2 Bayesian prediction, entropy, and option pricing

Australian Journal of Management, Dec, 2006 by F. Douglas Foster, Charles H. Whiteman

Abstract: This paper studies the performance of the Foster-Whiteman (1999) procedure for using a Bayesian predictive distribution for the future price of an asset to compute the price of a European option on that asset. A technical contribution of the paper is the description of a sequential importance sampling procedure for implementing an informative prior that reflects and rewards past option-pricing success. The risk-neutralization of the predictive distribution is accomplished by Stutzer's (1996) constrained KLIC-minimizing change of measure. The procedure is used in weekly pricing of July and November options on soybeans on the Chicago Board of Trade from 1993-1997, and produces option prices that mimic market prices much more closely than those of the Black model or those produced by risk-neutralizing a nonparametric predictive.

Keywords:

BAYESIAN PREDICTION; ENTROPY; OPTION PRICING; COMMODITIES.

1. Introduction

Options pricing techniques have been an important part of finance for some time. Most approaches specify a particular stochastic process to represent the price dynamics of the underlying asset and then derive an explicit pricing model. While this may be acceptable for standard financial assets, it may be problematic for commodities. Many commodities have significant seasonalities and require a far more elaborate time-series specification of the price dynamics of the underlying asset. Hence, it becomes difficult at best to derive explicit pricing formulae. Further, with the additional complexity of a rich time-series specification, estimation risk becomes a genuine concern. Finally, not all predictive information need be drawn from the historical time-series, so allowing for an explicit incorporation of nonsample beliefs could be important.

In this paper we suggest an alternative approach. We use numerical Bayes techniques to build a predictive density for the price of the underlying asset (for the example, in this paper we need to predict the soybean cash and futures prices at the option's expiration). Bayesian techniques allow for two very important additions. First, they enable us to integrate out any estimation risk. Second, they allow us to incorporate properly any non-sample information that we may have through an informative prior. Indeed, our informative prior is built sequentially by rewarding past option-pricing successes. Once the predictive density has been computed, we use a procedure proposed by Stutzer (1996) to translate this density to its risk-neutral form. With the risk-neutral density, pricing European options is very straightforward.

To illustrate this technique, we consider prices of options on soybean futures traded on The Chicago Board of Trade. We start with a simple vector autoregressive specification for the spot and futures prices. We enrich this predictive model to include weather data as well as futures market trading activity as evidenced by trading volume and open interest. We compare this procedure with traditional approaches as well as with a non-parametric procedure advocated by Stutzer (1996).

The paper is organized as follows. The next section describes the construction of a predictive distribution for the futures price at the time of expiration of the contract. Section 3 uses a procedure due to Stutzer (1996) to change the probabilities implicit in the numerical predictive distribution in such a way that they can be thought of as 'risk-neutral' probabilities. Using this transformed distribution, options can be priced by computing a partial expectation and discounting at the risk-free rate. Section 4 describes a study of the performance of the procedure in week-by-week pricing of soybean options from 1993-1997. Section 5 concludes.

2. Building the Predictive

The options to be priced are on soybean futures contracts, so the predictive distribution that will be needed is for the soybean futures price on the day the option contract expires, [F.sub.E], where F denotes the futures price and the E subscript refers to the future's expiration day. To predict the futures price we use a multivariate model describing the evolution of the (log of the) spot and futures prices, which are related via the cost of carry relation:

1n([F.su.t]) = 1n([S.sub.t]) [b.sub.t], (1)

where S denotes the spot price and b is the basis. The basis represents the percentage cost of carrying the spot commodity forward in time to the future's expiration date (which is not the same as the options expiration date).

The model we utilize is a kind of vector autoregression (VAR) for the spot and futures prices. This structure is convenient for our study, but it will become clear below that in principle we could use any model that can be simulated to produce a predictive distribution. Let [y.sub.t] denote the 4 x 1 vector containing the futures price, the basis, and open interest ([O.sub.t]) and volume ([V.sub.t]) in the futures market:

[y.sub.t] = (1n([F.sub.t]), [b.sub.t] 1n([O.sub.t]), 1n([V.sub.t]))' (2)

The trading volume and open interest variables and equations are included to pick up any volume / volatility relations. (1) Because reported trading volume gives the summed absolute value of trade sizes (it ignores whether the trade is buyer or seller initiated) we also include open interest. Open interest can be thought of as a signed summation of past trading volumes.