I'am wondering if there is a standard definition to the Vega of an exotic product when the underlying model is not Black-Scholes.
Let me give some examples :
What is the Vega if the price is obtained by a local volatility model. Is it obtained by a parallel shift of local vol ?
What is the Vega when the model is a stochastic vol ? Is it a sensitivity to spot vol ?
Or the Vega is obtained by parallel bumping implied volatility surface ?
In the first place I thought Vega was obtained with the following steps :
Price the exotic with the relevant (well calibrated) model.
Find the constant implied volatility of Black-Scholes model that gives the same price (let us call it the $\text{ExoIV}$)
Find the sensitivity of BS price to $\text{ExoIV}$.
But this might be quite complicated either numerically or merely because no close formula exists for the exotic derivative in Black-Scholes framework.