Regime-switching Market Research Articles

Mean-variance portfolio selection with an uncertain exit-time in a regime-switching market

In this paper, we deal with multi-period mean-variance portfolio selection problems with an exogenous uncertain exit-time in a regime-switching market. The market is modelled by a non-homogeneous Markov chain in which the random returns of assets depend on the states of the market and investment time periods. Applying the Lagrange duality method, we derive explicit closed-form expressions for the optimal investment strategies and the efficient frontier. Also, we show that some known results in the literature can be obtained as special cases of our results. A numerical example is provided to illustrate the results.

Optimal execution with regime-switching market resilience

In this paper, we study the optimal placement of market orders in a limit order book (LOB) market when the market resilience rate, which is the rate at which market replenishes itself after each trade, is stochastic. More specifically, we establish a tractable extension to the optimal execution model in Obizhaeva and Wang (2013) by modelling the dynamics of the resilience rate to be driven by a Markov chain. When the LOB replenishes itself stochastically through time, the optimal execution strategy becomes state-dependent, and is driven linearly by the current remaining position and the current temporary price impact, with their linear dependence based on the expectation of the dynamics of future resilience rate. A trader would optimally place more aggressive (respectively, conservative) market orders when the limit order book switches from a low to a high resilience state, (respectively, from a high to a low resilience state). Our cost saving analysis indicates that the incremental execution costs can be substantial when the agent ignores the stochastic dynamics of the market resilience rate by adopting the state-independent strategies.

Time-consistent and self-coordination strategies for multi-period mean-Conditional Value-at-Risk portfolio selection

The multi-period mean-Conditional Value-at-Risk (mean-CVaR) portfolio decision model is prone to time inconsistency problems that drive CVaR investors away from the pre-committed portfolio strategy, although this strategy is regarded as the global optimal strategy at the initial time point. In the existing literature, authors have proposed time-consistent and self-coordination strategies to solve the time inconsistency issue arising from other sequential decision problems. However, these strategies are rarely studied under the multi-period mean-CVaR portfolio decision framework. This work fills in these gaps by providing both computationally tractable methods and analytical solutions for these strategies. The revealed time-consistent strategy is a piecewise linear function of the wealth level, wherein the other parameters can be computed by solving a series of mixed-integer programming problems off line. The self-coordination strategy can be formulated as a convex program with a quadratic constraint. We also prove that the pre-committed strategy and the time-consistent strategy are the extreme cases of the self-coordination strategy. Furthermore, we extend our main findings to a regime-switching market setting.

Optimal mean–variance investment/reinsurance with common shock in a regime-switching market

In this paper, we consider the problem of optimal investment-reinsurance with two dependent classes of insurance risks in a regime-switching financial market. In our model, the two claim-number processes are correlated through a common shock component, and the market mode is classified into a finite number of regimes. We also assume that the insurer can purchase proportional reinsurance and invest its surplus in a financial market, and that the values of the model parameters depend on the market mode. Using the techniques of stochastic linear-quadratic control, under the mean–variance criterion, we obtain analytic expressions for the optimal investment and reinsurance strategies, and derive closed-form expressions for the efficient strategies and the efficient frontiers which are based on the solutions to some systems of linear ordinary differential equations. Finally, we carry out a numerical study for illustration purpose.

Optimal Dynamic Futures Portfolio in a Regime-Switching Market Framework

We study the problem of dynamically trading futures in a regime-switching market. Modeling the underlying asset price as a Markov-modulated diffusion process, we present a utility maximization approach to determine the optimal futures trading strategy. This leads to the analysis of the associated system of Hamilton-Jacobi-Bellman (HJB) equations, which are reduced to a system of linear ODEs. We apply our stochastic framework to two models, namely, the Regime-Switching Geometric Brownian Motion (RS-GBM) model and Regime-Switching Exponential Ornstein-Uhlenbeck (RS-XOU) model. Numerical examples are provided to illustrate the investor's optimal futures positions and portfolio value across market regimes.

