Implied Remaining Variance in Derivative Pricing

J ia n Su n is an executive director at a large financial institution in N ew York, NY. [email protected] C onsider a call swaption m aturing at tim e T > 0 w ritten on an underly ing swap m aturing at a later tim e T' > T. Let A t(T,T') be the spot price at time t e [0,T] o f a forward­ starting annuity whose payments begin at T and end at T'. Let F (T ,T ') be the forward swap rate at tim e t e [0 ,T ]. In what follows, we fix bo th T and T', so we henceforth denote the forw ard-starting annuity value by just A and likewise denote the forward swap rate by just F . The call swaption’s time T payoff in units o f the forw ard-starting annuity is (F1~K)+, where K> 0 is the strike price. It is well know n that the absence of arbitrage betw een swaptions o f m aturity T and swaps m aturing at T and T' implies the existence o f an equivalent martingale mea­ sure Q s, com m only called forw ard swap measure (see, e.g., W u [2009]). U nder the forward swap measure Q s., the forward swap rate F is a m artingale. O ne o f the simplest possible specifications o f this m artingale is driftless GBM, i.e.,