2 Bayesian prediction, entropy, and option pricing

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

whose Lagrange multiplier, [[gamma].sup.*], is found by solving:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

Finally, to price a European option on a future contract, use the risk-neutral density to compute the discounted expected value at the option's expiration. For a call option with a strike price of Xwe have:

C(T, E) = [summation over(i)]max[[F.sub.T][R.sup.i](E - T) - X, 0]/[r.sup.(E - T)][[??].sup.*](i), (23)

where [r.sup.(E-T)] is the risk-free discount factor from the current time, T, to the option expiration date E..

4. Data and Implementation (3)

For our analysis we use the soybean futures contract traded on the Chicago Board of Trade (CBOT) from 1993-1997. Each futures contract entitles the long to receive 5,000 bushels of soybeans. The deliverable grade is No. 2 Yellow at par and there can be delivery substitutions at differentials established by the CBOT (e.g. #1 soybeans at par $0.06 per bushel.). During the sample period, delivery sites were terminal elevators in the Chicago region. (Beginning in 2000, delivery may be made at any point along a 403-mile stretch of the Illinois and Mississippi rivers from Chicago to St. Louis, with location differentials per bushel established by the CBOT.) Prices for the futures are quoted in cents and quarter cents per bushel, with a tick size of $0.0025 ($12.50 per contract). The daily price limit is $0.30 per bushel above or below the prior day's price, with no price limit in the spot month. The contract year starts trading in September and each year there are futures that expire in September, November, January, March, May, July, and August. During the sample period, the last delivery day was the last business day of the delivery month; for 2000 and beyond, this has been changed to the 2nd business day following the last trading day.

Our histories of the soybean futures and cash prices extend back to January 1959. Figure 1 shows a plot of two key contract months, July and November. July is the last contract before the new crop is harvested, while November is the first futures contract where the details of the new crop are known. For these reasons we concentrate our efforts on pricing these two contract months. Note that there are a number of occasions where the two series diverge, indicating, in part, traders' beliefs about the quality of the incoming crop. Each series represents a joining of the underlying contract years. In producing this graph we have a simple rollover policy; use the price from the next year when a contract expires. In the predictive we use a dummy variable and adjust explicitly for the change in the contract year.

[FIGURE 1 OMITTED]

We also examine the cash price and the simple difference between the cash and futures prices (the basis). This is done in figure 2. The cash price series follows the same general pattern as the two futures. The bases have a saw-tooth pattern that is due to the rollover and subsequent decay to expiration as the futures and spot converge. Secondly, the November basis is more variable than the July, supportive of the view that the November basis is more driven by the uncertainty in the growth of the new crop.