Let FT,S t and T,S t denote the forward price and futures price at time t, respectively, for delivery at time T > t of a zero-coupon bond maturing at time S > T.

Under the assumptions of the Vasicek model introduced in Section 11.5.1, show that FT,S t = exp{−[A(S − t) − A(T − t)] − [Bκ (S − t) − Bκ (T − t)]rt},

T,S t = exp{−A˜(T − t, S − T) − [Bκ (S − t) − Bκ (T − t)]rt},

where A(·) and Bκ (·) are given by Eqs. (11.29) and (11.25), A˜(T − t, S − T) = A(S − T) + κrˆBκ (S − T)Bκ (T − t)
− σ 2 r 2 Bκ (S − T)
2 (
Bκ (T − t) − κ
2 Bκ (T − t)
2 )
and rˆ = r¯ − σrλ/κ. Compare the forward price and the futures price.