I've been speaking to someone who generates a volatility cube this way:- sabr calibration on selected maturities and tenors. Let's say maturities 2Y, 5Y,10Y,... and tenors 6M,1Y,2Y,5Y,10Y,...-linear interpolation of sabr parameters for other maturities and tenors (Beta being constant across maturities)From my point of view it makes no sense to interpolate the sabr parameters as I am not too sure of what is the relationship netween these parameters and maturity. I think there's no linear relationship there! But I may be wrong.I would rather interpolate the raw data and then try to fit a sabr smile on each maturity I need.Any point of views?

- SpreadRisk
**Posts:**1**Joined:**

I also wondered about that issue before.I guess the question is how you calibrate the parameters. When I calibrated the SABR parameters for swaptions I optimised them separately for each expiry&tenor, so there is no guarantee that linear interpolation makes sense.

golddigga, if data is not available its best to generate interpolated points for the vol cube first and then fit SABR on that rather than interpolate on the fitted parameters.

QuoteOriginally posted by: slackergolddigga, if data is not available its best to generate interpolated points for the vol cube first and then fit SABR on that rather than interpolate on the fitted parameters.Why is it better? My impression is that both in the cases in which we are pricing with SABR or with Black76, we are using different and independent models for each expiry and tenor (I think GoldDigga is suggesting that this is the issue).So neither interpolating on the parameters nor on the points for the vol cube has any strong (mathematical or financial) justification. Sadly, due to the limited liquidity of OTM swaptions, it's hard to test which is better... And this is the point at which i decided that the choice was probably due to aesthetical reasons... what did i miss?thanks

Last edited by gc on October 2nd, 2006, 10:00 pm, edited 1 time in total.

- Traden4Alpha
**Posts:**23951**Joined:**

If the second derivative of the true curve is non-zero, then linear interpolation will create a systematic bias. The sign of the bias will be the same as the sign of the second derivative. If you then fit a model to the interpolated points, you'll get biased parameters and that's not good.You might consider a higher-order interpolation function, e.g. parabolic or cubic. Of course, higher-order interpolation has its own problems as it uses data even further from the interpolation point. Moreover, if the data is noisy, the variance on the interpolated values can exceed the variance on the original data. Yet, I'd argue that a systematic bias is worse than the added variance.

Sure, SABR gives you independent prescriptions for each maturity/tenor.Though smoothing the raw data and then calibrating on desired points, I think, has some justification. Interpolating each parameter based on maturity time raises the possibility that all the parameters together do not generate vols which lie on some smooth mkt vol cube (hypercube??) Bye

- andreloeffler
**Posts:**33**Joined:**

interpolating parameters to estimate the vol of a caplet or a european swaption and interpolating directly the volatilities resulting from the known Sabr parameters give usually similar numbers.

Hello,I would like to implement the interpolation of the SABR model to incomplete market as written in the first message.But i can't find any information.Can someone give me the volatilities observed on the market for swaptions, let's say 2Y, 5Y,10Y,... and tenors 6M,1Y,2Y,5Y,10Y, ATM, OTM ,ITM.And then i'll try to interpolate the volatilities OTM and ITM for other swaptions, for which we only know the ATM volatility.Thank you.PS:- it doesn't matter if ur information is old - do u have somewhere a piece of a code to interpolate ?>David

- NorthernJohn
**Posts:**632**Joined:**

QuoteOriginally posted by: andreloefflerinterpolating parameters to estimate the vol of a caplet or a european swaption and interpolating directly the volatilities resulting from the known Sabr parameters give usually similar numbers.I think, though, that you are talking here about pulling a vl from a fully populated surface, for pricing.This is not the same as choice of whether to force the SABR variables to be smooth in an initial calibration.In general, as data is sparse, if we fit to the market alone, we will have a SABR set which contains some large jumps between adjacent points. This needs some smoothing, as it is not generally believed that, say, 5y 10y and 5y 12y swaptions can have alphas which are very different.

In relation to the sabr model, I have heard people say that 'alpha' is the volatility - how is this so? To me it is a model parameter we can find by calibration?

i think alpha is the initial volatility of the stochastic volatility process used to derive the model. btw, as i always understood that SABR is a short time approximation of parametrized implied vola skew backed out of a 2 factor model. how is this supposed to work well for long times to maturity?

- mekornilol
**Posts:**32**Joined:**

Reviving this thread 9 years later for all users who joined after the credit crunch...

Any expert views on which interpolation in the volatility cube with SABR interpolators produces better results? Interpolation on parameters or rather on volatilities?

I would also be very interested to be pointed to any relevant papers or studies that have been published on this recently. More generally, is it better to interpolate on parameters or rather interpolate on vol using our maturity-indexed vol model interpolators?

I am currently working on the issue and cannot find theoretical/practical justifications to prefer one approach over the other.

Thanks to all.

Any expert views on which interpolation in the volatility cube with SABR interpolators produces better results? Interpolation on parameters or rather on volatilities?

I would also be very interested to be pointed to any relevant papers or studies that have been published on this recently. More generally, is it better to interpolate on parameters or rather interpolate on vol using our maturity-indexed vol model interpolators?

I am currently working on the issue and cannot find theoretical/practical justifications to prefer one approach over the other.

Thanks to all.