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thedoc
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Joined: July 27th, 2010, 6:53 am

Basic Hull White Calibration

April 18th, 2018, 7:22 am

HI, I'm trying to calibrate a 1F HW model to the EUR swaptions market and am hitting a few problems.

I am using a matrix of standard atm swaptions premia, out to 10y (1y1y, 1y2y, ... 1y9y; 2y1y, ...2y8y; ... ; 9y1y) sourced from bberg.

The problem is that I can calibrate to small individual sets of swaptions (e.g. the 1y forward starting only) and match quite accurately to market vols, but then all other premia are mispriced (e.g. 2yf).

My model projects a distribution of yield curves at each point forward in time, but for each point (e.g. 1y) this distribution appears to be only a distribution of parallel shifts on the expected shape of the yield curve at this point, with the variation driven by the distribution of the short rate at the 1y point.

How can the HW model reflect he volatility structure at all forward points and tenors, if their is no stochastic variation of the forward curve: do I need to use a 2F mode? should I simply recalibrate for every forward point?

I'm fairly new to this model, so I feel like I'm missing something obvious. Any help appreciated.
 
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bearish
Posts: 5186
Joined: February 3rd, 2011, 2:19 pm

Re: Basic Hull White Calibration

April 18th, 2018, 10:12 am

HI, I'm trying to calibrate a 1F HW model to the EUR swaptions market and am hitting a few problems.

I am using a matrix of standard atm swaptions premia, out to 10y (1y1y, 1y2y, ... 1y9y; 2y1y, ...2y8y; ... ; 9y1y) sourced from bberg.

The problem is that I can calibrate to small individual sets of swaptions (e.g. the 1y forward starting only) and match quite accurately to market vols, but then all other premia are mispriced (e.g. 2yf).

My model projects a distribution of yield curves at each point forward in time, but for each point (e.g. 1y) this distribution appears to be only a distribution of parallel shifts on the expected shape of the yield curve at this point, with the variation driven by the distribution of the short rate at the 1y point.

How can the HW model reflect he volatility structure at all forward points and tenors, if their is no stochastic variation of the forward curve: do I need to use a 2F mode? should I simply recalibrate for every forward point?

I'm fairly new to this model, so I feel like I'm missing something obvious. Any help appreciated.
It all depends on what you are ultimately trying to do, but this model has always had problems when faced with the reality that interest rates do not mean revert on time scales up to a few years. The model will tell you that the max implied (basis point) vol in your matrix is at the 1x1 point, which is probably in fact the min. The two factor version will do much better, as will the slightly richer Gauss 2+ model described in Tuckman & Serrat, although you will have a hard time hitting all the vols at the same time. And if you do, you should still worry a little bit about the third dimension.
 
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thedoc
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Posts: 4
Joined: July 27th, 2010, 6:53 am

Re: Basic Hull White Calibration

April 18th, 2018, 10:44 am

HI, I'm trying to calibrate a 1F HW model to the EUR swaptions market and am hitting a few problems.

I am using a matrix of standard atm swaptions premia, out to 10y (1y1y, 1y2y, ... 1y9y; 2y1y, ...2y8y; ... ; 9y1y) sourced from bberg.

The problem is that I can calibrate to small individual sets of swaptions (e.g. the 1y forward starting only) and match quite accurately to market vols, but then all other premia are mispriced (e.g. 2yf).

My model projects a distribution of yield curves at each point forward in time, but for each point (e.g. 1y) this distribution appears to be only a distribution of parallel shifts on the expected shape of the yield curve at this point, with the variation driven by the distribution of the short rate at the 1y point.

How can the HW model reflect he volatility structure at all forward points and tenors, if their is no stochastic variation of the forward curve: do I need to use a 2F mode? should I simply recalibrate for every forward point?

