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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?

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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).

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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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    $\begingroup$ Just because other values for $\beta$ don't represent a special case that we attached a label to doesn't mean they are not admissible. $\endgroup$ – LocalVolatility Mar 13 '17 at 17:40
  • $\begingroup$ ok so if you are fine with them outside that range what is the question then? $\endgroup$ – Bond007 Mar 14 '17 at 10:18
  • $\begingroup$ @LocalVolatility Is it necessarily the case that the SDEs that define SABR have the same kind of solution as the usual solution when $\beta$ is between 0 and 1? I don't know, but it would be relevant I would imagine. Either way, not a huge fan of using SABR for smile dynamics - it has a major inherent flaw in that the stochastic volatility does not have a mean reverting component. $\endgroup$ – FinanceGuyThatCantCode Apr 12 '17 at 20:19
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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.

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