# Vega in Heston / Bates Model

Just a question regarding "convention": Is the Vega in Heston / Bates model the sensitivity with regards to $\sqrt v_0$ or a term of $\sqrt v_0$ and $\theta$ (Long term variance)?

Regards

## 1 Answer

Since $v_0$ and $\theta$ are responsible for the initial and long-term level of the variance,Zhu (2010) recommends basing vega on those two parameters. Both parameters represent variance, so to create measures of sensitivity to volatility, Zhu (2010) defines two vegas, one based on $\upsilon=\sqrt v_0$ and the other based on $\omega=\sqrt \theta$ for the call are, therefore, the derivatives \begin{align} \vartheta_1=\frac{\partial C}{\partial\upsilon}=\frac{\partial C}{\partial v_0}2\sqrt v_0 \end{align} and \begin{align} \vartheta_2=\frac{\partial C}{\partial\omega}=\frac{\partial C}{\partial \theta}2\sqrt \theta \end{align} The first vega is \begin{align} \vartheta_1=S\,e^{-q\tau}\frac{\partial P_1}{\partial v_0}2\sqrt{v_0}-K\,e^{-r\tau}\frac{\partial P_2}{\partial v_0}2\sqrt{v_0} \end{align} where, for $j=1,2$ \begin{align} P_j=\frac{1}{\pi}\int_{0}^{\infty}Re\left[\frac{e^{-i\phi\ln K}f_j(x_t,v_t,t;\phi)D_j(\tau;\phi)}{i\phi}\right]d\phi \end{align} such that $x_t=\ln S_t$ and \begin{align} &\\ &f_j(x_t,v_t,t;\phi)=exp[C_j(\tau;\phi)+D_j(\tau;\phi)v_t+i\phi x_t]\\ &\\ &D_j(\tau;\phi)=\frac{b_j-\rho\sigma i\phi+d_j}{\sigma^2}\left(\frac{1-e^{d_j\tau}}{1-g_j e^{d_j\tau}}\right)\\ &\\ &C_j(\tau;\phi)=r i\phi\tau+\frac{\kappa\theta}{\sigma^2}\left[(b_j-\rho\sigma i\phi+d_j)\left(\frac{1-g_{j}\,e^{d_j\tau}}{1- g_j}\right)\right]\\ \end{align} where

\begin{align} &d_j=\sqrt{(\rho\sigma i\phi-b_j)^2-\sigma^2(2\,u_{j}\,i\phi-\phi^2)}\\ &g_j=\frac{b_j-\rho\sigma i\phi+d_j}{b_j-\rho\sigma i\phi-d_j}\\ &b_1=\kappa+\lambda-\rho\sigma,b_2=\kappa+\lambda,u_1=-\frac{1}{2},u_2=\frac{1}{2} \end{align} for more details look at this