In this below, we explain Caplet pricing with LIBOR Market Model.
Functions to load: Bl(), Cpl()
Usage: Cpl(K, T, U, PT, PU, nu)
T: Start of Loan
U: End of Loan
PT: spot price of T-Bond
PU: spot price of U-Bond
nu: standard deviation of logarithm of LIBOR rate L(T, T, U)
Value: Caplet price
Denote by the price of -bond at . Then, for , the forward LIBOR rate for loan period at time is defined by
Due to the fact
Proposition 1 is a martingale with respect to the forward martingale measure .
LIBOR market model (LMM) assumes that
where is -dim BM and is an -dim deterministic process. Now, we want to compute caplet price at for the loan with strike , denoted by . R-code is written according to the following fact:
Ex. Spot prices for 1-year and 2-year zero-coupon bond are 0.945 and 0.90. Volatility of the forward LIBOR rate is given as constant For strike of the LIBOR rate , we can compute caplet price at as follows. With given parameters
one can compute caplet by