Calibrating SABR Parameters Across Expiries
Summary
The document asks whether one SABR parameter set can be fitted to an entire implied volatility surface, and whether the at-the-money volatility can still determine alpha when fitting across expiries. The accepted response says standard SABR describes one forward rate with parameters calibrated to a single expiry; making those parameters vary across expiries lacks theoretical justification within that model. It points to SABR LMM as a framework for representing a term structure of smiles, or to stripping cap and floor quotes into caplet and floorlet volatilities before calibrating separate SABR smiles by expiry.
A second response says a joint surface fit is mathematically possible but may be practically unhelpful: short-dated smiles can have much higher volatility of volatility than long-dated smiles, so one shared value may misprice wings across maturities. The responses differ on whether joint calibration is categorically wrong or simply not useful. The discussion offers no derivation or empirical study, so it gives guidance rather than a demonstrated calibration procedure.
Key ideas
- Standard SABR calibration describes a single forward and one expiry at a time.
- A single parameter set across expiries may fail to capture changes in smile shape.
- SABR LMM is suggested for modeling a term structure of smiles.
- Cap and floor quotes can be stripped into caplet and floorlet volatilities for separate expiry calibrations.
- A shared volatility-of-volatility parameter may misprice short- and long-dated option wings.
Tags
Full text
# Calibrate the SABR model to the implied volatility surface # 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: - First take $\beta$ from market data. 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 $\tau$? ## Answer by Hasek (score 2, accepted) https://quant.stackexchange.com/a/70725 > 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. ## Answer by kolbe (score 0) https://quant.stackexchange.com/a/72254 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. ## Answer by Riccardo (score -2) https://quant.stackexchange.com/a/72250 In this paper you might find some answers to your question: https://www.researchgate.net/publication/319205444_MANAGING_VOL_SURFACES
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