SERVING THE QUANTITATIVE FINANCE COMMUNITY

 
User avatar
Kiri
Topic Author
Posts: 11
Joined: October 15th, 2010, 7:17 am

Avoid negative interest rates in Hull White model

October 19th, 2010, 8:26 am

Hi everyone.I am using a one Factor HW model to simulate spot rates. Unfortunately the HW model leads to negative rates. I am planning to adapt the procedure I am currently using, changing to another model which doesn?t lead to negative interest rates.Current calibration method:? Calibrate to market spot rates? Calibrate to ATM Swaption implied volatilities using closed form solutions for HW swaption prices to find a and sigmaIs there a no-arbitrage model, which leads to non-negative interest rates and can be calibrated to the same market data (perhaps CIR or BK?)?Which adjustments would I have to make?For example, is there a closed form solution for swaption prices in the models mentioned above?I would be glad to receive any hints or proposals for literature.Thank you very much, Kiri
 
User avatar
Costeanu
Posts: 189
Joined: December 29th, 2008, 5:33 pm

Avoid negative interest rates in Hull White model

October 19th, 2010, 4:26 pm

Hi Kiri,The Hull-White model is also named the Gaussian short rate model. If you look in Andersen and Piterbarg's book, you will find a chapter about the "quasi-Gaussian" short rate model, which is a slight generalization. It is very well explained. This model is also known as Cheyette, or Ritchken-Sankarasubramanian, and is discussed in many places, but nowhere as clearly as in the book I mentioned. The one factor quasi-Guassian model allows you to specify a volatility and you can achieve two things: calibrate to the market smile and ensure positive rates. You have closed form formulas for zero-coupon bonds; they are the same as in the Hull-White model, if you express them in the correct form (formula 13.5 in the book). In particular the model automatically calibrates to the initial zero curve. There aren't closed form formulas for swaption implied vols, but there are excellent approximations, which are well covered in the book. The price you have to pay is that the 1-factor model becomes 2-dimensional (you need to keep track of an additional state variable called the "accumulated variance"). However, if speed is very important to you, and you want to use a 1-dimensional finite difference scheme, then there is a good projection of this model on a 1-dim process. You lose the closed form of the zero-bonds, but that's not so bad. You also lose the non-dependency of options with expiry in 10 years on the implied vol at year 15 for example. There is another, more subtle problem in the quasi-Gaussian model, related to non-negative rates. If you impose the floor of zero on the short rate, then you'll end up with a floor higher than zero for 30 year swap rates, let's say 2.5%. That's not negligible, and if these rates are important for your product, you will need to make a compromise, like allow short rates to be floored at -1% so that 30y swap rates are floored at 1.5%. Another model that allows you to specify positive rates is the CIR model. More generally, you can specify a floor in the rates in affine short rate models. These models are covered in chapter 10.2 in the A&P book. Yet another model is the quadratic Gaussian model (QGM), covered in 12.3. The QGM model has a lot of desirable properties, like analytic formulas for bonds (well, almost: you need to solve numerically some Ricatti equations), and some quasi-analytical formulas for swaptions. A 1-factor QGM is equivalent to a 2-factor affine model, so you gain the speed of a 1-factor model and the flexibility of a 2-factor one. However, it's more difficult to master. I don't know if the problem with a higher floor for longer tenor swaps happens for affine models or the QGM model, althogh I suspect it does (plausibility reason: the quasi-Gaussian model contains CIR as a particular case)Hope that helps,V.
 
User avatar
Kiri
Topic Author
Posts: 11
Joined: October 15th, 2010, 7:17 am

Avoid negative interest rates in Hull White model

October 20th, 2010, 1:13 pm

Hi Costeanu,thank you very much for your answer. But I still have lots of questions. So I would be happy about every answer I receive.The Cheyette Model sounds quite complicated to me and I have not found a lot about it in the web. So I would have to buy the A&P (Volume II, right?), which is not good because I just wanted to order the Brigo Do you think it can be implemented and calibrated within several days without knowing more than the Hull-White model and its calibration and only having the A&P as source? You mention a second dimension and "accumulated variance". Is this a complicated concept compared to other short rate models?The higher floor for long-term interest rates is not that bad.It seems to me that everytime I think I have found a suitable model, there appear new models with other advantages or disadvantages. I think the first goal is to find the short-rate model, which is the easiest to implement and which matches the starting zero curve and has non-negative interest rates. The calibration to market volatilities might follow in a second step.May I shortly summarize my current understanding. Any feedback is highly appreciated:-CIR-Model: In it's clasical form it does not match the starting zero curve ( can I use the four parameters to try to fit as good as possible and ignore volatility matching?). There exist closed form formulas for BondPrices. -In the extended CIR model I can match the staring zero curve (How would that work?), there still exist closed forms for the BondPrices, but calibrating volatility parameters is not that easy. Am I right?-Black-Karasinsky / Black-Derman-Toy: No closed form formulas exist neither for BondPrices nor for swaption prices. So calibration would have to be done on a numerical basis. BondPrices have to be calculated numerically, as well. This would mean, I have to build recombinig trees. Is this right? Are trees suitable if the goal is to construct zero curves for future times???Thank you very much for your answers.Kind Regards, Kiri
 
