A conditional higher-moment CAPM

This paper investigates whether dynamic and moment extensions to the traditional CAPM can improve its empirical performance and offer some alternative explanation to the cross-section of average returns on portfolios of stocks double sorted on book-to-market ratios and size. We consider three extensions. First, we introduce time-varying factor loadings obtained from a multivariate GARCH and dynamic conditional correlations. Second, we extend the model to a four-moment CAPM, which incorporates coskewness and cokurtosis. Finally, we allow for time-varying risk premia, based on a Markov-switching process. Our results confirm that the higher-moment CAPM does not perform well in its unconditional version, but its performance is significantly improved when we introduce a conditional version that accounts for both time-varying factor loadings and time-varying risk premia. The four-moment CAPM tests lead to a positive total risk premium estimate of 0.67% per month over the period 1926–2021, with all risk premia (beta, coskewness, and cokurtosis) exhibiting the expected theoretical signs.     

The conditional CAPM does not explain asset-pricing anomalies

Recent studies suggest that the conditional CAPM holds, period by period, and that time-variation in risk and expected returns can explain why the unconditional CAPM fails. In contrast, we argue that variation in betas and the equity premium would have to be implausibly large to explain important asset-pricing anomalies like momentum and the value premium. We also provide a simple new test of the conditional CAPM using direct estimates of conditional alphas and betas from short-window regressions, avoiding the need to specify conditioning information. The tests show that the conditional CAPM performs nearly as poorly as the unconditional CAPM, consistent with our analytical results.     

Option pricing under regime switching

Abstract

This paper develops a family of option pricing models when the underlying stock price dynamic is modelled by a regime switching process in which prices remain in one volatility regime for a random amount of time before switching over into a new regime. Our family includes the regime switching models of Hamilton (Hamilton J 1989 Econometrica 57 357–84), in which volatility influences returns. In addition, our models allow for feedback effects from returns to volatilities. Our family also includes GARCH option models as a special limiting case. Our models are more general than GARCH models in that our variance updating schemes do not only depend on levels of volatility and asset innovations, but also allow for a second factor that is orthogonal to asset innovations. The underlying processes in our family capture the asymmetric response of volatility to good and bad news and thus permit negative (or positive) correlation between returns and volatility. We provide the theory for pricing options under such processes, present an analytical solution for the special case where returns provide no feedback to volatility levels, and develop an efficient algorithm for the computation of American option prices for the general case.     

A Markov regime switching approach for hedging energy commodities

This paper estimates constant and dynamic hedge ratios in the New York Mercantile Exchange oil futures markets and examines their hedging performance. We also introduce a Markov regime switching vector error correction model with GARCH error structure. This specification links the concept of disequilibrium with that of uncertainty (as measured by the conditional second moments) across high and low volatility regimes. Overall, in and out-of-sample tests indicate that state dependent hedge ratios are able to provide significant reduction in portfolio risk.     

A PDE approach for risk measures for derivatives with regime switching

This paper considers a partial differential equation (PDE) approach to evaluate coherent risk measures for derivative instruments when the dynamics of the risky underlying asset are governed by a Markov-modulated geometric Brownian motion (GBM); that is, the appreciation rate and the volatility of the underlying risky asset switch over time according to the state of a continuous-time hidden Markov chain model which describes the state of an economy. The PDE approach provides market practitioners with a flexible and effective way to evaluate risk measures in the Markov-modulated Black–Scholes model. We shall derive the PDEs satisfied by the risk measures for European-style options, barrier options and American-style options.      

Optimal dividend distribution under Markov regime switching

We investigate the problem of optimal dividend distribution for a company in the presence of regime shifts. We consider a company whose cumulative net revenues evolve as a Brownian motion with positive drift that is modulated by a finite state Markov chain, and model the discount rate as a deterministic function of the current state of the chain. In this setting, the objective of the company is to maximize the expected cumulative discounted dividend payments until the moment of bankruptcy, which is taken to be the first time that the cash reserves (the cumulative net revenues minus cumulative dividend payments) are zero. We show that if the drift is positive in each state, it is optimal to adopt a barrier strategy at certain positive regime-dependent levels, and provide an explicit characterization of the value function as the fixed point of a contraction. In the case that the drift is small and negative in one state, the optimal strategy takes a different form, which we explicitly identify if there are two regimes. We also provide a numerical illustration of the sensitivities of the optimal barriers and the influence of regime switching.     

Dynamic optimal capital structure with regime switching

We investigate the optimal capital structure of a corporate when the dynamics of the assets (both growth rate and volatility) change following different states of the economy. Two structural models are examined in the paper. The first considers the case when the firm is not facing tax benefit and bankruptcy costs with a regime switching dynamics. This model extends the Black and Cox (J Financ 31:351–367, 1976) model to allow for regime switching risk. The second model incorporates both tax benefit and bankruptcy costs with a regime switching dynamics. This is is more realistic, and is an extension of the Leland (J Financ 49(4):1213–1252, 1994) model with regime switching risk. We obtain closed-form analytic solutions for the optimal capital structure and default barrier for both models.     

Value-at-Risk: a multivariate switching regime approach

This paper analyses the application of a switching volatility model to forecast the distribution of returns and to estimate the Value-at-Risk (VaR) of both single assets and portfolios. We calculate the VaR value for 10 Italian stocks and a number of portfolios based on these stocks. The calculated VaR values are also compared with the variance–covariance approach used by JP Morgan in RiskMetrics™ and GARCH(1,1) models. Under backtesting, the VaR values calculated using the switching regime beta model are preferred to both other methods. The Proportion of Failure and Time Until First Failure tests [The Journal of Derivatives (1995) 73–84] confirm this result.     

