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Time-Consistent Mean-Variance Pairs-Trading Under Regime-Switching Cointegration
... In practice, parameters and can be estimated using an online updated procedure combined with the maximum likelihood estimation (which is commonly used in calibrating stochastic volatility models, see Wang et al. [17] for more detail) rolling on the historical data set. We also refer to Chambers [2] for details about testing and estimating continuous-time cointegration models. ...
... In a number of companion papers (Leung andYan, 2018, 2019;Angoshtari and Leung, 2019a,b), the utility maximization approach is used to derive dynamic futures trading strategies under various stochastic models without regime switching. Also, we refer to Zhou and Yin (2003); Leung (2010); Chen et al. (2019) for continuous-time portfolio optimization with regime switching. In comparison to these studies, the current paper investigates the trading of derivatives (futures), instead of stocks, in a general regime-switching market framework that can be applied to an array of regime-switching models. ...
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.
... In a number of companion papers (Leung andYan, 2018, 2019;Angoshtari and Leung, 2019a,b), the utility maximization approach is used to derive dynamic futures trading strategies under various stochastic models without regime switching. Also, we refer to Zhou and Yin (2003); Leung (2010); Chen et al. (2019) for continuous-time portfolio optimization with regime switching. In comparison to these studies, the current paper investigates the trading of derivatives (futures), instead of stocks, in a general regime-switching market framework that can be applied to an array of regime-switching models. ...
... In a number of companion papers (Leung andYan, 2018, 2019;Angoshtari and Leung, 2019a,b), the utility maximization approach is used to derive dynamic futures trading strategies under various stochastic models without regime switching. Also, we refer to Zhou and Yin (2003); Leung (2010); Chen et al. (2019) for continuous-time portfolio optimization with regime switching. In comparison to these studies, the current paper investigates the trading of derivatives (futures), instead of stocks, in a general regime-switching market framework that can be applied to an array of regime-switching models. ...
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.
Cointegration analysis is an econometric tool used to identify equilibrium among assets and construct a pairs trading portfolio. The discrete-time vector error correction model (VECM) for identifying cointegration includes lag difference terms as explanatory variables, thus permitting delayed adjustment of the deviations from equilibrium. The continuous-time limit of the VECM becomes a stochastic delay differential equation. We investigate the dynamic open-loop equilibrium mean–variance pairs trading strategy under such a delayed cointegration model. The existence and uniqueness results for the equilibrium are offered. We prove in general that the equilibrium strategy can lead to statistical arbitrage under certain conditions that are related to the roots of the corresponding characteristic equation. We obtain an explicit solution for a case with distributed delay. Our empirical study demonstrates the superiority of our strategy over its Markovian counterpart when the model selection result prefers a high-order VECM.
The optimal retirement decision is an optimal stopping problem when retirement is irreversible. We investigate the optimal consumption, investment, and retirement decisions when the mean return of a risky asset is unobservable and is estimated by filtering from historical prices. To ensure nonnegativity of the consumption rate and the borrowing constraints on the wealth process of the representative agent, we conduct our analysis using a duality approach. We link the dual problem to American option pricing with stochastic volatility and prove that the duality gap is closed. We then apply our theory to a hidden Markov model for regime-switching mean return with Bayesian learning. We fully characterize the existence and uniqueness of variational inequality in the dual optimal stopping problem, as well as the free boundary of the problem. An asymptotic closed-form solution is derived for optimal retirement timing by small-scale perturbation. We discuss the potential applications of the results to other partial-information settings.
In this paper, we study a time consistent solution for a defined contribution pension plan under a mean-variance criterion with regime switching in a jump-diffusion setup, during the accumulation phase. We consider a market consisting of a risk-free asset and a geometric jump-diffusion risky asset process. Our solution allows the fund manager to incorporate a clause which allows for the distribution of a member’s premiums to his surviving dependents, should the member die before retirement. Applying the extended Hamilton-Jacobi-Bellman (HJB) equation, we derive the explicit time consistent equilibrium strategy and the value function. We then provide some numerical simulations to illustrate our results.