User avatar
Costeanu
Posts: 189
Joined: December 29th, 2008, 5:33 pm

Avoid negative interest rates in Hull White model

October 20th, 2010, 1:50 pm

Hi Kiri, The Cheyette model was explained extremely poorly in the "old times" (i.e. before this book). Don't be scared of it. It's a very slight generalization of the Hull-White model. Between buying the A&P book and the B&M one, I wouldn't hesitate one second (I have both, I lent B&M to a friend more than a year ago, and he keeps telling me he'll return it. I don't miss it that much. I wouldn't tolerate being away from the A&P book for more than a few days). It depends what you mean by implementation. If you want to implement it in Mathematica, you can probably do it in a day, even less. If you want to implement it in C++, even that wouldn't be too complicated, if what you have now is a Monte Carlo implementation. You just need to add one more state variable, and keep track of it. If currently you have a tree implementation, I don't know, it could take some time; essentially your tree becomes a local vol tree, and you have to be careful in the parts of the tree where the vol is high (you need to increase the step size in the short rate, or you'll get an instability). It would certainly be a more difficult project, and I would budget much more than just a few days. So, just to give you some peace of mind about the current state of the art in terms of models. There are 2 widespread ones, and another emerging one. - quasi-Guassian- Libor Market Models- quadratic GaussianThe affine class of models (CIR among them) is ok-ish, but you don't have the same flexibility with the smile. The quadratic Gaussian is quite new, and requries a lot of thinking until you become thouroughly familiar with it. It has the tremendous advantage of dimension reduction, so it certainly is worth a look. But don't start with it. The quasi-Gaussian is very convenient, being just a slight generalization of the Hull-White. It is very flexible, allows for all sorts of smiles and correlations, but may have some quirks, as the rates floor I mentioned. The LMM is the ultimate in flexibility, but having more factors is slow, and may not be very stable. These are all "honest" models. There are also some "hacks", where you start with some O.U. process, then get the short rate as a function of this process. You can run these only in trees. Their only advantage is dimension reduction, and therefore speed. Back to your observations on models: - CIR model: that's correct, you don't match the initial zero curve- CIR with time dependent parameters: you can match the initial zero curve. Again, a good description is in the A&P book. I don't think calibrating the vol is very hard. Whatever you use to calibrate Hull-White you can still use now- BK/BDT: correct, you need (recombining) trees to calculate both zero bonds and swaptions. Trees are suitable for constructing zero curve in the future, just pick a point in the future, and take the subtree that stems from that point. On that tree you can calculate whatever zero bonds you want. Best,V.
 
User avatar
Cuchulainn
Posts: 62916
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
Contact:

Avoid negative interest rates in Hull White model

October 20th, 2010, 2:35 pm

here is a book dedicated to Cheyette model. And a number of theses in the last few years.
Last edited by Cuchulainn on October 19th, 2010, 10:00 pm, edited 1 time in total.
Step over the gap, not into it. Watch the space between platform and train.
http://www.datasimfinancial.com
http://www.datasim.nl
 
User avatar
chocolatemoney
Posts: 322
Joined: October 8th, 2008, 6:50 am

Avoid negative interest rates in Hull White model

October 20th, 2010, 3:21 pm

Up to you the final decision to drop the HW model to avoid negative interest rates... but.. who says that negative IR is a negative feature in an model?I mean: history shows that a scenario where IR fall below 0% is highly unprobable but possible.
 
