealth level. Theorem 4.3 characterizes the optimal portfolio and efficient frontier for the original MV problem with the no-shorting constraint. The optimal portfolio, given by equation (4.6), is expressed in terms of the solutions to the Riccati equation. The efficient frontier, given by equation (4.7), is shown to be a half-line, providing a clear relationship between the expected return and the variance of the optimal portfolio.

Conclusion

This paper provides a comprehensive analysis of the MV portfolio selection problem in a regime-switching market with stock price shocks. The explicit solutions for the optimal portfolio and efficient frontier, derived under both unconstrained and no-shorting constrained scenarios, offer valuable insights for practitioners. The paper highlights the mathematical challenges introduced by regime-switching-induced stock price shocks and no-shorting constraints.

A potential avenue for future research is the extension of the model to non-Markovian settings, which would necessitate the paper of highly nonlinear BSDEs. Addressing the case of general stochastic coefficients, where the Riccati equations will be BSDEs, presents a significantly more complex challenge.