Longevity securitization enables insurers to manage longevity or mortality risk in the life market. Recent empirical studies identify long-range dependence (LRD) in mortality rates using life tables, which casts doubt on the adequacy of Markovian models for actuarial pricing and risk management. This paper derives the first time-consistent mean–variance longevity hedging strategy for insurers using a stochastic mortality model with LRD. We adopt the open-loop equilibrium control framework and derive an analytical solution to the hedging strategy. Our numerical examples show the relevance of LRD to longevity hedging and the cost of ignoring it.
This paper proposes a regression-based simulation algorithm for multi-period mean-variance portfolio optimization problems with constraints under a high-dimensional setting. For a high-dimensional portfolio, the least squares Monte Carlo algorithm for portfolio optimization can perform less satisfactorily with finite sample paths due to the estimation error from the ordinary least squares (OLS) in the regression steps. Our algorithm, which resolves this problem e, that demonstrates significant improvements in numerical performance for the case of finite sample path and high dimensionality. Specifically, we replace the OLS by the least absolute shrinkage and selection operator (lasso). Our major contribution is the proof of the asymptotic convergence of the novel lasso-based simulation in a recursive regression setting. Numerical experiments suggest that our algorithm achieves good stability in both low- and higher-dimensional cases.
We consider a discrete time pairs trading model which includes regime changes in the dynamics. The prices of the pair of assets, and so their difference or spread, depend on the state of the market, which in turn is modelled by a finite state Markov chain. Different states of the chain give rise to different parameters in the dynamics of the spread. However, the state of the chain is not observed directly but only through the prices or spread. Based on observations of the spread, this paper provides recursive estimates for both the state of the market and all coefficients in the model.
In this paper, which is a continuation of the previously published discrete time paper we develop a theory for continuous time stochastic control problems which, in various ways, are time inconsistent in the sense that they do not admit a Bellman optimality principle. We study these problems within a game theoretic framework, and we look for Nash subgame perfect equilibrium points. For a general controlled continuous time Markov process and a fairly general objective functional we derive an extension of the standard Hamilton-Jacobi-Bellman equation, in the form of a system of non-linear equations, for the determination for the equilibrium strategy as well as the equilibrium value function. As applications of the general theory we study non exponential discounting, various types of mean variance problems, a point process example, as well as a time inconsistent linear quadratic regulator. We also present a study of time inconsistency within the framework of a general equilibrium production economy of Cox-Ingersoll-Ross type.
This paper studies the problem of determining the optimal cut-off for pairs
trading rules. We consider two correlated assets whose spread is modelled by a
mean-reverting process with stochastic volatility, and the optimal pair trading
rule is formulated as an optimal switching problem between three regimes: flat
position (no holding stocks), long one short the other and short one long the
other. A fixed commission cost is charged with each transaction. We use a
viscosity solutions approach to prove the existence and the explicit
characterization of cut-off points via the resolution of quasi-algebraic
equations. We illustrate our results by numerical simulations.
This work develops Feynman-Kac formulas for a class of regime-switching jump diffusion processes, in which the jump part is driven by a Poisson random measure associated to a general L\'evy process and the switching part depends on the jump diffusion processes. Under broad conditions, the connections of such stochastic processes and the corresponding partial integro-differential equations are established. Related initial, terminal, and boundary value problems are also treated. Moreover, based on weak convergence of probability measures,
it is demonstrated that a sequence of random variables related to the regime-switching jump diffusion process converges in distribution to the arsine law.
In this paper we discuss the calibration of models built on mean-reverting processes combined with Markov regime-switching (MRS). We propose a method that greatly reduces the computational burden induced by the introduction of independent regimes and perform a simulation study to test its efficiency. Our method allows for a 100 to over 1000 times faster calibration than in case of a competing approach utilizing probabilities of the last 10 observations. It is also more general and admits any value of gamma in the base regime dynamics. Since the motivation for this research comes from a recent stream of literature in energy economics, we apply the new method to sample series of electricity spot prices from the German EEX and Australian NSW markets. The proposed MRS models fit these datasets well and replicate the major stylized facts of electricity spot price dynamics.
