I have a problem with the normal implied vols derived from SABR (http://www.math.nyu.edu/~alberts/spring07/Lecture2.pdf) Alpha is not proportional to ATM vol , as i increase alpha the ATM vol increases to a point and then decreases , similar issues with rho , is this a known issue ?Thanks

Anyone?I am sure all those using the normal SABR would be able to answer this I just want to confirm if i have made an implementation error or is it something known and corrected for> feesl like its a big practical issue to be ignored

From (36), I assume the leading term is sig(n,atm) = sig0 F0^betaIf you include the corrections, you havesig(n,atm) = sig0 F0^beta{ 1 + [c1 beta + c2 rho alpha + c3 (2 - 3 rho^2) alpha^2]},where c1,c2,c3 are independent of alpha and rho. So, the corrections are linear+quadratic in both rho and alpha.A linear+quadratic dependence on a parameter is a parabola and can behave just like you described: increase and then decrease (or vice-versa). So, I don't really see the issue.

QuoteOriginally posted by: AlanFrom (36), I assume the leading term is sig(n,atm) = sig0 F0^betaIf you include the corrections, you havesig(n,atm) = sig0 F0^beta{ 1 + [c1 beta + c2 rho alpha + c3 (2 - 3 rho^2) alpha^2]},where c1,c2,c3 are independent of alpha and rho. So, the corrections are linear+quadratic in both rho and alpha.A linear+quadratic dependence on a parameter is a parabola and can behave just like you described: increase and then decrease (or vice-versa). So, I don't really see the issue.Alan , the issue is that if the parameters are parabolic in terms of the vol , then there is issue with the dynamics , if i have to raise the straddle by x ticks , normally i would like to just move the alpha and keep the other three parameters same , but because of this parabolic relation this becomes difficult , also when the market risk reversal changes , you dont know what paramter to change !The lognormal version of sabr has well defined axis for the sabr skew paramters , so it becomes easier for risk management , but the problem with that is again you cannot get the current market skew , dynamics being a secondary issue !

I see. Well, instead of adjusting the parameters one by one, "by hand" as you seem to do, you couldwrite a simple optimizer to adjust them all at once for the best fit.

QuoteOriginally posted by: AlanI see. Well, instead of adjusting the parameters one by one, "by hand" as you seem to do, you couldwrite a simple optimizer to adjust them all at once for the best fit.An optimizer will definitely give me the best fit , but i would not be able to quantify my risks in terms of alpha (ATM vol) , rho(risk reversal risk) and nu (convexity) , I don't think any trader will like a model in which skew parameters loose the reference point even if it fits perfectly . Each parameter should have one principal component , if i want to move the risk reversal i should know what parameter to change , otherwise there is now way to gauge the risk of an option portfolio .I am posting my matlab code for the implementation of eqn 36 from the nyu notes and eqn A 59 in the appendix on managing smile paper by Hagan , i will be glad if you can find an implementation error function ret=SABRsigmaN(K,F) global alpha; global beta; global rho; global nu; global T; Fmid = (F * K).^.5; gamma1 = beta./Fmid; gamma2 = (-beta*(1-beta))./Fmid.^2; tau = nu*(F^(1-beta) - K .^(1-beta))./(alpha*(1-beta)); dgamma = log(((1-2*rho.*tau + tau.^2).^.5 + tau - rho )./(1-rho)); t1 = (2.*gamma2 - gamma1.^2).*((alpha.*(Fmid.^beta)).^2)./(24*nu^2); t2 = (rho.*gamma1.*alpha.*Fmid.^beta)./(4*nu); t3 = (2 - 3*rho^2)/24; ret = (nu.*(F - K)./dgamma).*(1+ (t1 +t2 +t3)*T*nu*nu);eqn a 59 in Hagan function ret=SABRsigmaNH(K,F) global alpha; global beta; global rho; global nu; global T; Fmid = (F * K).^.5; gamma1 = beta./Fmid; gamma2 = (-beta*(1-beta))./Fmid.^2; tau = (nu .*(F-K))./(alpha.*Fmid.^beta); dgamma = log(((1-2*rho.*tau + tau.^2).^.5 + tau - rho )./(1-rho)); t1 = (2.*gamma2 - gamma1.^2).*((alpha.*(Fmid.^beta)).^2)./(24); t2 = (nu*rho.*gamma1.*alpha.*Fmid.^beta)./(4); t3 = ((2 - 3*rho^2)*nu^2)/24; ret = ((nu.*((F - K).^2).*(1-beta))./((F^(1-beta) -K.^(1-beta)).*Fmid.^(1-beta).*dgamma)).*(1+(t1+t2+t3).*T);

Re a trader not liking a model, I can't argue with that.Re your code -- I am not a matlab programmer, but I am pretty confident it will fail badly if you set F=K