I have a problem with vega hedging.
Consider the management of an exotic derivative, such as Barrier option. Typically we do the following tasks:
- selecting a pricing model, say, a local volatility model such as CEV model.
- choosing a relevant European option implied volatility smile/skew as the calibration instrument.
- Calibrating the pricing model to the calibration instrument and use the calibrated model to find the value of exotic.
My question is how to find the vega greek? As it's the whole imp vol smile/skew that affects the value of the exotic, should I calculate vega(K) by bumping each imp vol with strike K one at a time, and construct the hedging portfolio by using vanilla options across all strikes? (I don't think it's going to happen in practice considering the illiquidity of OTM option.)
It would be best if anyone could provide some reference dealing with this issue. Thanks!!!