2 Bayesian prediction, entropy, and option pricing

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

f([B.sub.i])/[[summation].sup.N.sub.i=1]f([B.sub.i]).

Thus, for example, the posterior mean under the flat prior posterior would be calculated by

[N.sup.-1][[summation].sup.N.sub.i=1][B.sub.i],

while the posterior mean under the informative prior would be calculated by

[[summation].sup.N.sub.i=1][f([B.sub.i])/[[summation].sup.N.sub.j=1]f ([B.sub.J])][B.sub.i].

Under mild conditions (see Geweke 1989), estimates computed in this way converge almost surely to the population values. (See DeJong & Whiteman 1994 for an application of this procedure.)

3. The Risk-Neutral Density and Option Prices

Once we have computed the predictive density we need to risk-adjust the probabilities to form the risk-neutral or pricing density. To do this we use a procedure advocated by Stutzer (1996). This procedure uses the maximum entropy principle of information theory to transform the predictive density to its risk-neutral form. This section describes his basic approach.

Using the Monte Carlo sample from the predictive density for [F.sub.E] we compute a futures return factor, [R.sup.i](E-T), for each draw, i=1,2, ..., N:

[F.sup.i.sub.E] = [F.sub.T][R.sup.i](E-T), (18)

where T denotes the end of the sample (the 'current' date) and E is the options expiration date. In the flat-prior case, each of these drawings is assigned an equal weight of 1/N, whereas in the informative prior case the weights will differ according to (17) and the ensuing discussion. We now need to transform the Monte Carlo probabilities for each draw, [??](i) so that the resulting estimated risk-adjusted density, [[??].sup.*](i) prices the futures contract properly. That is, we require the true risk-adjusted density to satisfy the following:

[summation over (i)][[pi].sup.*](i)[R.sup.i](E-T) = 1. (19)

Equivalently, under the probabilities [[pi].sup.*], the expected value of the futures price at E is the current futures price. Thus {[[pi].sup.*](i), i = 1, ..., N} is the 'equivalent martingale measure' associated with arbitrage-free price system. (See Huang & Litzenberger 1988, chapter 8.).

Of course, there are many choices of the N-component vector [[pi].sup.*] satisfying expression (19). Following Stutzer (1996), we use an estimate, [[??].sup.*], satisfying expression (19) that is chosen to minimize the Kullback-Leibler Information Criterion (KLIC) divergence between the risk-adjusted probabilities and those from the predictive density formed with our numerical Bayes procedure. (2) This optimization is of the form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

When the prior is flat and the weights [[??](i)= [N.sup.-1] are equal for all N Monte Carlo draws, the constrained optimization in expression (20) is identical to a constrained maximization of Shannon entropy,

-[summation over ([for all]i)][[pi].sup.*](i)ln([[pi].sup.*](i)).

In the general case, using the Lagrange multiplier method gives the Gibbs canonical distribution:

[[??].sup.*](i) = [[??](i)exp([[lambda].sup.*][R.sup.i](E - T))/[summation over ([for all]i][??](i)exp([[lambda].sup.*][R.sup.i](E - T)), i = 1,2, ..., N, (21)