# 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?

## 1 Answer

The LMM is typically done with each rate having its own time dependent volatility function.

You can the same effective Black constant vols by taking the root mean square vol over the rate's life for each rate.

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

• Thanks Mark for your answer. The piecewise structure of quoted flat volatilities and the corresponding calculus is clear to me. My question was rather about the relation between such a piecewise calibrated Black76 model and the LMM.(i) How must be the LMM's covariance in order to replicate the Black76? (ii) What is the practical advantage of the LMM against a piecewise calibrated Black model (the answer is covariance, I guess).(iii) How to approach the LMM from the BM in an alternative way? If I add random jumps at tenor times (which decorrelate, so to say), will the BM come close to the LMM? – davidhigh Nov 6 '16 at 16:30
• ... I should possibly edit that into the OP. :-) – davidhigh Nov 6 '16 at 16:34
• well Black is a model about one rate, and LMM about several rates. So I really don't see the issue. Can I recommend my "more mathematical finance" where go I into all this stuff into gory detail? – Mark Joshi Nov 6 '16 at 23:07