# Calibrate the SABR model to the implied volatility surface

I'm currently trying to calibrate the SABR model. The question I have is that when I consider papers and other websites I only come across cases where the SABR parameters are calibrated to the implied volatility smile, thus for one specific time-to-maturity. However, I'm wondering if it is possible to just calibrate the SABR parameters to the entire volatility surface.

For example in the following way:

1. First take $$\beta$$ from market data.
2. Second solve for: $$\hat{\alpha}, \hat{\rho}, \hat{\nu} = min \sum_{(K,T)} (\sigma_{mkt} - \sigma_{SABR})$$

A follow up question if this is possible: In the literature I often read about not calibrating $$\alpha$$ but extracting it directly from the implied volatility from the ATM level. I'm assuming this is no longer possible if you calibrate the parameters to the entire surface since the ATM level changes depending on $$\tau$$?

• Yes you could calibrate it to the whole surface but the fit will be less good (which is why the calibration for each time slice is often done). And yes the ATM will vary with time so in your case alpha would be whatever value gives the best fit. Apr 29 at 14:00
• @FridoRolloos, you cannot calibrate vanilla SABR to the whole surface, this is a model for a single expiry. May 4 at 8:04
• @Hasek disagree, that would be the same as saying you cannot calibrate Heston to the whole surface. Of course you can, but the fit would be less good than fitting it to a single time slice (as I wrote above). May 4 at 11:15

> However, I'm wondering if it is possible to just calibrate the SABR parameters to the entire volatility surface

No, this is not how it supposed to work. SABR model describes dynamics of a single forward $$F$$, i.e. the parameters of the process can only be calibrated to a single expiry and they are not time-dependent. Doing what you suggested is plain wrong and has no theoretical justification.

There are several ways around it if you need to accomodate the term structure of the smile. In general it seems like what you are looking for is known as the SABR LMM -- this is the LIBOR market model with stochastic volatility. However if you're working with plain vanilla European interest rate caps/floors then you can do a caplet/floorlet volatility stripping from market quoted caps/floors and separately calibrate vanilla SABR on each time expiry.

• I think there is a difference between how a model is supposed to work in theory and what is done in practice. You wrote in your answer "This is not how it is supposed to work". On the contrary, the point of a SV model is, in theory, to describe the dynamics of a price process consistently, and that includes explaining the entire volatility surface. In practice, though, SABR is not used that way, and neither is Black Scholes. The model(s) are fudged. May 4 at 11:20

This is absolutely something you can do. However, I've never once seen a market in which this is practically a useful thing to do. When parameters get calibrated smile-by-smile, vol of vol is often far higher in the short term than the long term. So, if you're trying to use a single number for both, you end up overpricing the wings of long term options and underpricing the wings of short term options.

In this paper you might find some answers to your question: https://www.researchgate.net/publication/319205444_MANAGING_VOL_SURFACES

• It would be very nice if you could also include in your answer exactly how this paper was answering the question as well as a more detailed reference including title, authors, etc. These are important because should the link gets broken, future readers of your answer might not benefit much from it. Sep 20 at 17:50
• While this link may answer the question, it is better to include the essential parts of the answer here and provide the link for reference. Link-only answers can become invalid if the linked page changes. - From Review Sep 21 at 21:47