Serving the Quantitative Finance Community

 
krs
Topic Author
Posts: 20
Joined: August 20th, 2019, 12:40 pm

Calibration Errors

January 29th, 2021, 11:41 am

Hi all, I am working on several interest rate models, calibrating them to ATM swaption prices. I would like to know what is an acceptable range for relative errors in swaption prices? I observed a strange pattern in my analysis.

Hull-White 1 Factor model:- 35% average error

G2++ model:- 7% average error 

Displaced diffusion Libor Market Model: 80% average error

the objective function is minimizing the sum of squared relative errors in swaption prices. I am using Rebonato's approximation in Displaced Diffusion LMM, I think that this model should be able to outperform the other two model. 

Any insight into this?

thank you,
Krish
 
User avatar
bearish
Posts: 6451
Joined: February 3rd, 2011, 2:19 pm

Re: Calibration Errors

January 30th, 2021, 3:26 am

You are leaving out a rather important detail: what is your end goal? In the highly unlikely event that you plan to use your model to make markets in swaptions, you probably want a fit that is within the bid/ask spread. If you are looking to take outright swaption positions and use your model to guide that, you actually want some “errors”, as long as they are signals of mispricing and tend to revert to zero. Of course, none of the models you describe is likely to fit either of those criteria, so how about you tell us?
 
krs
Topic Author
Posts: 20
Joined: August 20th, 2019, 12:40 pm

Re: Calibration Errors

February 1st, 2021, 4:23 pm

Thank you, Bearish!

The end goal is to generate 'market consistent' interest rate scenarios for insurance analytics, so, I am not really concerned about the bid-ask spread. (not sure if I should be) The idea is to capture the current volatility structure to generate IR scenarios. I am calibrating the model in price space (first convert the vol into prices and then calibrate the model using Black-like formula for swaption pricing and Rebonato's approximation for the volatility). The errors that I mentioned in my previous comment are errors in pricing. I was expecting a better fit using displaced diffusion LMM compared to the other two models. Are those error values make any sense?

Please also elaborate on this. 

"In the highly unlikely event that you plan to use your model to make markets in swaptions, you probably want a fit that is within the bid/ask spread."

Aren't these models good for pricing swaption to take a position in the market? (what about hedging?) I am also working on LMM-SABR, but again everything applied to Insurance analytics. Would like to know what tools traders use :)

thank you very much,
Krish
 
User avatar
DavidJN
Posts: 1781
Joined: July 14th, 2002, 3:00 am

Re: Calibration Errors

February 1st, 2021, 11:16 pm

Not meaning to put words in bearish's mouth, but the fitting errors you are showing are simply way too large to be useful in pricing. You would get your face ripped off if you tried trading and hedging with such poor fitting.

There are a number of possibilities as to why your fit is so poor: 1) as bearish has suggested, the models you have tried may not be rich enough to capture the volatility surface; 2) there are errors in the data you are using, and 3) there are errors in your implementations.

It has been more than a decade since I have traded this stuff, so I can't realistically offer you much more advice other than to say I would start remediation by looking at the data. Where is your data set from? Are they real, consistent market quotes? Then review your implementations. 
 
 
krs
Topic Author
Posts: 20
Joined: August 20th, 2019, 12:40 pm

Re: Calibration Errors

February 2nd, 2021, 10:56 am

Thanks, DavidJN!

The errors that I reported are for the entire swaption surface, 110 swaptions in this case. Are those numbers too large to get a good fit for the vol surface? what is an acceptable range of errors in this case?

I started with a Milliman document that reports comparable error using the G2++ model.  

https://milliman-cdn.azureedge.net/-/me ... white.ashx

I just expected DD-LMM to give a better fit, maybe I need to switch to a multi-factor model or SV to get close to the market prices. 

when you talk about estimation/fitting of a volatility surface do you mean the entire vol surface or a part of it? I suppose that traders when trading/hedging a swaption are more concerned about the part of vol surface that actually impacts the dynamics of the underlying security.

thanks,
K

 
 
User avatar
DavidJN
Posts: 1781
Joined: July 14th, 2002, 3:00 am

Re: Calibration Errors

February 3rd, 2021, 7:34 pm

As unintuitive as it sounds, one does not calibrate to the entire vol surface. That likely explains a good part of your fitting errors.

Instead, one chooses the parts of the vol surface that logically span the particular deal you are pricing. And as unintuitive as it also sounds, this means one recalibrates using different parts of the vol surface for every different swaption one prices. Which is why the more complex models prove to be slow and cumbersome in practice. The more realistic the model, the slower it is.

Why not calibrate to the whole vol surface? Because much like people use different vols in the Black Scholes model for different options (e.g. ITM vs OTM) because the BS model simply does not quite capture reality, people calibrate different swaption deals using different parts of the vol surface because the yield curve models they are using also do not quite capture reality.

There are lots of questions and answers about this in the Wilmott Forums. Search the words calibration and swaptions.  
 
krs
Topic Author
Posts: 20
Joined: August 20th, 2019, 12:40 pm

Re: Calibration Errors

February 5th, 2021, 8:47 am

Thank you, David! Insightful comments. 
 
hs16022021
Posts: 2
Joined: February 18th, 2021, 4:52 pm

Re: Calibration Errors

February 19th, 2021, 2:52 pm

IMHO, both local calibrations (i.e. fitting to specific instruments) and global calibration (fitting to all swaptions simultaneously) are valid approaches and I've seen them used on a trading desk.  

Local calibration is (generally) easier to implement and allows you to reprice the instruments you are likely to use as hedges, with more accuracy than a global fit. Using different calibration sets for different trades means you are effectively using different models for different trades. Or as DavidJN puts it, you are using different models to describe different parts of the curve. There are endless debates on whether this is a good thing or not.

A global fit on the other hand, with a suitably high dimensional (e.g. 3 or more factors) should be able to capture the main dynamics of the curve, and be fitted to the entire swaption matrix. The degree of fitting success depends on the specification of the volatility function you use in your term structure model (e.g. separable, time-homogenous, etc), correlation structure (related to factor loadings in a LMM model) and the numerical scheme you use - e.g. global optimisation, cascade calibration, etc.

I would guess that for insurance purposes, having an exact fit to all parts of the swaption surface is not as important as capturing the overall  curve dynamics, especially if the resultant model is to be used for generating rates scenarios. I would not dismiss global calibration for this purpose.