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Topic Title: Calibration of Quadratic Gaussian model
Created On Wed Apr 27, 11 09:50 AM

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Jayshen
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Wed Apr 27, 11 09:50 AM

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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.

newbanker
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Wed Apr 27, 11 01:11 PM

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According to Dr. Wilmott's recent talk in Paris, you should first ask the modelers if they use their own model, before you try and calibrate it.

-------------------------
Pecunia ducat, non ratio.

Jayshen
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Wed Apr 27, 11 01:48 PM

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A super expert from Paris once told me this model has been used almost ten years in bank, but I don't know in which area, bond yield curve or option pricing.

piterbarg
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Thu Apr 28, 11 11:49 AM

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Quote
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

Vladimir

Paul
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Thu Apr 28, 11 02:03 PM

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Quote
Originally posted by: Jayshen A super expert from Paris once told me this model has been used almost ten years in bank, but I don't know in which area, bond yield curve or option pricing.

But the same bank as the inventors, by the inventors? That's the point! Lots of people will be implementing models while the inventors have moved on to other models (or just dumped the one they invented).

P

Edited: Thu Apr 28, 11 at 02:04 PM by Paul

Jayshen
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Fri Apr 29, 11 11:17 AM

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Thanks for your answer, Piterbarg.

As the short interest rate can be negtive, 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 negtive short rate is a big disadvantage? what's more, is it easy to hedge the caplet and swaption under this model?

Edited: Fri Apr 29, 11 at 12:13 PM by Jayshen

TinMan
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Fri Apr 29, 11 11:51 AM

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Quote
Originally posted by: Paul
Quote
Originally posted by: Jayshen A super expert from Paris once told me this model has been used almost ten years in bank, but I don't know in which area, bond yield curve or option pricing.
But the same bank as the inventors, by the inventors? That's the point! Lots of people will be implementing models while the inventors have moved on to other models (or just dumped the one they invented).P

Or didn't even invent the one people think they invented.

Jayshen
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Posts: 5
Joined: Apr 2010

Fri Apr 29, 11 12:15 PM

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Quote
Originally posted by: piterbarg
Quote
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

Vladimir

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?

Jayshen
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Posts: 5
Joined: Apr 2010

Fri Apr 29, 11 12:18 PM

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Quote
Originally posted by: TinMan
Quote
Originally posted by: Paul
Quote
Originally posted by: Jayshen A super expert from Paris once told me this model has been used almost ten years in bank, but I don't know in which area, bond yield curve or option pricing.
But the same bank as the inventors, by the inventors? That's the point! Lots of people will be implementing models while the inventors have moved on to other models (or just dumped the one they invented).P

Or didn't even invent the one people think they invented.

People should be critical with these models and don't fall in love with them

piterbarg
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Posts: 425
Joined: Oct 2002

Tue May 03, 11 09:59 AM

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Quote
Originally posted by: Jayshen
Thanks for your answer, Piterbarg.

As the short interest rate can be negtive, 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 negtive short rate is a big disadvantage? what's more, is it easy to hedge the caplet and swaption under this model?

I have never been much bothered by the possibility of negative rates in an interest rate model, to be honest. as long as you are careful and not payting for zero-strike puts what the model says they are worth, you should be alright

not sure what you mean by your hedging question

V

list
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Wed May 04, 11 05:11 PM

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If we dealing with interest rate models the fact of negative interest rates values itself does not say something too bad but if we note that the model admits with positive probability values of the bond more than its face value for any moment during the lifetime of the bond it calls for the question what is the definition of the bond we used. It might be relevant then to recall that bond and bond prices is the instruments issued by companies rather than symbol B ( t , T ). If the instrument price never was larger than its face value then B ( t , T ) with a chance be upper its FV cannot be called bond price. Nevertheless if the chance that B( t , T ) > 1 is sufficiently small we can interpret B as a function that sufficiently good approximate real bond values but this function does not good interpret with the formal definition of the bond. The problems with using such type of approximation come up with next constructions whether it is forwards or other derivatives based on function B. It would be more complicated to state about reasonable approximation without estimate of a chance when based on a model derivatives do not make sense.
Actually, if we deal with LIBOR rate models the main problem does not negative values. For example let us take a paper M. Rutkowski Market Models of LIBOR and Swap Rates Sydney Financial Math Workshop, 2005. With any forward LIBOR rate approach in section 2.2 producers have used unknown bond as the underlying. That is why some time ago I asked about underlying bond for LIBOR and people in wilmott kindly explained me that there is no bond to generate forward LIBOR. Now I have one more example that highlights the fact that the main problem with the use of probability in financial applications rather related to the people, how they use math but not to as how math say 'approximately' describe finance.

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