EXERCISE 11.1 (Caplets and options on zero-coupon bonds) Assume that the lognormal LIBOR market model holds. Use the caplet formula (11.19) and the relations between caplets, floorlets, and European bond options known from Chapter 6 to show that the following pricing formulas for European options on zero-coupon bonds are valid:

and vL(t, Ti −, Ti) is given by (11.22) in the one-factor setting and by (11.26) in the multi-factor setting.
Note that these pricing formulas only apply to options expiring at one of the time points T0, T1, . . . , Tn−1, and where the underlying zero-coupon bond matures at the following date in this sequence. In other words, the time distance between the maturity of the option and the maturity of the underlying zero-coupon bond must be equal to .

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where CKT-T (1-K)BN (e) - K[BB]N(e2i), KT-8=KB-- BN (-2)-(1-K)BN(-1), 1 e1 = | In UL(t,Ti-8,T) (1-K)B KIBB 17) + 1/1UL (t, T. - 8, T.), e2e1i vL (t, Ti-8,Ti),