Although models of cointegrated financial time series are now relatively common place in the literature their importance has, until very recently, been mainly theoretical. This is because the traditional starting point for portfolio risk management in practice is a correlation analysis of returns, whereas cointegration is based on the raw price, rate or yield data. In standard risk-return models these price data are differenced before the analysis is even begun, and differencing removes a-priori any long-term trends in the data. Of course these trends are implicit in the returns data, but any decision based on long-term common trends in the price data is excluded in standard risk-return modelling. Cointegration and correlation are related, but different concepts. High correlation of returns does not necessarily imply high cointegration in prices. An example is given in figure 1, with a 10-year daily series on US dollar spot exchange rates of the German Mark (DEM) and the Dutch Guilder (NLG) from 1975 to 1985. Their returns are very highly correlated: the correlation coefficient is approximately 0.98 (figure 1(a)). So also do the rates move together over long periods of time, and they appear to be cointegrated (figure 1(b)). Now suppose that we add a very small daily incremental return of, say, 0.0002 to NLG. The
The main objective of a hedge strategy is to generate positive returns irrespective of market conditions. This paper presents a classic hedge fund strategy: an investment vehicle whose key objective is to minimize investment risk in an attempt to deliver profits under all market circumstances. The hedge is designed to have minimal correlation with the market and, irrespective of market direction, the fund seeks to generate positive alpha. A significant difference between this model and more traditional hedge fund strategies is that portfolio optimization is based upon the cointegration of prices rather than the correlation of returns. Models that are based on mean-variance analysis seek portfolio weights to minimise the variance of the portfolio for a given level of return. The portfolio variance is measured using a covariance matrix and these matrices are notoriously difficult to estimate. Moreover the mean-variance criterion has nothing to ensure that trac king errors are stationary. Although the portfolios will be efficient, the tracking errors will in all probability be random walks. Therefore the replicating portfolio can drift very far from the benchmark unless it is frequently re-balanced. This paper shows that it is possible to devise allocations that have stationary tracking errors: any strategy that guarantees a stationary tracking error must be based on cointegration. Efficient long short hedge strategies may be achieved with relatively few stocks and with much lower turnover rates compared to traditional market neutral strategies.
In this paper, Hamilton's (1988, 1989) Markov-switching model is extended to a general state-space model. This paper also complements Shumway and Stoffer's (1991) dynamic linear models with switching, by introducing dependence in the switching process, and by allowing switching in both measurement and transition equations. Building upon ideas in Hamilton (1989), Cosslett and Lee (1985), and Harrison and Stevens (1976), a basic filtering and smoothing algorithm is presented. The algorithm and the maximum likelihood estimation procedure is applied in estimating Lam's (1990) generalized Hamilton model with a general autoregressive component. The estimation results show that the approximation employed in this paper performs an excellent job, with a considerable advantage in computation time.A state–space representation is a very flexible form, and the approach taken in this paper therefore allows a broad class of models to be estimated that could not be handled before. In addition, the algorithm for calculating smoothed inferences on the unobserved states is a vastly more efficient one than that in the literature.
Filtering and parameter estimation techniques from hidden Markov Models are applied to a discrete time asset allocation problem. For the commonly used mean-variance utility explicit optimal strategies are obtained.
This paper deals with error correction models (ECM's) and cointegrated systems that are formulated in continuous time. Problems of representation, identification, estimation and time aggregation are discussed. It is shown that every ECM in continuous time has a discrete time equivalent model in ECM format. Moreover, both models may be written as triangular systems with stationary errors. This formulation simplifies both the continuous and the discrete time ECM representations and it helps to motivate a class of optimal inference procedures. It is further shown that long run equilibria in the continuous system are always identified in the discrete time reduced form, so that there is no aliasing problem for these coefficients.
In this paper, we study the effects of cointegration on optimal investment and consumption strategies for an investor with exponential utility. A Hamilton-Jacobi-Bellman (HJB) equation is derived first and then solved analytically. Both the optimal investment and consumption strategies are expressed in closed form. A verification theorem is also established to demonstrate that the solution of the HJB equation is indeed the solution of the original optimization problem under an integrability condition. In addition, a simple and sufficient condition is proposed to ensure that the integrability condition is satisfied. Financially, the optimal investment and consumption strategies are decomposed into two parts: the myopic part and the hedging demand caused by cointegration. Discussions on the hedging demand are carried out first, based on analytical formulae. Then numerical results show that ignoring the information about cointegration results in a utility loss.
