2 Bayesian prediction, entropy, and option pricing

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

4.2 Informative Prior

In addition to the flat-prior procedure described above, we implemented an informative prior procedure that embodied a type of 'learning' over the contract year. In our analysis, we valued calls and puts for July and November contracts for 1993-1997. For each contract, we priced the option on 26 consecutive Fridays beginning 27 weeks prior to expiration. In each case, with the passage of each week, we added that week's data to the data set, and updated the posterior distribution. Updating the posterior distribution associated with the flat prior thus incorporates a very crude sort of learning--the additional data is used as it becomes available, but in precisely the same way (i.e. via a flat prior) as the initial sample. A more natural type of learning would respect the success of the option pricing procedure itself. Indeed, our informative prior was built iteratively: in each contract year, we would begin 27 weeks prior to expiration with the basic near VAR specification and the data, but no other prior information. After pricing options in that first week with the flat prior specification, we would also determine the KLIC-closest predictive distribution satisfying the no-arbitrage condition that also priced three options correctly: the at-the-money, and just-in and just-out of the money options. The reweighting of sample values of the parameters implicit in this calculation was used as a prior distribution for the subsequent week's calculation.

Implementing this prior requires special care because of its non-conjugate nature. To see why, consider the situation at week t, somewhere in the midst of the 26-week prediction period for a particular contract. Suppose that the current sample from the posterior is {[[theta].sub.i]} for i = 1, ..., N. Suppose further that the posterior distribution can be represented by a set of probability density values [[omega].sub.i,t] = f([[theta].sub.i],t). After constructing the sample from the predictive density and risk-neutralizing (reweighting) via the Stutzer procedure, we also computed a different reweighting that priced the three options correctly. This reweighting can be represented by [??]*(i), which is calculated from [[omega].sub.i,t] in the same manner as [??]*(i) is calculated from [??](i), except that in addition to the no-arbitrage constraint, three additional constraints are imposed to ensure that the three nearest-the-money options are priced correctly in week t. Entering week t 1, the [??]*(i) distribution plays the role of the prior for that week.

The difficulty with this procedure is that the mapping from [[theta].sub.i] to [??]*(i) is unknown. Thus it is not possible to sample from the posterior distribution given by the product of the likelihood for week t 1 and the prior just derived the week before. Fortunately, this difficulty can be finessed because it is possible to construct an importance sample for which it is necessary only to know the weights [??]* (i) themselves rather than the mapping from [[theta].sub.i] to [??]* (i). This in turn requires that the values of the drawings {[theta].sub.i]} not change from week to week. To carry this out, we used the flat prior posterior from the first week of our pricing exercise (for each of the 5 years) as an importance sampler. That is, in week 1 of each contract year, we drew a very large number (N=50,000) of [[theta].sub.i]'s. (5) These drawings were then used in each of the subsequent 25 weeks. When used in week t, for example, the drawing [[theta].sub.i] was associated with a 'weight' not, of course, equal to 1/N, but rather equal to the product of the [??]*(i) calculated at the end of week t-1 and an additional weight proportional to the likelihood of [[theta].sub.i] using data up to and including week t. As the weeks passed, the prior weights and likelihood values changed, and the priors and posteriors updated sequentially (6,7).