Relating Time-Varying Black Volatility to Libor Market Model Covariance

Summary

The document compares a single-forward-rate Black model with a Libor market model that evolves multiple correlated forward rates. It asks how stepwise time-dependent volatility in the Black setting relates to a model with constant volatilities, and what correlation structure would reproduce the Black behavior. The discussion notes that each rate in an LMM can have its own time-varying volatility function; an effective constant volatility over a rate’s life can be represented by a root-mean-square average.

For two rates, the covariance accumulated over a time interval is determined by integrating their correlation multiplied by their respective volatility functions. This identifies the quantities that must be matched when comparing the models. The note does not provide a full construction of a correlation matrix or address its validity constraints, so it offers a compact covariance relation rather than a complete equivalence procedure.

Key ideas

• Black76 models a forward rate with lognormal dynamics, while an LMM models several correlated forward rates.
• An LMM can assign each forward rate its own time-dependent volatility function.
• An effective constant volatility can be formed from the root-mean-square volatility over the rate’s life.
• Pairwise covariance accumulates by integrating correlation times both rates’ volatilities over time.
• Matching covariance alone does not specify all constraints needed for a valid LMM correlation matrix.

Tags

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


# 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?

## Answer by Mark Joshi (score 2)

https://quant.stackexchange.com/a/30883

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 $$ \int_{s}^{t} \rho_{ij} \sigma_i(r) \sigma_j(r) dr. $$

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.