Swaption Skew Calibration in a Shifted LMM and ATM Volatility Mismatch

Summary

The document describes a calibration issue in a LIBOR market model after introducing skew through a shifted diffusion. Instead of modeling each forward rate with volatility proportional to the rate itself, the author uses volatility proportional to the forward rate less a shift parameter. Although paths are simulated under the spot measure, the shifted-model volatilities are calibrated following a referenced textbook procedure.

The reported diagnostic is a mismatch between swaption volatility smiles: the shifted model produces a skewed curve that remains above the nearly flat curve from the non-skew model, rather than intersecting it at the at-the-money forward strike. Caplet implied volatility curves, by contrast, are reported as almost identical, as expected from the calibration. The author suspects the shifted-model volatilities may be too high but offers no resolution or numerical details. The passage is therefore useful as a calibration question and diagnostic contrast, not as a completed method; it does not provide enough information to identify the cause or reproduce the result.

Key ideas

• The shifted-diffusion LMM models forward rates with volatility applied to the rate minus a shift parameter.
• The author simulates paths under the spot measure and calibrates shifted-model volatilities using a referenced procedure.
• The reported swaption skew remains above the non-skew curve instead of meeting it at the at-the-money forward strike.
• Caplet volatility curves are reported as nearly matching, while the cause of the swaption mismatch remains unresolved.

Tags

# Calibration Problem in the LMM-Skew (Shifted Diffusion) Model


# Calibration Problem in the LMM-Skew (Shifted Diffusion) Model












I have implemented the LIBOR market model (LMM) and I am quite satisfied with the results. I have now added a skew to the model as described in 10.1 of Brigo/Mercurio. That is, I have replaced the SDE

$dF_{k}(t) = \sigma_{k}(t) F_{k}(t) dW_{t}$

for the forward rate $F(t, T_{k-1}, T_{k})$ with

$dH_{k}(t) = v_{k}(t) (H_{k}(t) - \eta) dW_{t}$.

In the actual path simulation I do not simulate the above equations but rather the SDEs under the spot measure. The volatilities $v_{k}$ for the skew case are calibrated as stated in 10.1 of Brigo/Mercurio.

My problem is the following. When I use the simulated paths to price swaptions and back-out the implied swaption volatilities I indeed get an almost flat curve in the non-skew case and a skew curve in the skew case. However, the two implied volatility curves (implied volatility plotted against strike) do not intersect at the at-the-money strike. The skew curve is always above the non-skew curve. When I do the same for caplets I get two almost identical curves (as I would expect from the calibration procedure). I have included an example plot here. The green line is what I would expect and the red-brown line is what I get. The x-axis is moneyness with respect to the ATM forward rate.

My suspicion is that I am missing something simple and that my calibrated volatilities for the skew case are too high. However, I calibrate exactly as suggested by Brigo/Mercurio in 10.1. I would appreciate any hint!

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