Caplet pricing with LIBOR market model

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)
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 {P(t,T)} the price of {T}-bond at {t}. Then, for {t< T< U}, the forward LIBOR rate for loan period {[T, U]} at time {t} is defined by

\displaystyle L(t) := L(t, T, U) = \frac{1}{U-T} \Big(\frac{P(t,T)}{P(t, U)} - 1 \Big).

Due to the fact

Proposition 1 {\{L(t): t\le T\}} is a martingale with respect to the forward martingale measure {\mathbb P_{U}}.

LIBOR market model (LMM) assumes that

\displaystyle d L(t) = L(t) \nu(t) \cdot d W^{U}(t),

where {W^{U}} is {n}-dim {\mathbb P_{U}-}BM and {t\mapsto \nu(t) = \nu(t, U, T)} is an {n}-dim deterministic process. Now, we want to compute caplet price at {t} for the loan {[T, U]} with strike {K}, denoted by {Cpl_{t}}. R-code is written according to the following fact:

Proposition 2

\displaystyle Cpl_{t} = (U-T) P(t, T) Bl(K, L(t), \bar \nu, +1).

where {\bar \nu^{2} = \int_{t}^{T} |\nu(s)|^{2}ds}.

Ex. Spot prices for 1-year and 2-year zero-coupon bond are 0.945 and 0.90. Volatility of the forward LIBOR rate {L(t, 1, 2)} is given as constant {0.3.} For strike {K = 0.05} of the LIBOR rate {L(1, 1, 2)}, we can compute caplet price at {t= 0} as follows. With given parameters

\displaystyle K= .05; T = 1; U=2; PT=.945; PU = .9; nu = .3*sqrt(T)

one can compute caplet by

\displaystyle v = Cpl(K, T, U, PT, PU, nu) = 0.005365592.

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