Interpreting Multifactor Structure in LMM Swaption Calibration

Summary

The document asks how to interpret Rebonato’s approximation for swaption volatility when using a factor-reduced or multifactor Libor Market Model. It questions whether each factor should be represented by a separate volatility function tied to a tenor, with its own calibration parameters. The author suspects this interpretation is wrong because a one-factor setup would then appear to assign the same approximate volatility to swaptions sharing an expiry.

The text raises a useful modeling question about how factor structure and tenor-specific volatility loadings relate to swaption prices. It does not provide a proposed calibration method, derivation, data, or resolution. The concern is therefore a starting point for understanding LMM factor reduction rather than a validated conclusion about the model. Readers would need additional sources to clarify how factors are defined and how the covariance structure affects swaptions with different underlying swap tenors.

Key ideas

• The document asks how Rebonato’s swaption volatility approximation should be interpreted in a multifactor LMM.
• It questions whether factors correspond to separate tenor-specific volatility functions with independent calibration parameters.
• The author worries this interpretation would give swaptions with the same expiry identical volatility in a one-factor model.
• No answer or supporting calibration evidence is included.

Tags

# LMM multifactor swaption calibration


# LMM multifactor swaption calibration












Brigo and Mercurio give Rebonato's approximation for Black-like swaption volatility as

$(v^{LFM}_{\alpha,\beta})^2=\sum^\beta_{i,j=\alpha+1}\frac{w_i(0)w_j(0)F_i(0)F_j(0)p_{i,j}}{S_{\alpha,\beta}(0)^2}\int^{T_a}_0\sigma_i(t)\sigma_j(t)dt$

How would I interpret this result in a "factor reduced" aka "multi-factor" or "n-factor" context?

Assuming I have a functional form for $\sigma_i(t)$, do I define my $n$ factors in terms of various $\{T_i\}$ such that $\alpha \leq i \leq \beta$, and also such that there are $n$ separate $\sigma_i(t)$, each with their own calibration parameters?

This does not seem like the correct approach to me, since if I had, for example, one factor, then any two swaptions with expiry at time $i$ would have the same approximated volatility.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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