Markov switching GARCH models for Bayesian hedging on energy futures markets
Introduction
The recent volatile nature of energy commodity (e.g. crude oil, heating oil and gasoline) prices has caused a lot of concern among policy makers, investors, traders and companies operating in energy markets because of the important role it plays in modern economy and the potential losses that can be generated by price fluctuations. In line with this, a number of recent studies on how to hedge energy price risk have emerged in the literature. Hedging is an investment position taken to mitigate the adverse effect arising from changes in the price of a companion investment. A crucial issue is the determination of the optimal hedge ratio, i.e. the number of derivative contracts to buy (or sell) for each unit of the underlying asset on which the investor bears risk (see Chen et al. (2003) for a review). In this paper we focus on crude oil market and contribute to recent studies (see, for example, Alizadeh et al., 2008, Chang et al., 2010 Lee, 2010, Wang and Wu, 2012, Pan et al., 2014) dealing with hedging crude oil risk. We show how designing effective hedging strategies on oil spot and futures markets can help in mitigating against the adverse effect of volatility and volatility transmission across commodity markets. Since volatility regimes and correlation changes have been documented in other commodity markets, such as copper, gold and silver (Bhar and Hammoudeh, 2011), corn, and wheat (Creti et al., 2013), our hedging strategy can also be applied successfully to these markets.
In this paper, we propose a new hedging model based on minimizing the risk of a hedged portfolio. The result of this minimization exercise is the Minimum Variance (MV) hedge ratio defined as the ratio of the covariance between the underlying spot and futures returns to the variance of the futures return (see Johnson (1960)). To apply this optimum hedge ratio in practice, Ederington (1979) suggests regressing the underlying spot returns against the futures returns, and using the estimate of the slope as an MV hedge ratio. This approach has been widely criticized on the grounds that some of the well known stylized facts about asset returns are ignored. Accordingly, to improve the hedging performance, time-varying hedge ratios have been proposed in the literature and its estimation have been developed along two major lines. The first approach involves the estimation of the conditional second order moments of the underlying spot and futures returns. The Generalized Autoregressive Conditional Heteroscedasticity (GARCH) models have been proposed for this by Haigh and Holt, 2002, Chang et al., 2010, Cifarelli and Paladino, 2015, Abul-Bashera and Sadorsky, 2016, Lai and Lein, 2017, among others. The later approach treats the hedge ratio as a time-varying regression coefficient, and focuses on the estimation of such a parameter (e.g. see Alizadeh and Nomikos, 2004, Lee et al., 2006, and Chang et al. (2010)). Note that this hedging strategy works by re-balancing the hedged portfolio on a period-by-period basis. As this may involve huge transaction costs, it may not be worthwhile to use this particular instrument for hedging. It has also been well documented in the empirical literature that the class of GARCH models exhibits high persistence of the conditional variance, i.e. the process is close to being nearly integrated. In view of this, some authors allow the optimal hedge ratio to be state-dependent. Alizadeh et al., 2008, Lee and Yoder, 2007a, Lee and Yoder, 2007b Dark, 2015, Philip and Shi, 2016, Su, 2017, among others, propose various multivariate Markov-switching (MS) GARCH (MS-GARCH) models. More precisely, due to the path dependence problem of MS-GARCH models, these authors implement the multivariate extension of Gray (1996) model with different characterizations of the time-varying covariance matrix. While Gray's model is attractive, its analytical intractability is a drawback since it cannot be derived using any standard analytical approximation technique.
Our hedging model builds on the Billio et al. (2016) approach to MS-GARCH modeling and inference by extending it to a bivariate GARCH model with multiple and possibly dependent MS processes (multichain MS). More specifically, we assume a system of simultaneous equations modeling both return dynamics on the hedged portfolio and futures in lieu of the traditional spot-futures framework. Each component of this system is characterized by a path dependent MS-GARCH process. This specification firstly makes it possible to take care of the estimation risk of the parameters in the hedging model, which has not gained much attention in literature and secondly allows for a better understanding of the cross dynamics of futures and hedging portfolios and their effect on the optimal hedging decisions. Our modeling framework is close to Alizadeh and Nomikos (2004), but we differ in two ways. The first difference lies in the characterization of the time-varying variance process. While Alizadeh and Nomikos (2004) consider a time-varying variance defined by an exponential function of the lagged 4-week moving average of the difference between the logarithm of the underlying and the logarithm of the futures, we consider a MS-GARCH model. The second difference relates to the properties of the underlying hidden process governing the observable processes. Alizadeh and Nomikos (2004) either assume that the conditional variances of futures returns are regime independent or that the hidden process characterizing the dynamics of the hedged portfolio is independent of the one influencing the futures returns process. We account for these limitations in our econometric framework. Still regarding the MS-GARCH framework, Sheu and Lee, 2014, Lai et al., 2017 argue that the dependence of both the derivative and the spot on the same hidden state process might be inappropriate. Thus, the authors propose the use of a multichain Markov regime switching GARCH (MCSG) model. In this paper, we also extend the work of Sheu and Lee (2014) by allowing for simultaneous dependence between the Markov chains of the MSCG model.