When a financial institution has a difficulty to close a position in an illiquid asset, a practical solution is to hedge the risk of the position with a related liquid asset. This may frequently occur in insurance institutions who diversify asset allocation by illiquid investment and face high transaction costs when try to close the illiquid asset position. A conventional approach constructs optimal hedging based on the correlation between the liquid and illiquid assets. We show, however, that ignoring the serial dependence through cointegration significantly reduces the hedging effectiveness. This paper investigates the time-consistent mean-variance optimal hedging using cointegration for the risk of a non-traded portfolio. Specifically, only the investment amount in the liquid asset is the control variable. The mathematical challenge stems on the derivation of the optimal equilibrium strategy when we have to deal with cointegration, illiquidity and time-inconsistency simultaneously. Using a stochastic differential game approach, we characterize the equilibrium hedging strategy from a system of partial differential equations that involves an HJB equation for the time-consistent problem with cointegration. Novel solutions to the equilibrium strategy and the efficient frontier are obtained in explicit closed-form formulas. The significance of cointegration in the corresponding hedging problem is evidenced with real empirical data.
This paper studies equilibrium in a pure exchange economy with unobservable Markov switching growth regimes and beliefs-dependent risk aversion (BDRA). Risk aversion is stochastic and depends nonlinearly on consumption and beliefs. Equilibrium is obtained in closed form. The market price of risk, the interest rate, and the stock return volatility acquire new components tied to fluctuations in beliefs. A three-regime specification is estimated using the generalized method of moments (GMM). Model moments match their empirical counterparts for a variety of unconditional moments, including the equity premium, stock returns volatility, and the correlations between stock returns and consumption and dividends. Dynamic features of the data, such as the countercyclical behaviors of the equity premium and volatility, are also captured. Model volatility provides a good fit for realized volatility. A new factor, the information risk premium, is found to be a strong predictor of future excess returns. These results are obtained with an estimated risk aversion fluctuating between 1.44 and 1.93.
This paper investigates the robust optimal pairs trading using the concept of equivalent probability measures and a penalty function associated with the confidence in parameter estimates when the parameters in the drift term of the continuous-time cointegration model are estimated with errors. A closed-form solution is derived for the robust pairs trading rule. We compare the robust pairs trading rule against its non-robust counterpart using simulations and real data. The robust strategy is empirically more stable and less volatile.
We introduce a regime-switching Ornstein–Uhlenbeck (O–U) model to address an optimal investment problem. Our study gives a closed-form expression for a regime-switching pairs trading value function consisting of probability and expectation of the double boundary stopping time of the Markov-modulated O–U process. We derive analytic solutions for the homogenous and non-homogenous ODE systems with initial value conditions for probability and expectation of the double boundary stopping time, and translate the solutions with boundary value conditions into solutions with initial value conditions. Based on the smoothness and continuity of the value function, we can obtain the optimum of the value function with thresholds and guarantee the existence of optimal thresholds in a finite closed interval. Our numerical analysis illustrates the rationality of theoretical model and the shape of transition probability and expected stopping time, as well as discusses sensitivity analysis in both one-state and two-state regime-switching models. We find that the optimal expected return per unit time in the two-state regime-switching model is higher than that of one-state regime-switching model. Likewise, the regime-switching model’s optimal thresholds are closer and more symmetric to the long-term mean.
Local regime-switching models are a natural consequence of combining the concept of a local volatility model with that of a regime-switching model. However, even though Elliott et al. (2015) have derived a Dupire formula for a local regime-switching model, its calibration still remains a challenge, primarily due to the fact that the derived volatility function for each state involves all the state price variables whereas only one market price is available for model calibration, and a direct implementation of Elliott et al.’s formula may not yield stable results. In this paper, a closed system for option pricing and data extraction under the classical regime-switching model is proposed with a special approach, splitting one market price into two “market-implied state prices”. The success of our approach hinges on the recovery of the two local volatility functions being transformed into an optimal control problem, which is solved through the Tikhonov regularization. In addition, an efficient algorithm is proposed to obtain the optimal solution by iteration. Our numerical experiments show that different shapes of local volatility functions can be accurately and stably recovered with the newly-proposed algorithm, and this algorithm also works quite well with real market data.
