Hi,I am wondering about the following:Say I am using a forward rate model:[$]df_i(t)/f_i(t) = \mu_i(t)dt + \sigma_i(t)dW_i(t), \quad i = 1, ..., 10[$]with stochastic volatilities[$]d\sigma_i(t)/\sigma_i = \mu_i^\sigma(t)dt + \nu_i(t)dZ_i(t), \quad i = 1, ..., 10.[$]Now I want to vega hedge a swaption according to this model. How would I do it?Idea: Take ten caplets or ten additional swaptions. Bump each of the ten [$]\sigma_i[$]s and revalue the swaption to be hedged as well as each of the caplets/swaptions used as hedging instruments. This gives me ten vegas for the swaption and 10x10 vegas for the hedging instruments. The solution to the corresponding linear system of equations yields the positions in each product to make the whole portfolio vega neutral.(Afterwards I would add discount bonds to make the portfolio delta neutral as well.)Is this correct?

Ok, I think I have misunderstood the meaning of a "vega hedge". I thought is was about the model-specific volatility risk but now I talked to someone and was told it is computed by shifting the implied volatility surface, recalibrating the model to the new volatility surface and repricing the product at hand. I also found this method in a paper.But now I am wondering the following: Say I am considering a particular swaption and I am also calibrating the model to this swaption (and a few more). Also assume that the model is capable of perfectly recovering the input swaption prices used for calibration. In this case the above method (bumping the vola surface, recalibrating, repricing) would simply lead to the Black vega. This feels wrong to me.