Consider a Constant Maturing Swap (CMS) caplet whose payoff at payment date T p takes the form where the par swap rate over the tensor T 0 , ,T n is set at T 0 ,T p T 0 and s cap is some pre set constant cap value As usual, T 0 , ,T n 1 are the reset dates at which the relevant LIBOR L 0 , ,L n1 ar...

Lets denote the timet value of the CMS caplet by VCMS t To T where the payoff at payment date T take ...

Consider a Constant Maturing Swap (CMS) caplet whose payoff at payment date T p takes the form

Question:

Consider a Constant Maturing Swap (CMS) caplet whose payoff at payment date T_p takes the form

max(KT [To, Tnl - Scap, 0), where the par swap rate over the tensor {T₀, ··· ,T_n} is set at T₀,T_p > T₀ and s_capis some pre-set constant cap value. As usual, {T₀, ··· ,T_n₋₁} are the reset dates at which the relevant LIBOR {L₀, ··· ,L_n−1} are determined. Using the annuity B̂(t; T₀,T_n) as the numeraire, show that the time-t value of the CMS caplet is given by = VCMS(t; To, Tn) B(t; To, Tn) Eso.n max(KTo [To, Tn] - Scap, 0)B(To, Tp) - B (To: To, Tn)

where Q_s0,n is the swap measure with the annuity numeraire. Using the relation

Qso.n B(, Tp) B(To; To, Tn) = B(t, Tp) B(t; To, Tn) express V_CMS(t; T₀,T_n) in terms of the price function of a European swaption plus a convexity adjustment term due to payment of the caplet payoff at a later date T_p. Determine the form of this convexity adjustment.