This paper discusses an optimal investment–consumption problem in a continuous-time co-integration model, where an investor
aims to maximize an expected, discounted utility derived from intertemporal consumption and terminal wealth in a finite time
horizon. Using the dynamic programming principle approach, we obtain an Hamilton–Jacobi–Bellman equation related to the problem.
In each of the power and logarithmic utility cases, we obtain semi-closed-form expressions of the investment–consumption strategy
and the value function. We also provide some numerical examples to illustrate these theoretical results.
We study the optimal trading policy of an arbitrageur who can exploit temporary mispricing in a market with two convergent assets. We build on the model of Liu and Timmermann (2013) and include transaction costs, which impose additional limits to the implementation of such convergence trade strategy. We show that the presence of transaction costs could reveal an endogenous stop-loss concern in a certain economy, which affects the optimal policy of the arbitrageur in significant ways. Using pairs of Chinese stock shares that are dual-listed in Shanghai and in Hong Kong, we show that our co-integration strategy is generally superior to the relative-value pairs trading strategy studied in Gatev et al. (2006). Several extensions of our model are also discussed.
Cointegration is a useful econometric tool for identifying assets which share a common equilibrium. Cointegrated pairs trading is a trading strategy which attempts to take a profit when cointegrated assets depart from their equilibrium. This paper investigates the optimal dynamic trading of cointe-grated assets using the classical mean-variance portfolio selection criterion. To ensure rational economic decisions, the optimal strategy is obtained over the set of time-consistent policies from which the optimization problem is enforced to obey the dynamic programming principle. We solve the opti-mal dynamic trading strategy in a closed-form explicit solution. This ana-lytical tractability enables us to prove rigorously that cointegration ensures the existence of statistical arbitrage using a dynamic time-consistent mean-variance strategy. This provides the theoretical grounds for the market belief in cointegrated pairs trading. Comparison between time-consistent and pre-commitment trading strategies for cointegrated assets shows the former to be a persistent approach, whereas the latter makes it possible to generate infinite leverage once a cointegrating factor of the assets has a high mean reversion rate.
This paper investigates the time-consistent dynamic mean–variance hedging of longevity risk with a longevity security contingent on a mortality index or the national mortality. Using an HJB framework, we solve the hedging problem in which insurance liabilities follow a doubly stochastic Poisson process with an intensity rate that is correlated and cointegrated to the index mortality rate. The derived closed-form optimal hedging policy articulates the important role of cointegration in longevity hedging. We show numerically that a time-consistent hedging policy is a smoother function in time when compared with its time-inconsistent counterpart.
In this article, we provide the first study in the time consistent solution of the mean–variance asset–liability management (MVALM). The framework is even considered under a continuous time Markov regime-switching setting. Using the extended Hamilton–Jacobi–Bellman equation (HJB) (see Björk and Murgoci (2010)), we show that the time consistent equilibrium control is state dependent in the sense that it depends on the uncontrollable liability process, which is in substantial contrast with the time consistent solution of the similar problem in Björk and Murgoci (2010), in which it is independent of the state. Finally, we give a numerical comparison between our work with the corrected version (as obtained here) of pre-commitment strategy in Chen et al. (2008).
We propose a model for analyzing dynamic pairs trading strategies using the stochastic control approach. The model is explored in an optimal portfolio setting, where the portfolio consists of a bank account and two co-integrated stocks and the objective is to maximize for a fixed time horizon, the expected terminal utility of wealth. For the exponential utility function, we reduce the problem to a linear parabolic Partial Differential Equation which can be solved in closed form. In particular, we exhibit the optimal positions in the two stocks.
This paper considers the continuous-time mean-variance portfolio selection problem in a financial market in which asset prices are cointegrated. The asset price dynamics are then postulated as the diffusion limit of the corresponding discrete-time error-correction model of cointegrated time series. The problem is completely solved in the sense that solutions of the continuous-time portfolio policy and the efficient frontier are obtained as explicit and closed-form formulas. The analytical results are applied to pairs trading using cointegration techniques. Numerical examples show that identifying a cointegrated pair with a high mean-reversion rate can generate significant statistical arbitrage profits once the current state of the economy sufficiently departs from the long-term equilibrium. We propose an index to simultaneously measure the departure level of a cointegrated pair from equilibrium and the mean-reversion speed based on the mean-variance paradigm. An empirical example is given to illustrate the use of the theory in practice.