Multi-Period Mean-Variance Portfolio Selection with State-Dependent Exit Probability and Bankruptcy State

Based on the mean-variance portfolio selection under multi-period criterion, this paper focuses on the study of the uncertain time horizon and the regime-switching market including the bankruptcy state, where the conditional distribution of exit time is followed by the market state. When the market enters the bankruptcy state, investors are assumed to get back δ part of the wealth from the bankrupt company, where δ refers to the retrieval rate. By introducing the Lagrange multiplier λ, we create an innovative expression for the wealth process and the iterative representation of the value function to obtain the analytical expression of the optimal strategy and the corresponding efficient frontier. Besides, some special cases and numerical examples are presented to demonstrate the effects of state-dependent exit probability and bankruptcy state on the investment strategy.

Optimal Portfolio Selection with Regime-Switching Hamilton-Jacobi-Bellman (HJB) Equation and Maximum Value-at-Risk (MVaR) Constraint

The forming of portfolio is necessary to determine the decision of the best investment so as to investors can identify the securities and determine the allocation of asset to obtain an optimal portfolio. The problem in forming optimal portfolio is the determination of the proportion which is allocated at investment assets in order to maximize expected return with certain risks. The model contains regime-switching market models, which states are interpreted as the states of economy. The risk measuring instrument used is VaR and MVaR is defined as the maximum value of the VaRs in all economy states. The optimal proportion formula is sought by using the stochastic optimal control theory with the aim of maximizing the discounted utility of consumption over a finite time horizon. We use regime-switching Hamilton-Jacobi-Bellman (HJB) equation and then derive a system of coupled HJB equation corresponding to the economy states. Lagrange multiplier method is used to solve the optimization problem with the constraint. We apply Kuhn-Tucker conditions due to MVaR is an inequality function, so that we derive the optimal investment and the optimal consumption. Finally, numerical examples are investigated, and the effect of parameter on the optimal investment and on the optimal consumption are studied.

Optimal investment and dividend for an insurer under a Markov regime switching market with high gain tax

<p style='text-indent:20px;'>This study examines the optimal investment and dividend problem for an insurer with CRRA preference. The insurer's goal is to maximize the expected discounted accumulated utility from dividend before ruin and the insurer subjects to high gain tax payment. Both the surplus process and the financial market are modulated by an external Markov chain. Using the <i>weak dynamic programming principle</i> (WDPP), we prove that the value function of our control problem is the unique viscosity solution to coupled <i>Hamilton-Jacobi-Bellman</i> (HJB) equations with first derivative constraints. Solving an auxiliary problem without regime switching, we prove that, it is optimal for the insurer in a multiple-regime market to adopt the policies in the same way as in a single-regime market. The regularity of the viscosity solution on its domain is proved and thus the HJB equations admits classical solution. A numerical scheme for the value function is provided by the Markov chain approximation method, two numerical examples are given to illustrate the impact of the high gain tax and regime switching on the optimal policies.

Optimal switching under a hybrid diffusion model and applications to stock trading

This paper is concerned with the optimal switching problem under a hybrid diffusion (or, regime switching) model in an infinite horizon. The state of the system consists of a number of diffusions coupled by a finite-state continuous-time Markov chain. Based on the dynamic programming principle, the value function of our optimal switching problem is proved to be the unique viscosity solution to the associated system of variational inequalities. The optimal switching strategy, indicating when and where it is optimal to switch, is given in terms of the switching and continuation regions. In many applications, the underlying Markov chain has a large state space and exhibits two-time-scale structure. In this case, a singular perturbation approach is employed to reduce the computational complexity involved. It is shown that as the time-scale parameter ε goes to zero, the value function of the original problem converges to that of a limit problem. The limit problem is much easier to solve, and its optimal switching solution leads to an approximate solution to the original problem. Finally, as an application of our theoretical results, an example concerning the stock trading problem in a regime switching market is provided. It is emphasized that, this paper is the first time to introduce the optimal switching as a general framework to study the stock trading problem, in view of their inherent connection. Both optimal trading rule and convergence result are numerically demonstrated in this example.