Another aim of the paper is to develop a robust hedging approach within the MS-GARCH framework. In practice, the parameters in the optimal hedge ratio are unknown, thus optimal hedge ratios are estimated by replacing the unknown parameters by their corresponding estimates. This approach is referred to in the literature as the “plug-in” or Parameter Certainty Equivalent (PCE) principle. Generally speaking, decision makers are left to provide, using an estimation technique of their choice, estimates of the model parameters, and to substitute them directly in the theoretical model. One of the problems with this approach is that it completely ignores estimation risk. Depending on the econometric specification considered for estimating the optimal hedge ratio, large differences are observed in the estimated MV hedge ratios on the same commodity. This observation further suggests that it may be very costly to ignore estimation risk. Moreover, possible relevant non-sample information (such as insider information or subjective prior) could be available to the hedger but discarded in the decision making process.
We thus recast the MV hedging model as an expected utility model and deal with the estimation risk problem within this framework. It may be argued that a rational decision maker would choose an action that maximizes its expected utility over the unknown parameter space. Early studies on this problem have been pursued by Raiffa and Schlaifer, 1961, DeGroot, 2005, among others. A review of the application of this theory to portfolio choice, prior to 1978, is provided in Bawa et al. (1979). A more recent application can be found in Kan and Zhou (2007). As appealing as the expected utility theory sounds, it is laden with a number of computational issues. In many empirical analysis, analytical solutions to either the optimization exercise and/or the integration problem are often not achievable. Accordingly, alternative solutions, such as approximation or simulations, are called for. Müller et al. (2004), among others, propose simulation-based approaches to the expected utility optimization problem. In this paper, we propose a robust hedging ratio that accounts not only for parameter uncertainty, but also for different states of the market. We follow a Bayesian decision rule (see, for example, Lence and Hayes, 1994a, Lence and Hayes, 1994b) to account for parameter uncertainty in the definition of optimal hedging strategies.
The structure of the paper is as follows. In the next section, we present the conventional MV hedge ratio as well our revised approach and the Bayesian hedging strategy. In Section 3, we discuss the empirical application of our proposed model to West Texas Intermediate (WTI) crude oil spot and futures prices and compare the result with the conventional OLS method proposed by Ederington (1979). Section 4 concludes the paper.
Section snippets
Bayesian optimal hedging
Let be a probability observation space, with {Pθ} θ∈Θ a parametric family of probability distributions and θ a parameter in the measurable parameter space . Let , be an observable process, where RSt, RFt, respectively, correspond to returns on the underlying and returns on the derivative (e.g., option, futures) at time t. Let us define the information set available at time t, as the σ-algebra generated by yt, t = 1,…,T and denote with ys:t = (ys,…,yt)
An application to energy markets
The goal of this section is twofold. First, we aim to apply our model to provide empirical evidence of the effects of the recent financial crisis on the crude oil markets. Second we want to assess the efficiency of the proposed hedging models and to compare them.
Conclusion
In this paper we propose a new Bayesian multichain MS-GARCH model with dependent chains. We apply the model to hedging in energy markets, thus extending the existing literature on MV hedging. The proposed parameterization of the multichain MS-GARCH model allows for a straightforward interpretation of the parameters of the models as level-shift and variance-covariance hedging components. Both the Bayesian model and the inference approach allow us to easily account for parameter uncertainty in
Acknowledgments
We thank the seminar participants at the SIS 2013 Statistical Conference, Brescia, 2013. The authors' research is supported by funding from the European Union, Seventh Framework Programme FP7/2007-2013 under grant agreement SYRTO-SSH-2012-320270, by the Institut Europlace of Finance, “Systemic Risk grant”, the Global Risk Institute in Financial Services, the Louis Bachelier Institute, “Systemic Risk Research Initiative”, and by the Italian Ministry of Education, University and Research (MIUR)
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