A Bayesian Implementation of the Standard Optimal Hedging Model: Parameter Estimation Risk and Subjective Views

We propose a Bayesian implementation of the standard optimal hedging model that effectively and practically accommodates estimation errors and subjective views regarding both the expectation vector and the covariance matrix of asset returns. Numerical examples show that subjective views have a substantial impact on a hedger’s optimal position and that the impact of views regarding the direction of future price changes far outweighs that of views regarding the standard deviation of future price changes. keywords: optimal hedging, parameter estimation risk, subjective views, Bayesian Decision Theory. A Bayesian Implementation of the Standard Optimal Hedging Model: Parameter Estimation Risk and Subjective Views The standard optimal hedging model (Johnson, 1960; Stein, 1961; Anderson and Danthine, 1980) has been the preferred theoretical model of normative hedging behavior for some time. In empirical applications, the model is often implemented with a Parameter Certainty Equivalent (PCE) procedure, which directly substitutes sample estimates for the model’s parameters in determining the optimal hedging position. However, the PCE procedure completely ignores parameter estimation risk, i.e., the estimation errors in the expectation vector and covariance matrix of returns on the assets involved in the hedging decision. Furthermore, the PCE procedure cannot accommodate hedgers’ subjective views, which refer to hedgers’ opinions (“views”) regarding the direction of market returns of the assets involved in hedging decisions. The problem of decision making in the presence of parameter uncertainty has long been recognized and has been analyzed in a Bayesian decision theory framework. Within a portfolio optimization context, the Bayesian framework has been used to accommodate parameter estimation risk and subjective views (e.g., Brown, 1979; Jorion, 1985, 1986; Frost and Savarino, 1986; Black and Litterman, 1990, 1992; Polson and Tew, 2000; Pastor, 2000). Since optimal hedging can be considered a special case of portfolio optimization, the Bayesian framework can be applied to optimal hedging to accommodate the problems identified with the PCE procedure. In the first formal hedging applications, Lence and Hayes (1994a,b) develop a Bayesian optimal hedging model that can effectively accommodate parameter estimation risk. However their model can only accommodate subjective views under restrictive and unrealistic assumptions due to its underlying “pure” Bayesian approach. The pure Bayesian approach requires hedgers to calibrate their prior distribution with non-sample information including subjective views. However, in practice most hedgers are unlikely to have subjective views on more than one or two parameters of the prior distribution or only the relative relation of the parameters of the prior distribution. Thus, it is unlikely that hedgers can calibrate the entire prior distribution with only non-sample information. Shi and Irwin (2005) argue that the Bayesian framework should be implemented with an “empirical” Bayesian approach when applied to optimal hedging. The reason is that with an empirical Bayesian approach hedgers calibrate the prior distribution with sample data, which, compared with non-sample information, should contain enough information regarding all the parameters of the prior distribution. Furthermore, with an empirical Bayesian approach the number and type of subjective views that hedgers can express is quite flexible. For example, hedgers can have one or more subjective views that may be in the form of “absolute” or “relative” views regarding expected asset returns. However, Shi and Irwin (2005) only consider estimation risk and subjective views regarding the expectation vector of asset returns, ignoring those regarding the covariance matrix of asset returns. Empirical work clearly indicates that time-varying volatility prevails in many economic and financial time series, and conditional volatility models such as GARCH/ARCH-type models have been widely used in estimating the volatilities and correlations of asset returns. In addition, other methods such as implied volatility, factor models, exponential weighting methods and Bayesian shrinkage estimators have also been developed to estimate the covariance matrix. Given the large array of estimation methods and potentially different data sets available, hedgers may obtain estimates of volatility and correlation quite different from those obtained by most other market participants (market consensus). The differences in covariance matrix estimates may have a significant impact on hedgers’ optimal hedging positions. For example, Myers (1991), Lien and Luo (1994), and Kroner and Sultan (1993) model the behavior of spot and futures prices with bivariate GARCH models and propose various dynamic hedging strategies. In this study, we propose a Bayesian implementation of the standard optimal hedging model that accommodates estimation risk and subjective views regarding both the expectation vector and the covariance matrix of asset returns. The Bayesian framework is applied to optimal hedging with an empirical Bayesian approach in order to accommodate subjective views in a practical and realistic manner. Compared with Bayesian models proposed in previous studies, the new Bayesian model solves the problems identified with the PCE procedure in a more satisfactory and complete manner. Theoretical Model To begin, we assume that (1) the hedger has no other investment opportunities and does not borrow or lend, (2) markets are frictionless, which means no commissions, no margin requirements and no lumpiness due to standardization of futures contracts, (3) asset price changes follow a multivariate normal distribution, (4) the hedger has a long position in the