Defining Vega for Exotic Options Across Pricing Models
Summary
The discussion examines what “vega” should mean when an exotic option is priced with a model beyond Black–Scholes. It raises several possible bump conventions: shifting local volatility, changing stochastic volatility parameters, or bumping an implied volatility surface. It also considers an indirect approach: find the Black–Scholes implied volatility that reproduces the exotic’s model price, then measure sensitivity to that equivalent volatility.
The answers describe conventions rather than a single universal definition. One approach in interest-rate models is to scale relevant volatilities, either at their input quotes or after interpolation. For the Heston model, the cited reference defines sensitivities with respect to square roots of the initial variance and long-run variance, illustrating that model-specific parameter choices can produce different vegas. The discussion does not compare these conventions empirically or prescribe one for all products; the chosen risk measure depends on the model and the volatility exposure being reported.
Key ideas
- Exotic option vega depends on how volatility risk is defined for the pricing model.
- A volatility bump can be applied to quoted inputs or to interpolated model volatilities.
- A Black–Scholes equivalent volatility offers one possible way to define sensitivity, but may be difficult to compute.
- Heston sensitivities can be expressed with respect to transformed variance parameters.
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# Vega of exotic options
# 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.
## Answer by dm63 (score 3)
https://quant.stackexchange.com/a/40995
In the interest rate world, the vega of an exotic is usually defined by bumping all the relevant volatilities by a multiplicative factor , typically 1.01, 1.05 or 1.10. This could be done in at least two ways (1) bump all the input volatilities or (2) bump directly all the volatilities in the model. To illustrate the difference: some models take as inputs a set of standard swaptions. These are then used by some interpolation scheme to calculate all other required volatilities. In (1), we bump the inputs. In (2), we bump the outputs of the interpolation. There shouldn't be much difference.
## Answer by Mehdi (score 1)
https://quant.stackexchange.com/a/41034
In this book : Zhu, J. (2010). Applications of Fourier Transform to Smile Modeling: Theory and Implementation. Zhu defines two vegas for the Heston model :
$ \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.
You can see more about the greeks in the Heston model in this book : Fabrice D. Rouah The Heston Model and Its Extensions in Matlab and C#Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.