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SABR
G'day!I am in need for help. So far, I have been trying to apply the SABR model to estimate swaption volatility smiles. To get a better fit to the market, I would like to estimate beta anew for each option maturity before re-applying the model.I am getting the swap forward rates from a collegue and now try to set up a programm that will calculate the distribution of the betas. Does anyone know how to go about this? I got a hint that there is a way to do it in "Paul Wilmott on Quantitative Finance", but my mathematicall knowledge is a bit limited as that I could do it all by myself.I'm open for any suggestions!cheers,Julz
SABR
fiveone: (assuming my version of the paper is the same as yours, and thus our equation numbering is the same)Try making the assumption that the local vol is off the form: sigma_loc (F) = A + B(F-f0)where f0 is the forward value when the implied vol was obtained from the market. Writing 2.8 as sigma_B(K,f0) = simga_loc(0.5[f0+K)) + (1/24)*sigma_loc''(f0-K)^2yields sigma_B(f0,K) = A + (1/3)*B*(f0-K)^2To make sigma_B(K) = alpha + beta*(K-f0)^2we need to choose A to alpha and B = 3*beta, so sigma_loc (F) = alpha + 3*beta(F-f0)Now supose that the market moves from f0 to f. Then using this sigma _loc in sigma_B(K,f) = simga_loc(0.5[f+K)) + (1/24)*sigma_loc''(f-K)^2 yields the result
SABR
If you are calibrating rho and volvol, you really shouldn't calibrate beta. That said, one can fint the best fit CEV model by making a linear regression of log vol_ATM versus log (forward) to get the exponent beta. Once beta is chosen, one can go ahead and fit the other paramets for each swaption in the matrix
SABR
Dear Pat,thank you for the idea of estimating beta via linear regression. I have attempted this a while ago but the results were not really desirable. I got values for beta that varied between -1 and +3. This cannot be right. From what I learned in our econometrics classes, a linear regression will produce ambiguous results. Does the problem of such a linear regression lie in the time series data for the forwards? wouldn't it make more sense to run a GMM or ML like regression?julia
SABR
If we take a snapshot of the market on a given day, both the beta and rho contribute to the skew, so many people feel that it isn't advisable (in terms of getting stable parameter values) to calibrating them both from the smile.The only way to effectively distinguish between the two is to note that the atm vol goes like sigma(F,F) = a/F^(1-b) * { 1 + trash}That said, the data is equivocal because sigma (that is, a) can go up or down all by itself in addition to going up or down because F changes, so studies of log{sigma(F,F)} versus log{F} are somewhat (how do I phrase this?) enigmatic. Many firms simply choose beta (invariably at either b=1 or b=1/2 or b=0, except one firm uses b=0.10) and then fit a, volvol and rho.One idea is to pick the beta that fits the smile best, and then fit the other parameters, but I'm generally not in favor of this approach since it is essentially picking the beta so that rho is as close to zero as possible. I'm lacking an insight as to why this is desireable
SABR
I understand that not both beta and rho should be calibrated. I am just trying to get a feeling for the parameters. Also, I have heard from several traders and financial engineers that the value of beta used in the market lies around 0.5. So I assumed that there must be a way of showing that this is correct. It is a bit hard to convince your thesis referee of the beta value in use if you cannot prove it to him. )
SABR
A paper by Berestycki, Busca and Florent, "Computing the implied volatility in stochastic volatility models", (2004), claims different results compared to that published by Hagan et al. (see the remark at the end of section 6.3) although both start from the same problem. Does anyone have any comments on that? (the math is too complicated for me....)
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- StanTheMan
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SABR
Hi Julz,If you want to get a rough idea of your Beta, you can start from the following approximation of the dynamics of the ATM vols (in the SABR framework) - which is a log_derivation (+approximation: get rid of the trash!) of Pat's formulae:dSigma_ATM dForward---------------- = (Beta - 1) * ------------ Sigma_ATM ForwardSo one can compute a simple linear regression from the variation of the ATM vols and the variation of the forward rate.This will give you a good idea of your Beta...But if one wants to be more precise he has to consider the relative volatility of F and Sigma and then include the correlation between the two...Stan