Hi All, I would need some pointers to the shifted SABR model expansion. That is, the process of the forward is dF = alpha . F^beta. dW is changed to dF = alpha. (F-S)^beta.dW, where S is the shift (a constant, user defined). The process for alpha remains same. I need the expression for the implied volatility sigma(K,F) (as in the Eqn 2.17 in the original paper) for this shifted model. Will much appreciate any reference. Thanks vm

Hi,Hagan gives implied volatility formula for local volatility C(F), just apply it to C(F) = (F - S)^beta. See formulas (A.59) in Managing Smile Risk.QuoteOriginally posted by: prodiptagHi All, I would need some pointers to the shifted SABR model expansion. That is, the process of the forward is dF = alpha . F^beta. dW is changed to dF = alpha. (F-S)^beta.dW, where S is the shift (a constant, user defined). The process for alpha remains same. I need the expression for the implied volatility sigma(K,F) (as in the Eqn 2.17 in the original paper) for this shifted model. Will much appreciate any reference. Thanks vm

thank, this looks like close to what I wantednever read the appexdix before!!

did you make any progress on this? why do you want to consider this variant? negative strikes?

yup, I consider more or less done on this, the objective was getting meaningful smiles for low strikes

is it the case like in deterministic vol shifted lognormal, that all you need to do is to shift the forward and strike and then reuseyour existing lognormal black calculators? In other words, the SABR expansion is applicable as such without further modification?

No, the expansion is slightly more elaborate than that, the ref below (A.59 in original paper) shows the expnasion for a generic case which can be used to get the results. the idea is to shift the smile and hit the mkt prices at the same time. the absorption at zero in normal sabr is not always very great in an environment of near zero rates, so you just shift the absorption boundary

why do you need a more elaborate expansion? A couple of people seem to be using a similar result to what is being used for shifted lognormal. Look at it as follows:d F_t = v_t (F_t + s) dW1d v_t = n v_t dW2Is that the system you are considering, with s being the shift and the usual SABR assumptions on n (vol of vol) and rho?Then consider the following transformation X_t = F_t + s.Now we haved X_t = v_t X_t dW1d v_t = n v_t dW2This system is the lognormal SABR system, which fits straight into the proposed Hagan expansion. A call option on the forward with payoff at t = T can be written in terms of X_T as follows:p_T = (F_t - K)^+ = [X_T - (K + s)]^+. When the dust settles, then it should be possible to price the option using the regular SABR expansionequivalent vol. Do you see anything wrong with that reasoning? I believe that this is also what Kienitz and Wittke do in some of their papers on CMS spread options.

perfect, nothing wrong I see for your case, I use a beta params that makes the transformation a bit nasty - plus I get the equivalent normal vol first and then convert it to a lognormal blend to get black vol, so it is not a simple case of shifting the forwards and strikes. refer to the original paper and also "equivalent black volatilites" by pat hagan for details

right, so you are doing something more general. Are you treating the shifted CEV SABR model?also, the normal expansion suck as far as I know. Even for low strikes, I don't see what it buys you overthe lognormal expansion.

Hi,I'm also trying to figure out how the Hagan's formula would change when you shift the SABR model. At first sight it seems to me that you can just substitute the F and K by F+theta and K+theta in the original formula, because the function C(f) (see formula A.58 in Managing Sile Risk) changes brom C(f) = f^beta to (f+theta)^beta, so the derivatives are the same.