I'm fairly new to this model, so I feel like I'm missing something obvious. Any help appreciated.
It all depends on what you are ultimately trying to do, but this model has always had problems when faced with the reality that interest rates do not mean revert on time scales up to a few years. The model will tell you that the max implied (basis point) vol in your matrix is at the 1x1 point, which is probably in fact the min. The two factor version will do much better, as will the slightly richer Gauss 2+ model described in Tuckman & Serrat, although you will have a hard time hitting all the vols at the same time. And if you do, you should still worry a little bit about the third dimension.
Hi, thanks for the response. 
I'm trying to build a monte-carlo rate path/yield curve generator which will be used to value CVA, so I basically need a way to map market vols to rate path distributions. So I don't care so much about implied vols, just that I can validate market premia. 
I guess my main problem with my work so far on HW 1F is that there is no variation in the shape of the the yield curve at each future point on the path of the short rate. I don't know whether this is a feature of the model, or a mistake I'm making.
What do you mean by third dimension?
 
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bearish
Posts: 5186
Joined: February 3rd, 2011, 2:19 pm

Re: Basic Hull White Calibration

April 18th, 2018, 2:03 pm

HI, I'm trying to calibrate a 1F HW model to the EUR swaptions market and am hitting a few problems.

I am using a matrix of standard atm swaptions premia, out to 10y (1y1y, 1y2y, ... 1y9y; 2y1y, ...2y8y; ... ; 9y1y) sourced from bberg.

The problem is that I can calibrate to small individual sets of swaptions (e.g. the 1y forward starting only) and match quite accurately to market vols, but then all other premia are mispriced (e.g. 2yf).

My model projects a distribution of yield curves at each point forward in time, but for each point (e.g. 1y) this distribution appears to be only a distribution of parallel shifts on the expected shape of the yield curve at this point, with the variation driven by the distribution of the short rate at the 1y point.

How can the HW model reflect he volatility structure at all forward points and tenors, if their is no stochastic variation of the forward curve: do I need to use a 2F mode? should I simply recalibrate for every forward point?

I'm fairly new to this model, so I feel like I'm missing something obvious. Any help appreciated.
It all depends on what you are ultimately trying to do, but this model has always had problems when faced with the reality that interest rates do not mean revert on time scales up to a few years. The model will tell you that the max implied (basis point) vol in your matrix is at the 1x1 point, which is probably in fact the min. The two factor version will do much better, as will the slightly richer Gauss 2+ model described in Tuckman & Serrat, although you will have a hard time hitting all the vols at the same time. And if you do, you should still worry a little bit about the third dimension.
Hi, thanks for the response. 
I'm trying to build a monte-carlo rate path/yield curve generator which will be used to value CVA, so I basically need a way to map market vols to rate path distributions. So I don't care so much about implied vols, just that I can validate market premia. 
I guess my main problem with my work so far on HW 1F is that there is no variation in the shape of the the yield curve at each future point on the path of the short rate. I don't know whether this is a feature of the model, or a mistake I'm making.
What do you mean by third dimension?
The third dimension would be how the vol does (or, in this case, does not) vary with the level of rates. Equivalently, how well you fit the vol skew for a given maturity and tenor. Your observation that the curve shape is constant suggests that you are using a mean reversion speed close to zero. In that case there should be a very slight slope change arising from a convexity effect, but perhaps too small for you to notice. For a stronger mean reversion you should see a curve flattening as the short rate rises and a steepening as it goes down, since the forward rate volatility is an exponentially declining function of maturity.
 
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thedoc
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Joined: July 27th, 2010, 6:53 am

Re: Basic Hull White Calibration

April 18th, 2018, 2:53 pm

OK understood on the third dimension - I was leaving that until after I got the basics right.

Just to clarify, the curve structure isn't flat, it's fairly steep. The the problem I have is is that each MC path generates a curve which just looks like a parallel shift of the expected shape of the curve. As such, I can match my 1y1y swaption premium easily to calibrate my 1y vol, but then using the same vol, my 1y9y premium is way too low. Struggling to make sense of it tbh.
 
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bearish
Posts: 5186
Joined: February 3rd, 2011, 2:19 pm

Re: Basic Hull White Calibration

April 18th, 2018, 4:19 pm

OK understood on the third dimension - I was leaving that until after I got the basics right.

