Recall that a rational log-normal model is defined by where M is a log-normal martingale under Z, a measure locally equivalent to the ‘real world’ measure P, M0 = 1, and A and B are deterministic...

Recall that a rational log-normal model is defined by

where M is a log-normal martingale under Z, a measure locally equivalent to the ‘real world’ measure P, M₀
= 1, and A and B are deterministic positive absolutely continuous functions decreasing to zero at infinity.

(i) Show that a rational log-normal model is in the class (FH). (See Chapter 8, page 190, for a definition of the class (FH).)

(ii) In the definition of an (FH) model suppose that

where W is a one-dimensional Brownian motion under Z. Use this rational lognormal model to find the time-zero value of a payers swaption having cash flows at times

and fixed rate K.