User avatar
Kiri
Topic Author
Posts: 11
Joined: October 15th, 2010, 7:17 am

Avoid negative interest rates in Hull White model

October 21st, 2010, 7:07 am

Thank you so much for your answers.I think, I will buy the A&P Vol. II and the book suggested by Cuchulainn and go for the Cheyette model implementation.My new plan is to implement the model completely new in VBA or Matlab using parts of my old HW-implementation. I would then follow the following steps:1) Calibrate drift parameters to match current yield curve (therefore I will need closed form solutions for Bond prices as of t=0 and compare them to observed zero coupon yields. Am I right in assuming that those closed form prices for t=0 are independent of the volatility parameters?)2) Calibrate (constant) volatility parameters to swaption prices, using observed option prices on the one hand and approximated swaption prices of the cheyette model as they are discribed in the A&P on the other hand. Run a minimization routine to minimize squared error.3) Use Euler-Discretisation to simulate r_t for about 1000 scenarios and say next 200 quarters.4) Calculate Zero Coupon Rates using closed form solutions for bond prices depending on r_t, t and T.Is this a suitable way for the implementation or are there any difficulties I haven't seen yet?Thank you very much. Kiri
 
User avatar
Costeanu
Posts: 189
Joined: December 29th, 2008, 5:33 pm

Avoid negative interest rates in Hull White model

October 21st, 2010, 9:25 am

Hi Kiri,1) You don't need to calibrate any drift parameters in the quasi-Gaussian model (I prefer to use this term, since it was used by Jamshidian before Cheyette) The drift calibration is automatic in this model2) The vol calibration is a bit more involved. I'll describe it separately3) Correct4) Correct (except that bond prices will also depend on one more variabe, let's call it y - the accumulated variance)The vol calibration: there are several types of vol calibration, depending what you plan to use it for. In general there are 3 uses for a model (aside from learning): 1. price and hedge exotic derivatives, 2. project scenarios in order to calculate various risks, 3. proprietary tradingFor the first purpose, usually people calibrate exactly to some swaption volatilities. In order to do that the volatility in the model is time dependent. You also want to fit some smiles as closely as possible. So you use some smile parameters, which may also be time-dependent. The approximated swaption prices described in A&P are used for initial guesses; since you want exact calibration, you then run your Monte Carlo several times with different parameters until you fit the market; you use a multi-dimensional Newton solver (Broyden for example). For the second and third purposes, a least square approximation is ok. And in any case, until you get familiar with the model, you shouldn't bother with an exact calibration. Good luck
 
User avatar
Kiri
Topic Author
Posts: 11
Joined: October 15th, 2010, 7:17 am

Avoid negative interest rates in Hull White model

October 23rd, 2010, 7:17 am

Hi Costeanu,Thanks again for your answer.I have ordered the book now and will start implementing the Model very soon. Perhaps there may arise new questions. I would be happy if you could help me again once new problems appear.Regards, Kiri
 
User avatar
Cuchulainn
Posts: 62916
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
Contact:

Avoid negative interest rates in Hull White model

October 24th, 2010, 3:50 pm

Here is an article on FDM for the Hull-White modelIf the scheme is monotone in general, the approximate results will be positive. But that's different from HW model itself. What's the rationale for step 3)? //An off note is instead of truncating PDE to [-1/2, 1/2] it is better to transform to [-1,1], apply finance logic to find the boundary conditions and price with your favourite FD scheme,if you decide to go the pde route, that is.
Last edited by Cuchulainn on October 23rd, 2010, 10:00 pm, edited 1 time in total.
Step over the gap, not into it. Watch the space between platform and train.
http://www.datasimfinancial.com
http://www.datasim.nl
 
User avatar
Kiri
Topic Author
Posts: 11
Joined: October 15th, 2010, 7:17 am

Avoid negative interest rates in Hull White model

November 3rd, 2010, 7:25 am

Hi everyone and especially Hi Costeanu,I have started to implement the quasi Gaussian model and I have come to a point, where I can get no further. I cannot figure out, how to force interest rates to be non-negative. Could you please help me in the following thread:http://wilmott.com/messageview.cfm?cati ... 80342Thank you very much,Kiri
 
User avatar
gerbijoe
Posts: 3
Joined: August 20th, 2015, 9:19 am

Avoid negative interest rates in Hull White model

August 21st, 2015, 1:31 pm

hi Costeanu do you have any advice about hw to implement a Hull & white with volatility as a step function?Thank youManuel
 
User avatar
Costeanu
Posts: 189
Joined: December 29th, 2008, 5:33 pm

Avoid negative interest rates in Hull White model

August 25th, 2015, 10:47 am

Can you be more precise about this? Is the volatility a step function of time, or of the short rate, or something else?
ABOUT WILMOTT

PW by JB

Wilmott.com has been "Serving the Quantitative Finance Community" since 2001. Continued...


Twitter LinkedIn Instagram

JOBS BOARD

JOBS BOARD

Looking for a quant job, risk, algo trading,...? Browse jobs here...


GZIP: On