Just to clarify, the curve structure isn't flat, it's fairly steep. The the problem I have is is that each MC path generates a curve which just looks like a parallel shift of the expected shape of the curve. As such, I can match my 1y1y swaption premium easily to calibrate my 1y vol, but then using the same vol, my 1y9y premium is way too low. Struggling to make sense of it tbh.
The mispricing you describe is exactly what you should expect. In this particular model, the short rate is the most volatile, and longer rate vols drop off exponentially (this is an exact result for instantaneous forward rates, and a reasonable approximation for swap rates). In the real world, short rates are pretty constant most of the time and the most volatile part of the term structure is somewhere in the belly, say in the 2-10 year range, depending on currency and era. The most parsimonious way of modeling this in a Gaussian framework would be the Gauss 1+ model, but you can also generate that sort of dynamics in the 2 factor HW model (i.e. with the short rate being the sum of two OU processes with similar vol level, fairly different mean reversion speeds, and strongly negative instantaneous correlation).

When I am referring to curve steepening and flattening, it is the shape of the additive change to your starting curve, regardless of its shape.
 
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thedoc
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Joined: July 27th, 2010, 6:53 am

Re: Basic Hull White Calibration

April 19th, 2018, 7:52 am

OK understood on the third dimension - I was leaving that until after I got the basics right.

Just to clarify, the curve structure isn't flat, it's fairly steep. The the problem I have is is that each MC path generates a curve which just looks like a parallel shift of the expected shape of the curve. As such, I can match my 1y1y swaption premium easily to calibrate my 1y vol, but then using the same vol, my 1y9y premium is way too low. Struggling to make sense of it tbh.
The mispricing you describe is exactly what you should expect. In this particular model, the short rate is the most volatile, and longer rate vols drop off exponentially (this is an exact result for instantaneous forward rates, and a reasonable approximation for swap rates). In the real world, short rates are pretty constant most of the time and the most volatile part of the term structure is somewhere in the belly, say in the 2-10 year range, depending on currency and era. The most parsimonious way of modeling this in a Gaussian framework would be the Gauss 1+ model, but you can also generate that sort of dynamics in the 2 factor HW model (i.e. with the short rate being the sum of two OU processes with similar vol level, fairly different mean reversion speeds, and strongly negative instantaneous correlation).

When I am referring to curve steepening and flattening, it is the shape of the additive change to your starting curve, regardless of its shape.
OK, thanks for all your help. Much appreciated. I'll have a crack at the 2F model, and also the Gauss 1+ you mention. Do you know of any good resources online or in print that might help me with the implementation of those?
 
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bearish
Posts: 5186
Joined: February 3rd, 2011, 2:19 pm

Re: Basic Hull White Calibration

April 19th, 2018, 10:31 am

OK understood on the third dimension - I was leaving that until after I got the basics right.

Just to clarify, the curve structure isn't flat, it's fairly steep. The the problem I have is is that each MC path generates a curve which just looks like a parallel shift of the expected shape of the curve. As such, I can match my 1y1y swaption premium easily to calibrate my 1y vol, but then using the same vol, my 1y9y premium is way too low. Struggling to make sense of it tbh.
The mispricing you describe is exactly what you should expect. In this particular model, the short rate is the most volatile, and longer rate vols drop off exponentially (this is an exact result for instantaneous forward rates, and a reasonable approximation for swap rates). In the real world, short rates are pretty constant most of the time and the most volatile part of the term structure is somewhere in the belly, say in the 2-10 year range, depending on currency and era. The most parsimonious way of modeling this in a Gaussian framework would be the Gauss 1+ model, but you can also generate that sort of dynamics in the 2 factor HW model (i.e. with the short rate being the sum of two OU processes with similar vol level, fairly different mean reversion speeds, and strongly negative instantaneous correlation).

When I am referring to curve steepening and flattening, it is the shape of the additive change to your starting curve, regardless of its shape.
OK, thanks for all your help. Much appreciated. I'll have a crack at the 2F model, and also the Gauss 1+ you mention. Do you know of any good resources online or in print that might help me with the implementation of those?
The Gauss+ family of models is given a full textbook treatment in the excellent book by Tuckman and Serrat. The generic 2 factor, 2 state variable Gaussian model calibrated to an arbitrary initial yield curve (aka HW) is actually a limiting case of the Gauss 2+ model with the short rate mean reversion speed (to a moving target) going to infinity. Pat Hagan has written quite a bit on this model under the headline of Linear Gauss Markov, and it is definitely covered in the Andersen Piterbarg magnum opus.