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I am trying to calibrate the 1 factor Hull White model to ATM swaptions. The strategy which I use is to minimise the sum of squared difference between model and market prices for the swaptions on the diagonal of the swaption matrix. I am using only swaption maturities till 10 years. So the swaptions which I am using for the calibration are 10x1, 9x2, ..., 1x10. And I am running a DE algorithm for jointly calibrating both the mean reversion parameter and volatility parameter (both assumed constant).

Now I understand that a lot of practitioners fix the value of the mean reversion parameter and only calibrate for the volatility parameter. However I have not been able to find any references for how a "best value" for the mean reversion parameter is decided. So it would be great if someone could shed some light on what exactly are the steps to decide on a "best value" for the mean reversion.

The currency for which I am trying to calibrate is GBP.

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For short time horizons, the mean-reversion term will be drowned out by the volatility term, but over long horizons it's the opposite, so depending on what you want to do with your calibrated Hull-White model, a correct calibration of the mean reversion parameter might be more or less important.

With the need for long term horizon simulations (CVA, insurance, etc.) this topic has been getting some attention. It is not a simple topic though, for example, there has been some evidence that a correlation exists between the mean reversion level and the level of interest rates, and in low rates regimes, historical calibration (e.g. using MLE) may lead to a negative mean reversion...

A good reference for such topics would be Sokol's book: https://www.amazon.com/Long-Term-Portfolio-Simulation-Liquidity-Regulatory/dp/1782720952/ref=sr_1_1?s=books&ie=UTF8&qid=1525553478&sr=1-1&refinements=p_27%3AAlexander+Sokol

Usually, a value around 5% is used, and according to the book, over the past, both historical (rates time series) and risk-neutral calibration (swaptions volatilities) give mean reversion estimates of around 3% to 10% for most currencies.

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For the HW (Hull-White) 1F model, the piecewise constant short rate volatility parameter $\sigma_i$ is the most influential one. It is used to match the market volatilities that matter the most.

For Bermudan pricing, the most relevant instruments are the $n$ coterminal swaptions corresponding to call dates. These can be solved using $\sigma_i$ for a given $\lambda$.

The mean reversion level $\lambda$ only allows to calibrate one additional European swaption besides the diagonal of coterminal. Since a Bermudan option is the sum of the max of the coterminal European options and a switch option, the mean reversion should be used to calibrate to a swaption that is most likely to represent the switch option.

So the typical pattern is to use $\sigma_i$ to match a diagonal of the volatility surface, and to use $\lambda$ to fit one additional point, e.g, the $(3, 2)$ year European swaption for a $10$ year Bermudan.

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