Explicit solutions to utility maximization problems in a regime-switching market model via Laplace transforms

We study the problem of utility maximization from terminal wealth in which an agent optimally builds her portfolio by investing in a bond and a risky asset. The asset price dynamics follow a diffusion process with regime-switching coefficients modeled by a continuous-time finite-state Markov chain. We consider an investor with a Constant Relative Risk Aversion (CRRA) utility function. We deduce the associated Hamilton–Jacobi–Bellman equation to construct the solution and the optimal trading strategy and verify optimality by showing that the value function is the unique constrained viscosity solution of the HJB equation. By means of a Laplace transform method, we show how to explicitly compute the value function and illustrate the method with the two- and three-states cases. This method is interesting in its own right and can be adapted in other applications involving hybrid systems and using other types of transforms with basic properties similar to the Laplace transform.

Pricing derivatives in a regime switching market with time inhomogenous volatility

ABSTRACTThis paper studies pricing derivatives in a componentwise semi-Markov (CSM) modulated market. We consider a financial market where the asset price dynamics follows a regime switching geometric Brownian motion model in which the coefficients depend on finitely many age-dependent semi-Markov processes. We further allow the volatility coefficient to depend on time explicitly. Under these market assumptions, we study locally risk minimizing pricing of a class of European options. It is shown that the price function can be obtained by solving a non-local Black–Scholes–Merton-type PDE. We establish existence and uniqueness of a classical solution to the Cauchy problem. We also find another characterization of price function via a system of Volterra integral equation of second kind. This alternative representation leads to computationally efficient methods for finding price and hedging. An explicit expression of quadratic residual risk is also obtained.

Portfolio selection in a regime switching market with a bankruptcy state and an uncertain exit-time: multi-period mean–variance formulation

This paper studies three versions of the multi-period mean–variance portfolio selection problem, that are: minimum variance problem, maximum expected return problem and the trade-off problem, in a Markovian regime switching market where the exit-time is uncertain and exogenous. The underlying Markov chain contains an absorbing state, which denotes the bankruptcy state. When the Markov chain switches to this state, the investors only get a random fraction, known as the recovery rate, taking values in [0, 1] of their wealth. Asset returns, as well as recovery rate, depend on the market state. Dynamic programming and Lagrange duality method are used to derive analytical expressions for optimal investment strategies and the mean–variance efficient frontier. It is shown that portfolio selection models with no bankruptcy state and certain exit-time can be considered as special cases of our model. Some numerical examples are provided to demonstrate the effect of the recovery rate and exit-probabilities.

Polynomial upper and lower bounds for financial derivative price functions under regime-switching

We present a new approach to bounding financial derivative prices in regime-switching market models from both above and below. We derive sufficient conditions under which a particular class of functions act as bounds for the prices of financial derivatives in regime-switching market models. Using these sufficient conditions, we then formulate, in a general setting, optimization problems whose solutions can be identified with tight upper and lower bounds. The problems are made numerically tractable by imposing polynomial structures and employing results from the theory of sum-of-squares polynomials to arrive at a semidefinite programming problem that is implementable by existing software. The bounds obtained take the form of smooth polynomial functions and are valid for a continuous range of initial times and states. Moreover, they are obtained without recourse to sample path simulation or discretization of the temporal or spatial variables. We demonstrate the effectiveness of the proposed method on European-, barrier- and American-style options in several regime-switching settings with and without jumps.

Option pricing under regime switching: Integration over simplexes method

This paper aims to develop an alternative method for pricing European options under regime-switching market conditions by representing their values as a sum of integrations over simplexes. We calculate the integrations by using the method of Grundmann and Möller (1978). The method is applicable to the valuation of European-type options written on underlying assets whose prices follow a regime-switching mean-reverting process as well as a conventional regime-switching geometric Brownian motion. Numerical examples provide evidence that this method can be a powerful tool for practitioners in option pricing.

Regime-switching pure jump processes and applications in the valuation of mortality-linked products

ABSTRACTIn this paper, we study the pricing of longevity bonds and an insurance contract on multiple lives in a regime-switching market driven by an underlying continuous-time Markov chain. For modeling dependent mortality, we make use of a Markov chain and some shot noise processes with regime switching. By using a martingale method, we give semi-analytical expressions for the price of longevity bonds and the premium of an insurance contract on the kth person to die.

