Joined: Apr 2010
Fri Apr 29, 11 12:15 PM
Originally posted by: piterbarg
Originally posted by: Jayshen
Hi everyone, I have some questions about quadratic Gaussian model for swaption pricing.
I try to implement the QG model in Andersen and Piterbarg's book, 'interest rate modeling', 2010, chapter 12.3, where the short interest rate is formulated as follows,
The following questions are based on the 2-factor case.
1, On modeling. One of the problem in their modelling I'm wondering is that, the symmetric matrix in the formula of short interest rate , , is not positive definite. From mathematical point of view, this formula can't keep positive for all of the state factor , which means the short rate can be negtive.
2, On calibration. I have tried to calibrate the two factor model to a volatility smile curve and the result is good. However, I feel the volatility of the second factor, , is superfluous, which means it's uncertain to chose what value it should be. For example, I can chose either 0.4 or 0.3 for , then calibrate the other three parameters, i.e., , which will result different parameters.
3. how to keep the short interest rate positive in calibration? Is there a floor for the interest rate in this model? If it is, how to get it?
I hope someone here can give me suggestions. Thanks a lot.
1. In a parameterization that we choose the rate can indeed go negative. In fact it can be negative for any set of parameters, except possibly for correl=1 or -1 and some combination of other parameters
2. If sigma_22 is constant then, I believe, it is superfluous. If, however, it is non-constant in time, then, I believe, it can control something that looks like a mean reversion of volatility -- ie you can control how the curvature of a smile changes through time. In some places I know, they set (or used to set) sigma_22 = sigma_11. but I don't think it is a great idea
3. This sounds like question 1 to me; if you mean something else please explain. Unless rho=1 or -1, there is no floor on the short rate. if you want a floor or a positive rate then you should use a different parameterization of the quadratic form (positive definite) -- but I don't know of good ones to use
re: newbanker, we did use the model in production for some time
Thanks for your answer, Piterbarg.
As the short interest rate can be negative, then it's not convincing to describe the change of bond price and interest rate term structure underlying this model, although it can fit the volatility smile of swaption market. I expected that this model can incorporate some advantage of quadratic term structure model which is claimed to be better than affine term structure model. Don't you think the negative short rate is a big disadvantage? what's more, is it easy to hedge the caplet and swaption under this model?