برآورد پارامتر در بازار کالا : یک روش فیلتر شده

The application of Kalman filtering methods and maximum likelihood parameter estimation to models of commodity prices and futures prices has been considered by several authors. The usual method of finding the maximum likelihood parameter estimates (MLEs) is to numerically maximize the likelihood function. We present, as an alternative to numerical maximization of the likelihood, a filter-based implementation of the expectation maximization (EM) algorithm that can be used to find the MLEs. Finite-dimensional filters are derived that allow the MLEs of a commodity price model to be estimated from futures price data using the EM algorithm without calculating Kalman smoother estimates.

The purpose of this paper is to provide a demonstration of the main steps of a filtering approach to parameter estimation for a model of commodity spot prices (such as crude oil) using futures price data. A one-factor model for the spot commodity price is used to illustrate the approach. This is done to avoid drawing the focus away from the new filters and the overall approach which are our main contribution. However, the methodology can be used for multi-factor models with some modifications. The characteristic shared by many of the continuous-time futures, forward, and bond price models discussed in the finance literature is that under a log-transformation and once appropriately discretized their state-space representations belong to the class of affine Gaussian state-space models. We shall consider only commodity price models and futures prices. The commodity price models of Gibson and Schwartz (1990), Schwartz (1997), Schwartz (1998), Schwartz and Smith (2000), and Manoliu and Tompaidis (2002) are examples with different parameterizations but an affine Gaussian form. Complicating the implementation of commodity market models is that one or more of the factors may be unobservable. In practice meaningful spot prices for some commodities are not available. What is often reported as the ‘price’ of an asset is the futures price of the contract closest to maturity. In a model with two factors, such as the spot price and the convenience yield, both may be unobservable while what is observed is the term structure of futures prices for contracts with a quantity of the physical asset underlying. Both the problems of calibration and estimation of unobservable quantities fit naturally into the framework of filtering. The Kalman filter has been applied to both of these problems by Schwartz (1997), Schwartz and Smith (2000), and Manoliu and Tompaidis (2002) for various multi-factor models to both calibrate the model parameters to market data and to estimate the time series of the unobservable factors. The method of estimating model parameters in all of these previous studies was the direct approach where the likelihood function itself was computed and maximized numerically. Rather than computing the maximum likelihood parameter estimates (MLE) by direct maximization of the likelihood function the expectation maximization (EM) algorithm can be used. The EM algorithm is a general iterative algorithm for maximizing the likelihood. Each iteration consist of two steps: expectation (E-step) and maximization (M-step). Following the techniques of Elliott and Krishnamurthy (1999) we derive explicit finite-dimensional filters necessary to obtain maximum likelihood estimates of the model parameters via the EM algorithm. The filters derived in this paper allow the E-step to be done without calculating Kalman smoother estimates normally used to implement the EM algorithm. As in Elliott and Krishnamurthy (1999) the filter-based approach to the EM algorithm used in this paper has certain advantages over the standard smoother-based implementation of the EM algorithm. For example, since only a forward pass through the data is required the filter-based approach will be at least twice as fast as a smoother-based approach which requires a forward-backward pass scheme. Further, the EM algorithm is well suited to parallel implementation on a multi-processor computer system (see, Elliott and Krishnamurthy, 1999). Other, possibly model specific, computational advantages may be realized using a filter-based implementation of the EM algorithm rather than a smoother-based implementation. For example, for the model considered in this paper and other constant coefficient models, the steady state properties of the Kalman filter can be exploited by a filter-based implementation. This is not possible for a smoother-based implementation. Other authors have considered the application of the EM algorithm to parameter estimation. Analogues of the filters derived by Elliott and Krishnamurthy (1999) for continuous-time linear Gaussian systems were presented in Elliott and Krishnamurthy (1997). Online algorithms based on the EM algorithm to calculate parameter estimates of nonlinear Gaussian state-space models were presented in Andrieu and Doucet (2003). Filtering methods and the EM algorithm have also been applied to hidden Markov models in a variety of contexts (see Elliott et al., 1995). The remainder of this paper is organized as follows: Section 2 reviews the Kalman filter for discrete-time affine Gaussian state-space models and parameter estimation for state-space models using the EM algorithm. Section 3 begins with some basic facts about commodity markets and futures prices. In Section 3.1, we give the model used to illustrate the filter-based approach. In Section 3.2, we discuss an empirical model and one of two main results: the EM parameter estimate updates for the model. In Section 3.3 we present the second main result: finite-dimensional filters for implementing the EM algorithm. Section 4 provides the results of a numerical implementation, using simulated data, of the filter-based EM for the commodity market model. Conclusions are presented in Section 5 and Appendix A provides an outline of the derivation of the finite-dimensional filters.

We have demonstrated the main steps of a filtering approach to parameter estimation of a model for the spot price of a commodity using futures price data. A filtering approach is especially appealing since the spot price is usually unobservable. Following Elliott and Krishnamurthy (1999) finite-dimensional filters were derived that allow a filter-based EM algorithm to be used to estimate model parameters. A similar procedure can be used for other commodity price models or general affine Gaussian state-space models. The approach to maximum likelihood parameter estimation given in this paper has some advantages over other approaches which we have not discussed in any detail here. The advantages of the filter-based EM algorithm over standard smoother-based EM algorithms outlined for the general case by Elliott and Krishnamurthy (1999) will be applicable, particularly: 1. the filter-based algorithm will be at least twice as fast as existing smoother-based EM-algorithms since only a forward pass is required for the filter-based algorithm; 2. the filter-based algorithm is well suited to parallel implementation on a multiprocessor system. As well, computational advantages may be realized which are model specific. For example, the filter-based algorithm for the constant coefficient model of this paper can be adjusted to use the steady-state properties of the Kalman filter. A Monte Carlo study comparing the performance of our approach to other equivalent approaches such as the smoother-based EM algorithm or direct maximization of the likelihood would be of interest. From a financial point of view it is of also of interest to apply the filter-based EM algorithm to models with multiple factors, for example the models of Gibson and Schwartz (1990), Schwartz (1997), Schwartz and Smith (2000), Manoliu and Tompaidis (2002), that better approximate real data as well as studying the performance of the filter-based approach. The extended (nonlinear) Kalman filter has been applied to parameter estimation problems related to bond and commodity markets by several researchers (see Duffee and Stanton, 2004; Javaheri et al., 2003, for example). It would be of interest to investigate how the filter-based EM algorithm might be applied to nonlinear models and the extended Kalman filter. The causal nature of the filter-based approach might also be of interest in control problems associated with finance.