Không giống như mô hình Ho-Lee trình bày trong Chương 11, sự biến động của tỷ giá ngắn trong phương trình () phụ thuộc vào thời gian. Nếu, ví dụ, chức năng σ (t) như vậy là σ (1) =. 0126 và σ (2) =. 0120, sau đó sự biến động của | Futures on Rates in a Term Structure Model 349 of futures contracts the value of a futures contract after its mark to market payment must equal zero. Putting these two facts together .5 x p2uu - F1 .5 x p2 d - F __1 1_L 0 u 1 r Then solving for the unknown futures price _ u uu ud . F .5 x P2 .5 x P2 Since the same logic applies to the down state of date 1 Fd .5 x P2 ud .5 x P2dd As of date 0 setting the expected discounted mark-to-market payment equal to zero implies that .6 F - .FF - .4 F1 - F 0 0 1 r0 Or f o .6 x Fu .4 x Fd Substituting and into f o .3 x P2 uu .3 x P2 ud .2 x P2 ud .2 x P2 dd .3 x P2 uu .5 x P2 ud .2 x P2 dd In words under the risk-neutral process the futures price equals the expected price of the underlying security as of the delivery date. More generally f o e p m FUTURES ON RATES IN A TERM STRUCTURE MODEL The final settlement price of a Eurodollar futures contract is 100 minus the 90-day rate. Therefore the final contract prices are not Puu Pud and P2 d as 350 EURODOLLAR AND FED FUNDS FUTURES in the previous section but rather 100-r 100-r and 100-rịd. Following the logic of the previous section after this substitution the futures price equals the expected value of these outcomes. Denoting the rate on date M by r M and the futures price based on the rates as FR 0 FR 0 e 100 - r M l J r 100 - E r M J Defining the futures rate on date 0 rfut to be 100 minus the futures price rfM 100 - FR 0 E r M THE FUTURES-FORWARD DIFFERENCE This section brings together the results of Chapter 16 and of the two previous sections to be more explicit about the difference between forward and futures prices and between futures and forward rates. By the definition of covariance for two random variables G and H Cov G H e G X H - e g X e h Letting G P M and H 1 n 1 rm equation becomes . Cov p m I p m - E P M E n 1 J E In words this covariance equals the expected discounted
