# 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?

## 3 Answers

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).

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.

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

• 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. – LocalVolatility Mar 13 '17 at 17:40
• ok so if you are fine with them outside that range what is the question then? – Bond007 Mar 14 '17 at 10:18
• @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. – FinanceGuyThatCantCode Apr 12 '17 at 20:19