2 Bayesian prediction, entropy, and option pricing

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

In addition to the variables in y, we take as exogenous a set of [sigma] variables [d.sub.t] which includes seasonal dummy variables and variables describing the weather, including rainfall and temperature data at various locations around the soybeangrowing region of the Midwestern United States. The model is a near-VAR because it does not describe the evolution of these variables. The near-VAR can be written

[y.sub.t] = C [Dd.sub.1] A(L)[y.sub.t-1] [v.sub.t] [v.sub.t] ~ iid N(O,[SIGMA]), (3)

where C and D are vectors of constants and A(L) is a vector of [lambda]-degree polynomials in the lag operator. Henceforth, we shall refer to the parameters in C, D, and A(L) as [mu], and write [theta] = ([mu][SIGMA]).

Our approach to prediction is Bayesian. Thus we treat the unknown parameters [theta] as random, and we condition our analysis on the observed data. We first need to describe how observed data modify our subjective views about the unknown parameters through the posterior distribution, and then how the posterior is used to build a predictive distribution for future values of the spot and futures prices. Given the simple structure of our model, this is quite standard, and readers familiar with such derivations may wish to skip to the description of the informative prior case.

2.1 Flat Prior Posterior

Our description of the posterior distribution under a flat prior follows Foster and Whiteman (1999) closely. Let Y denote the Txn matrix with t-th row given by [y'.sub.t] , and let X denote the T x(2 n2) matrix with t-th row given by (1, [d.sub.t]', [y.sub.t-1]'). Using the independence of the [v.sub.t]'s and noting that the Jacobian of the transformation from v to y is unity, the sampling density of Y conditional on [lambda], initial values, is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

or

P(Y|[theta]) [varies] [[absolute value of [summation]].sup.-(T-[lambda]-1)/2] exp[-1/2 tr (Y - XB)'(Y - XB)[[summation].sup.-1]] (5)

where tr denotes the trace operator. That is, the VAR can be seen to be a version of the standard multivariate regression model:

Y = XB V, (6)

where the (1 [delta] n[lambda])x n matrix B contains the VAR coefficients, and the rows of V are iid N(0, [summation]).

To begin, we adopt an 'uninformative' prior. There are many interpretations that can be given to the term 'uninformative'; we use the standard 'flat' prior:

P(B, [summation]) [varies [[absolute value of [summation].sup.(n 1)/2] (7)

(see Zellner 1971). The posterior distribution of the parameters is the product of the likelihood and the prior, or

P(B, [summation]|Y, X) [varies] [[absolute value of [summation].sup.-(T-[lambda] n)/2] exp[-1/2 tr (Y -XB)'(Y-XB)[[summation].sup.-1]]. (8)

Analysis of this expression is simplified by rewriting using the least squares estimate of B, [??], and the sum of squares matrix,

S = (Y-X[??])'(Y-X[??]), (9) yielding

P(B, [summation]|Y, X) [varies] [[absolute value of [summation].sup.-(T-[lambda] n)/2]exp{-1/2 tr [S (B -[??]) 'X'X(B - [??])][[summation].sup.-1]}. (10)