This paper formulates the differential equations typical of a Markov problem in system-reliability theory in a systematic way in order to generate computer-oriented procedures. The coefficient matrix of these equations (the transition rate matrix) can be obtained for the whole system through algebraic operations on component transition-rate matrices. Such algebraic operations are performed according to the rules of Kronecker Algebra. We consider system reliability and availability with stress dependence and maintenance policies. Theorems are given for constructing the system matrix in four cases: * Reliability and availability with on-line multiple or single maintenance. * Reliability and availability with system-state dependent failure rates. * Reliabilityand availability with standby components. * Off-line maintainability. The results are expres § ed in algebraic terms and as a consequence their implementation by a computer program is straightforward. We also obtain information about the structure of the matrices involved. Such information can considerably improve computational efficiency of the computer codes because it allows introducing special ideas and techniques developed for large-system analysis such as sparsity, decomposition, and tearing.
This paper introduces the concept of statistical arbitrage, a long horizon trading opportunity that generates a riskless profit and is designed to exploit persistent anomalies. Statistical arbitrage circumvents the joint hypothesis dilemma of traditional market efficiency tests because its definition is independent of any equilibrium model and its existence is incompatible with market efficiency. We provide a methodology to test for statistical arbitrage and then empirically investigate whether momentum and value trading strategies constitute statistical arbitrage opportunities. Despite adjusting for transaction costs, the influence of small stocks, margin requirements, liquidity buffers for the marking-to-market of short-sales, and higher borrowing rates, we find evidence that these strategies generate statistical arbitrage.
This paper introduces an EM algorithm for obtaining maximum likelihood estimates of parameters for processes subject to discrete shifts in autoregressive parameters, with the shifts themselves modeled as the outcome of a discrete-valued Markov process. The simplicity of the EM algorithm permits potential application of the approach to large vector systems.
This work is concerned with the stability of a class of switching jump-diffusion processes. The processes under consideration can be thought of as a number of jump-diffusion processes modulated by a random switching device. The motivation of our study stems from a wide range of applications in communication systems, flexible manufacturing and production planning, financial engineering, and economics. A distinct feature of the two-component process (X(t),α(t)) considered in this paper is that the switching process α(t) depends on the X(t) process. This paper focuses on the long-time behavior, namely, stability of the switching jump diffusions. First, the definitions of regularity and stability are recalled. Next it is shown that under suitable conditions, the underlying systems are regular or have no finite explosion time. To study stability of the trivial solution (or the equilibrium point 0), systems that are linearizable (in the x variable) in a neighborhood of 0 are considered. Sufficient conditions for stability and instability are obtained. Then, almost sure stability is examined by treating a Lyapunov exponent. The stability conditions present a gap for stability and instability owing to the maximum and minimal eigenvalues associated with the drift and diffusion coefficients. To close the gap, a transformation technique is used to obtain a necessary and sufficient condition for stability.
A continuous-time version of the Markowitz mean-variance portfolio selection model is proposed and analyzed for a market consisting of one bank account and multiple stocks. The market parameters, including the bank interest rate and the appreciation and volatility rates of the stocks, depend on the market mode that switches among a finite number of states. The random regime switching is assumed to be independent of the underlying Brownian motion. This essentially renders the underlying market incomplete. A Markov chain modulated diffusion formulation is employed to model the problem. Using techniques of stochastic linear-quadratic control, mean-variance efficient portfolios and efficient frontiers are derived explicitly in closed forms, based on solutions of two systems of linear ordinary differential equations. Related issues such as a minimum-variance portfolio and a mutual fund theorem are also addressed. All the results are markedly different from those for the case when there is no regime switching. An interesting observation is, however, that if the interest rate is deterministic, then the results exhibit (rather unexpected) similarity to their no-regime-switching counterparts, even if the stock appreciation and volatility rates are Markov-modulated.