Estimating a regime switching pairs trading model

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.     

Option pricing and Esscher transform under regime switching

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).We would like to thank the referees for many helpful and insightful comments and suggestions.Correspondence to: R. J. Elliott     

On real interest rate dynamics and regime switching

We find evidence of regime switching dynamics in the USA and the UK real interest rates over the period 1881–2003. For the UK, there is a regime in which the real interest rate displays a relatively stronger mean-reversion and a regime in which it displays a relatively weaker mean-reversion. The former regime is characterized by a relatively larger error in the estimation of the reversion parameter, and higher volatility. For the USA, the two regimes differ in volatility. The probability of transition from one regime to another is found to be significantly related to the inflation rate regime, and to the political regime. The results highlight the importance of regime switching in the dynamics of the real interest rate, as well as the role of inflation and political regimes in explaining this switching.     

Value-at-risk in a market subject to regime switching

Many empirical researches report that value-at-risk (VaR) measures understate the actual 1% quantile, while for Inui, K., Kijima, M. and Kitano, A., VaR is subject to a significant positive bias. Stat. Probab. Lett., 2005, 72, 299–311. proved that VaR measures overstate significantly when historical simulation VaR is applied to fat-tail distributions. This paper resolves the puzzle by developing a regime switching model to estimate portfolio VaR. It is shown that our model is able to correct the underestimation problem of risk.     

Learning about beta: Time-varying factor loadings, expected returns, and the conditional CAPM

We amend the conditional CAPM to allow for unobservable long-run changes in risk factor loadings. In this environment, investors rationally “learn” the long-run level of factor loadings from the observation of realized returns. As a consequence of this assumption, we model conditional betas using the Kalman filter. Because of its focus on low-frequency variation in betas, our approach circumvents recent criticisms of the conditional CAPM. When tested on portfolios sorted by size and book-to-market, our learning-augmented conditional CAPM passes the specification tests.     

A stochastic differential game for optimal investment of an insurer with regime switching

We introduce a model to discuss an optimal investment problem of an insurance company using a game theoretic approach. The model is general enough to include economic risk, financial risk, insurance risk, and model risk. The insurance company invests its surplus in a bond and a stock index. The interest rate of the bond is stochastic and depends on the state of an economy described by a continuous-time, finite-state, Markov chain. The stock index dynamics are governed by a Markov, regime-switching, geometric Brownian motion modulated by the chain. The company receives premiums and pays aggregate claims. Here the aggregate insurance claims process is modeled by either a Markov, regime-switching, random measure or a Markov, regime-switching, diffusion process modulated by the chain. We adopt a robust approach to model risk, or uncertainty, and generate a family of probability measures using a general approach for a measure change to incorporate model risk. In particular, we adopt a Girsanov transform for the regime-switching Markov chain to incorporate model risk in modeling economic risk by the Markov chain. The goal of the insurance company is to select an optimal investment strategy so as to maximize either the expected exponential utility of terminal wealth or the survival probability of the company in the ‘worst-case’ scenario. We formulate the optimal investment problems as two-player, zero-sum, stochastic differential games between the insurance company and the market. Verification theorems for the HJB solutions to the optimal investment problems are provided and explicit solutions for optimal strategies are obtained in some particular cases.     

The frequency of regime switching in financial market volatility

The mechanism of risk responses to market shocks is considered as stagnant in recent financial literature, whether during normal or stress periods. Since the returns are heteroskedastic, a little consideration was given to volatility structural breaks and diverse states. In this study, we conduct extensive simulations to prove that the switching regime GARCH model, under the highly flexible skewed generalized t (SGT) distribution, is remarkably efficient in detecting different volatility states. Next, we examine the switching regime in the S&P 500 volatility for weekly, daily, 10-minute and 1-minute returns. Results show that the volatility switches regimes frequently, and differences between the distributions of the high and low volatility states become more accentuated as the frequency increases. Moreover, the SGT is highly preferable to the usually employed skewed t distribution.     

Optimal investment-reinsurance with dynamic risk constraint and regime switching

We study an optimal investment–reinsurance problem for an insurer who faces dynamic risk constraint in a Markovian regime-switching environment. The goal of the insurer is to maximize the expected utility of terminal wealth. Here the dynamic risk constraint is described by the maximal conditional Value at Risk over different economic states. The rationale is to provide a prudent investment–reinsurance strategy by taking into account the worst case scenario over different economic states. Using the dynamic programming approach, we obtain an analytical solution of the problem when the insurance business is modeled by either the classical Cramer–Lundberg model or its diffusion approximation. We document some important qualitative behaviors of the optimal investment–reinsurance strategies and investigate the impacts of switching regimes and risk constraint on the optimal strategies.     

Pricing of discount bonds with a Markov switching regime

We consider a Markov switching regime and price a discount bond using a CIR-type short rate model. An explicit formula is obtained for the bond price which includes the solution of a matrix ODE. Our model is easy to calculate and captures the effect of regime uncertainty in the price and term structure.     

Moments of multivariate regime switching with application to risk-return trade-off

We use a Fourier transform to derive multivariate conditional and unconditional moments of multi-horizon returns under a regime-switching model. These moments are applied to examine the relevance of risk horizon and regimes for buy-and-hold investors. We analyze the impact of time-varying expected returns and risk (variance and covariance) on portfolio allocations' “term structure”—portfolio allocations as a function of the investment horizon. Using monthly observations on S&P composite index and 10-year Government Bond, we find that the term structure of the optimal allocations depends on market conditions measured by the probability of being in bull state. At short horizons and when this probability is low, buy-and-hold investors decrease their holdings of risky assets. We also find that the conditional optimal portfolio performs quite well at short and intermediate horizons and less at long horizons.