Libor market model is an interest rate model used in finance to value different complex interest rate derivatives. It considers a set of forward rates that can be observed in market prices. The model captures the dynamic of the entire yield curve by building blocks of forward rates (usually called labor rates).
Here, we state the mathematical theory of the model briefly.
Consider \( 0= T_0 < T_1 < ... <T_n \) and let the tenor be \( \tau=T_{n+1}-T_n \), where \( n=0,...,N-1 \). Also suppose that \( L_n \) denotes the time \( t \) forward Libor rate for \( \tau_n \) period. We aim to study the dynamic of \( L_n \) which is as follows
$$dL_n(t) = \text{drift} + S_n(t) \, dW_t,$$
$$S_n(t) = \sigma_n (t) L_n(t),$$
where \( W_t \) denotes a Weiner process.
For \( n \geq q(t) \), we know that \( L_n(t) \) is martingale under \( T_{n+1} \)-forward measure \( Q^{n+1} \). Furthermore we know that the forward rate dynamic is drift-less under terminal measure, i.e.,
$$dL_n(t)= \sigma_n(t)^T L_n(t) dW^{n+1}(t).$$
The dynamic of forward rate is then
$$1+\sigma_n L_n(t) dW^{n+1}(t) = \frac{B_n(t)}{B_{n+1}(t)},$$
$$dL_n(t) = \frac{1}{\sigma_n} d(\frac{B_n(t)}{B_{n+1}(t)}).$$
Considering $$\frac{dB_n(t)}{B_n(t)} = \mu _n(t) \,dt + \beta_n(t) \, dW_t.$$
Using Ito's formula and after some calculations we obtain
$$(\beta_n(t) - \beta_{n+1}(t)) = \frac{\sigma_n S_n(t)}{1+\sigma_n L_n(t)}.$$
Applying Monte Carlo method and Euler discretization, one may calculate the forward rates from the key formula above.
To calibrate LMM model, we aim to estimate the volatility functions \( \sigma_i(t) \). We choose to use an approach that has been introduced by Rebonato, and is based on the fact that the instantaneous volatility function should be invariant under \( \tau = (T-t) \). An appropriate functional is then
$$g(t,T)= \sigma(T) \eta(t) f(T-t).$$
Moreover, there are some desired properties for this functional such as, time homogeneity, square integrability and ability to link the parameters of the function to observed economic situation. Rebonato considered the following parametrization function (also called abcd function)
$$g(\tau) = (a+b \tau) e^{-c\tau} +d.$$
As we see for a longer time period, i.e., \( \tau \rightarrow \infty \), the volatility asymptotically tends towards the parameter "\( d \)".
According to Solvency II framework, the ultimate forward rate (UFR) is 4.2 for all European Economic Area. The directive reflects for the Swedish Krona a convergence period of 10 years and last liquid point (LLP) of 10 years, which is equivalent that the forward rate will converge to its ultimate level UFR from total 20 years maturity onward, i.e., the volatility asymptotically goes to zero. This implies that in LMM calibration function, Rebonato, the parameter "\( d \)" should be zero.