We consider the bond valuation problem when the short rate process is described by a Markovian regime-switching Hull-White model or a Markovian regime-switching Cox-Ingersoll-Ross model. In each of the two short rate models, we establish a Markov-modulated exponential-affine bond price formula with coefficients given in terms of fundamental matrix solutions of linear matrix differential equations.
We solve the dynamic mean-variance portfolio problem and derive its time-consistent solution using dynamic programming. Previous
literature, in contrast, only determines either myopic or precommitment (committing to follow the initially optimal policy)
solutions. We provide a fully analytical simple characterization of the dynamically optimal mean-variance portfolios within
a general incomplete-market economy. We also identify a probability measure that incorporates intertemporal hedging demands
and facilitates tractability. We illustrate this by easily computing portfolios explicitly under various stochastic investment
opportunities. A calibration exercise shows that the mean-variance hedging demands are economically significant.
This study applies a cointegration system that considers regime shifts in order to study the long-run relationship between the stock index and stock index futures markets. The MSCI Taiwan, Nikkei 225, Hong Kong Hang-Seng, and Singapore Exchange (SGX) Straits Times indices are examined. The empirical evidence shows that the cointegration system with regime shifts performs better than the usual cointegration system without considering regime shifts.
Summary We consider the option pricing problem when the risky underlying assets are driven by Markov-modulated Geometric Brownian Motion (GBM). That is, the market parameters, for instance, the market interest rate, the appreciation rate and the volatility of the underlying risky asset, depend on unobservable states of the economy which are modelled by a continuous-time Hidden Markov process. The market described by the Markov-modulated GBM model is incomplete in general and, hence, the martingale measure is not unique. We adopt a regime switching random Esscher transform to determine an equivalent martingale pricing measure. As in Miyahara [33], we can justify our pricing result by the minimal entropy martingale measure (MEMM).
This paper examines, from a market efficiency perspective, the performance of a simple dynamic equity indexing strategy based on cointegration. A consistent 'abnormal' return in excess of the benchmark is demonstrated over different time horizons and in different real world and simulated stock markets. A measure of stock price dispersion is shown to be a leading indicator for the abnormal return and their relationship is modelled as a Markov switching process of two market regimes. We find that the entire abnormal return is associated with the high volatility regime as the indexing model implicitly adopts a strategic position that pays off during market crashes, whilst effectively tracking the benchmark in normal market circumstances. Therefore we find no evidence of market inefficiency. Nevertheless our results have implications for equity fund managers: we show how, without any stock selection, solely through a smart optimization that has an implicit element of market timing, the benchmark performance can be significantly enhanced. Copyright © 2005 John Wiley & Sons, Ltd.
The relationship between cointegration and error correction models, first suggested by Granger, is here extended and used to develop estimation procedures, tests, and empirical examples. A vector of time series is said to be cointegrated with cointegrating vector a if each element is stationary only after differencing while linear combinations a8xt are themselves stationary. A representation theorem connects the moving average , autoregressive, and error correction representations for cointegrated systems. A simple but asymptotically efficient two-step estimator is proposed and applied. Tests for cointegration are suggested and examined by Monte Carlo simulation. A series of examples are presented. Copyright 1987 by The Econometric Society.
We study a discrete-time version of Markowitz's mean-variance portfolio selection problem where the market parameters depend on the market mode (regime) that jumps among a finite number of states. The random regime switching is delineated by a finite-state Markov chain, based on which a discrete-time Markov modulated portfolio selection model is presented. Such models either arise from multiperiod portfolio selections or result from numerical solution of continuous-time problems. The natural connections between discrete-time models and their continuous-time counterpart are revealed. Since the Markov chain frequently has a large state space, to reduce the complexity, an aggregated process with smaller state-space is introduced and the underlying portfolio selection is formulated as a two-time-scale problem. We prove that the process of interest yields a switching diffusion limit using weak convergence methods. Next, based on the optimal control of the limit process obtained from our recent work, we devise portfolio selection strategies for the original problem and demonstrate their asymptotic optimality.
- G Yin
- C Zhu
G. Yin and C. Zhu, Hybrid Switching Diffusions: Properties and Applications, Stoch. Model. Appl.
Prob. 63, Springer, New York, 2010.