Valuation of CDS counterparty risk under a reduced-form model with regime-switching shot noise default intensities

We study the counterparty risk for a credit default swap (CDS) in a regime-switching market driven by an underlying continuous-time Markov chain. We model the default dependence via some correlated Cox processes with regime-switching shot noise intensities containing common shock. Under the proposed model, the general bilateral counterparty risk pricing formula for CDS contracts with the possibility of joint defaults is presented. Based on some expressions for the conditional Laplace transform of the integrated intensity processes, semi-analytical solution for the bilateral credit valuation adjustment (CVA) is derived. When the model parameters satisfy some conditions, explicit formula for the bilateral CVA at time 0 is also given.

Optimal investment–consumption strategy with liability and regime switching model under Value-at-Risk constraint

This paper considers the optimal investment–consumption problem with liability subject to a maximum Value-at-Risk (denoted by MVaR) constraint. The model also contains regime-switching market modes, whose states are interpreted as the states of the economy. In each state of economy state, we constrain a VaR value for the portfolio in a short time duration, and MVaR is defined as the maximum value of the VaRs in all economy states. We suppose that both the price dynamics of the risky asset and the liability value process are governed by a Markov-modulated geometric Brownian motion. With the objective of maximizing the discounted utility of consumption, we obtain a system of HJB equations corresponding to the economy states by using the dynamic programming principle. Then, by adopting the techniques of Chen et al. (2010), we get explicit expressions of value functions. Moreover, with the help of Lagrange multiplier method, we derive the optimal investment and the optimal consumption. Finally, a numerical example is investigated, and the effects of many parameters on the optimal investment, on the optimal consumption and on VaR value are studied. Furthermore, we also explore how VaR value affects the optimal investment and the optimal consumption.

Dual control Monte-Carlo method for tight bounds of value function in regime switching utility maximization

In this paper, we study the dual control approach for the optimal asset allocation problem in a continuous-time regime-switching market. We find the lower and upper bounds of the value function that is a solution to a system of fully coupled nonlinear partial differential equations. These bounds can be tightened with additional controls to the dual process. We suggest a Monte-Carlo algorithm for computing the tight lower and upper bounds and show the method is effective with a variety of utility functions, including power, non-HARA and Yaari utilities. The latter two utilities are beyond the scope of any current methods available in finding the value function.

Portfolio Selection with Capital Gains Tax, Recursive Utility, and Regime Switching

Capital gains taxation has important implications for investors’ portfolio choice decisions. To explore these implications, we develop a continuous time investment and consumption model with capital gains tax, Epstein–Zin recursive utility, and regime switching. We find that various factors, such as tax rate, risk aversion, interest rate, stock return, and volatility, jointly affect optimal portfolio allocation, whereas intertemporal substitution does not. In a regime switching market, investors may trade or stop trading purely because of a change in regime, and there is a distinct cross-regime effect on optimal portfolio allocation. In particular, investors tend to raise stock investment in a bear regime so as to reduce potential tax payments upon regime switching. Given reasonable parameter values, regime switching has a greater impact on optimal portfolio allocation in a bear regime than in a bull regime. This paper was accepted by Neng Wang, finance.

Multi-period defined contribution pension funds investment management with regime-switching and mortality risk

Using mean–variance criterion, we investigate a multi-period defined contribution pension fund investment problem in a Markovian regime-switching market. Both stochastic wage income and mortality risk are incorporated in our model. In a regime-switching market, the market mode changes among a finite number of regimes, and the market state process is modeled by a Markov chain. The key parameters, such as the bank interest rate, or expected returns and covariance matrix of stocks, will change according to the market state. By virtue of Lagrange duality technique, dynamic programming approach and matrix representation method, we derive expressions of efficient investment strategy and its efficient frontier in closed-form. Also, we study some special cases of our model. Finally, a numerical example based on real data from the American market sheds light on our theoretical results.