Hello,I'm trying to understand the calibration procedure given in FX Volatility Smile Construction by Wystup but have an issue with the premium-adjusted delta case.In his theorem 1 (p20-21) it refers to a variable a, used in eq 35 : "variable a is the difference of a call delta, corresponding to a -0.25 put delta and the -0.25 put delta for any delta type" (I assume d=0.25 for instance).When considering regular deltas this expression is 1 or exp(-rf.T) so no problem. However when we consider premium-adjusted deltas, this variable a thus depends of the strike K of the -0.25 delta put, and this strike is not known since we don't know yet the volatility of the -0.25 put (which would be given by the calibrated function). I can extract the strikes for the market strangle by 20-21 but I thought I had to wait for the calibrated function for calculating the K25C and K25P with equations on p14.So how can we carry on the calibration procedure ?

I would assume the following but would like to have a confirmation :- at each step of the root search I have a new sigma_s (vol from the smile strangle)- I deduce the sigma_25P from eq. 32 on p18- I find K_25P since I now have sigma_25P and the delta (unique solution since this is a put)- I use this K_25P for calculating a- I use the parametrization with this a and deduce the strangle vols for the strangle strikes- I can then calculate the price of the strangleIs this the right way to proceed ?

Sent you a PM.

WillK, this is exactly the way to proceed. As you have a starting parameter (i.e. quoted strangle)you'll have the vols at each time of the calibration, consequently the 25D strikes and the parametera.

If you work in the strike space, then the variable "a" can be calculated explicitly since the strike would be known.Once you've constructed the vol surface in the strike space, converting to delta space should be straight forward.