hi, i have been reading hagan's paper "managing smile risk" on the sabr model and all examples mentioned in the article and all postings on this forum seem to quote values of beta between [0,1]. Is that a constraint on the model? Is it possible (and sensible) to want to use beta > 1 or possibly < 0?Thanks

If you condition on the volatility, I believe your question becomes: for what beta is dS = a S^beta dB(t), with `a' fixed, sensible?First, for beta > 1, S is not a true martingale, just a local martingale, so you probably don't want that. Then, for beta <= 1, there are different types of hitting behavior at S=0. For beta = 1, the point S = 0 is unreachable: this is well-known from the Black-Scholes theory.For 1/2 <= beta < 1, the point S=0 is an exit boundary, which is reachable but no boundary conditions can be supplied.An exit boundary is a trap state.For -infinity < beta < 1/2, the point S = 0 is always hit and is a regular point. The only sensible boundary condition (without a cash flow) for this last hitting case would be to kill the process when it hits. It you were talking about an interest rate, for example, then reflection would be ok also.I'm going to fix on stock prices.So, it seems that beta < 1 always forces a bankruptcy-type event, where S=0 is reached with probability onein finite time and becomes a trap from then on.If this argument is correct, it reinforces the caution associated with this model that it should only be applied to equities close toexpiration. One reason for that is the absence of volatility drift. A second reason would be that you need a time much smaller than the first expected hitting time of S = 0. But, I am not a SABR expert, so my argument is a conjecture that conditioning on the volatility path yields a reliable conclusion. Perhaps one of the SABR experts on the board could confirm/deny this.For more details on dS = S^beta dB(t), see Ch. 9 of my "Option Valuation under Stochastic Volatility" book.I give a simple formula there for the strictly positive probability of hitting S = 0 prior to time t for any beta with: -infinity < beta < 1. regards,

hi Alan,Thank you very much for the comprehensive reply and for the chapter reference (we're having a look right now)!Regards,J

HiTo estimate beta parameter i used least square method for the equation log(sigmaATM)=log(alpha)-(1-beta)*log(Forward rate ) for swaption but i find somme values <0 and >1can you give me other method to estimate this parameter thanks in advance

What you're suggesting is that sigmaATM = alpha*F^(beta-1) - while it would be nice if thiswere the case, the fact that dS = alpha*S^beta dB doesn't imply this at all.If you need to estimate beta, try fitting it like you fit the other parameters.

alpha=sigmaATM*forward^(1-beta) is just the approximation of SigmaATM when time to expiry is near to zero (for this to estimate this parameter i used swaption with short maturity)Thanks in advance

Ah alright, then it would depend on how good the approximation is...Nevertheless, I guess you want one parameter beta for more maturities - so just do a global fit.

From a practical point of view, calibration tests give negative values (~-2) for beta (in particular for short maturities).