# Relation between Libor market model and Black76 with time-dependent vola

The Black76 model uses a lognormal process to model the forward rate $L_1(t)$ from $T_1$ to $T_2$ at time $t$,

$$dL_1(t) \ = \ \mu(t) L_1(t) dt + \sigma(t) L_1(t) dW_t$$

By switching to the $T_2$-forward measure, one can then get rid of the drift term and set up easy pricing formulas.

The Libor market model, on the other hand, uses several such forward rates at tenors $T_1< \cdots < T_N$, where the brownian motions in these are usually correlated. As in the Black model, the set of forward rates is brought to a common measure and simulated thereafter.

Question:

• If one assumes a time-dependent vola in the Black model which is stepwise constant between the tenor dates, what is the relation to a Libor market model (where the volas are assumed to be constant)? Asked differently, what is the correlation matrix in the LMM in order to reproduce the Black model with stepwise constant vola?

The covariance between two rates $i$ and $j$ if we do a step from s to t is $$\int_{s}^{t} \rho_{ij} \sigma_i(r) \sigma_j(r) dr.$$