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SABR Beta Outside the Usual Range and Martingale Conditions

Article Quant Q&A · Author: Janthelme
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Summary

The document asks whether SABR can be used for foreign-exchange and equity underlyings when beta, estimated from a regression of at-the-money implied volatility against the forward, falls below zero or above one. The responses distinguish parameter fitting from the mathematical behavior of the model. They state that negative beta is compatible with a strict martingale, while beta above one generally leaves only a local martingale, with possible finite-time explosion. At beta equal to one, the stated martingale condition depends on the correlation parameter.

Another response relates beta below one to a negative relationship between at-the-money volatility and the forward, and notes that a positive relationship for beta above one may conflict with observed patterns. These are replies to a narrow modeling question, not a broad validation study. The document supplies no market data or calibration comparison, so it does not establish whether a fitted SABR model is suitable for a particular underlying.

Key ideas

  • The question concerns applying SABR to equity and foreign-exchange underlyings with fitted beta outside the usual interval.
  • The responses state that negative beta can still yield a strict martingale.
  • For beta above one, the process is described as a local martingale that may explode in finite time.
  • At beta equal to one, the stated martingale condition depends on correlation.
  • The implied-volatility relationship discussed is not evidence of model fit for a specific market.

Tags

Derivatives pricingOptionsForexEquities
Full text
# SABR beta range


# SABR beta range












I am thinking of using SABR for non-rate underlyings (eg FX and equity underlyings).

Typically one finds the beta via a regression of historical implied vols vs forwards, since $$\ln(\textrm{atm vol}) = \ln(\alpha) - (1-\beta) \times \ln(\textrm{forward}).$$ However for FX and equity underlyings, it is not uncommon to find a resulting beta either negative or above 1.

My question : is the SABR model still valid for beta values outside the typical [0,1] range?

## Answer by andrew (score 7)

https://quant.stackexchange.com/a/36393

The SABR process is a strict martingale for all values of beta < 1 (in particular, negative betas are fine). If beta = 1, the process is a strict martingale if and only if rho < 0. Under all other circumstances, i.e. beta > 1, or beta = 1 and rho >= 0, the SABR process is a local martingale but not a martingale (it may explode in finite time).

## Answer by Yanyi Yuan (score 1)

https://quant.stackexchange.com/a/48753

Given your regression relationship between atm IV and forward price, as long as beta <1, atm IV and forward price are negatively correlated which is usually consistent with the market observations - the higher the forward price (longer maturity), the lower the atm IV. If beta is greater than 1, rather, ATM IV and forward price are positive correlated, which is abnormal.

## Answer by Bond007 (score 0)

https://quant.stackexchange.com/a/32999

The beta is handing the underlying distribution ie Beta = 0 as Stochastic Gaussian ( Normal ) Model, as 0.5 for Stochastic CIR model and 1 as Stochastic Lognormal Model. So outside of this witch kind of distribution you will have?

Never heard about this

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.