HiIn a calibrated SABR model, if I make beta, rho, and vov all Zero... I should be left with a normal model?And if I make beta=1, rho=vov=0, the resulting should be just logonormal model?Are these true?Thanks a lot

More specifically, in the formula, sigma(K,f)= alpha/(fK)^( (1-beta)/2) * (z/x(z))....if I just substitute beta=0, rho=0, vov=0 is the resulting vol the same as the one I would have got under a normal model?

If vov = 0, and beta = 1, the model produces the Black Scholes volatility given by alpha. if vov =0, and beta = 0, the model produces the normal distribution volatility given by alpha which has to be converted to its equivalent Black Scholes volatility if a Black Scholes pricer is used.Rho have no meaning as alpha is now deterministic.

Yes ... one can get the equivalent Black vol (or equivalent normal vol) for all the CEV models that way ... or if one backs up in the derivation (to before CEV is introduced) one can set volvol to zero and get the equivalent vol for "local vol" models dF = A(F)dWThis is very helpful if one wants to use a yield model for bonds, dy = {drift}dt + A(y)dWwhere drift is whatever's needed to make the model arb-free. Since one know's the bonds forward price F as a function of the forward yield y, F = G(y)one can deduce the model dF = A(y)F'(y)dWwhere y is determined implicitly by F=G(y) ... if this is run through the formulas, some rather spectacular cancelation occurs ....

Thanks to bothbut, how do you deal with the z/x(z) term which goes to zero/zero form... does it go to 1 in the limit?also, I compared the black vol formula under the normal model with the black vol under sabr (with vov=beta=0), and they look a bit different... is there an approximation?

1) You may face a problem related to machine precision as both terms approach zero. I would suggest implementing a explicit check in your program (an "if" statement) for cases where vov = 0 and beta is 0/1. For these cases, you can churn out the known BS vol immediately instead of plugging it into the SABR model formula. 2) Are you referring to the black vol formula under the normal model for ATM strikes? For out-of/in the money strikes, a convexity adjustment needs to be made.