# Vega of exotic options

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.

• It depends what you define as the Vega. If you are looking for the "Black-Scholes" Vega of an exotic product priced under another specific model then it would indeed be the sensitivity of the latter price to a full parallel shift of the IV surface used to calibrate your exotic model in the first place. – Quantuple Jul 30 '18 at 8:39
• If I understand you well, you believe the steps I mentioned about ExoIV are equivalent to compute Vega by bumping IV surface ? – Jiem Jul 31 '18 at 18:18

$\nu_1 = \frac{ \partial C}{ \partial v} = \frac{ \partial C}{ \partial v_0} 2 \sqrt{v_0}$ and $\nu_2 = \frac{ \partial C}{ \partial w} = \frac{ \partial C}{ \partial \theta } 2 \sqrt{\theta}$
With $v = \sqrt{v_0}$ and $w = \sqrt{\theta}$ , $v_0$ and $\theta$ being respectively the mean reversion level and the initial level of variance in the Heston model.