Valuation Commentary - February '06

Volatility Models and Mean Reversion
by Alex Levin

During the end of January, I received several calls from clients questioning the calibration of the mean reversion parameter for the Black-Karasinski model. They were alarmed because the mean reversion fell to zero and asked for a verification of this outcome. I explained that the zero is our lower bound (floor) when calibrating the mean reversion; therefore, the odds of seeing this bound reached are not small. Why did the mean reversion become zero and continue to remain at zero? The answer is related to the dynamics of the yield curve, volatility curve and to the model selection itself. In this piece, I will cover these inter-related topics.

Volatility specification: normal, squared Gaussian, lognormal

In a number of publications, we argue that the market’s view of interest rates is much more normal than log-normal. We back this conclusion with three research methods:

• Investigating actual rate statistics
• “Measuring” swaption volatility skew, and
• Observing the behavior of “volatility indices” and their relationships to interest rates.

Apparently, most researchers agree with us. The Hull-White model, despite its analytical simplicity, is used most frequently. For a long time, I have not seen a single paper advocating lognormality for interest rates. Those who use “shifted lognormal” models actually set their shift parameter to fairly high, “normal,” levels. Even sympathizers of “market” models, lognormal at their origin (like the BGM model), employ modifications that are closer to normal views.

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