Why Hull–White Calibration Often Fixes Mean Reversion
Summary
The document explains why Hull–White interest-rate model calibration often fixes mean reversion while fitting volatility. With mean reversion held constant and volatility represented as a stepwise or piecewise-linear function, volatility can be bootstrapped to vanilla options grouped by expiry. The answer compares this with bootstrapping a discount curve: the procedure is presented as fast and stable.
For a Bermudan swaption, the example fits volatility to coterminal European swaption prices, which also serve as natural hedges. That makes the Bermudan valuation consistent with those European prices. Mean reversion remains a free parameter for marking the Bermudan. The discussion gives a practical calibration rationale, but it does not compare alternative joint or reversed calibrations, quantify errors, or prescribe how to choose the fixed mean-reversion value.
Key ideas
- Fixing mean reversion allows a term structure of volatility to be bootstrapped to vanilla option prices by expiry.
- The answer describes this volatility bootstrap as stable and fast.
- Coterminal European swaptions can calibrate volatility for valuing a Bermudan swaption.
- Mean reversion remains a free parameter that can affect the Bermudan valuation.
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Full text
# why calibrate volatility and fix the mean reversion # why calibrate volatility and fix the mean reversion I have had a few experiences or chats with teammates about the Hull-White model. The famous model has 2 parameters : - The volatility - The mean reversion Very often I hear that the mean reversion has been fixed and that the calibration is only done on the volatility. Why do that ? Why not fix the volatility and optimize on the mean reversion since both parameters have influence on the vanilla products ? Moreover, why no optimize on both parameters simultaneously ? Thanks a lot in advance for the context or opinions that help me to understand what are the justification of these practices. ## Answer by Antoine Conze (score 4, accepted) https://quant.stackexchange.com/a/44058 Fixing the mean reversion, and parameterizing the volatility as a step function or as a piecewise linear function, the volatility can be bootstrapped exactly to a set of vanilla options sorted by expiries. This is a very stable and fast procedure, akin to the bootstrapping of a discount curve onto rate instruments. For instance when pricing a bermuda swaption with the HW model, a mean reversion is first choosen and the volatility is then bootstrapped on the coterminal european swaptions market prices. Hence the bermuda swaption is priced in a manner consistent with the coterminal european swaptions prices (the coterminal swaptions are also the natural hedge to the bermuda swaption). The remaining degree of freedom, the mean reversion, becomes a parameter to mark the bermuda swaption (not sure if it is still the case, but I think at some point it was even contributed to Markit's Totem).
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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.