This function is available in R-code (here). See also its upper level page (here).

In this below, we explain Caplet pricing with LIBOR Market Model.

**Functions to load**: Bl(), Cpl()

**Usage**: Cpl(K, T, U, PT, PU, nu)

**Arguments**:

K: strike

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 1is 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:

Proposition 2

where .

**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