Defining Heston Vega with Initial and Long-Run Variance
Summary
The document addresses which parameters should define vega in Heston or Bates option pricing. It describes an approach attributed to Zhu (2010): because initial variance and long-run variance both shape the variance level, sensitivity can be reported separately with respect to the square roots of each parameter. The resulting measures are derivatives of the option price with respect to initial volatility and long-run volatility, obtained by applying the chain rule to sensitivities with respect to variance.
For the initial-variance measure, the text gives a call-price expression using derivatives of the model’s characteristic-function-based probabilities. It also lists parts of the Fourier pricing setup. This gives a convention and mathematical outline rather than a full derivation. The excerpt does not provide the corresponding expanded formula for long-run variance sensitivity, numerical examples, or guidance on comparing vegas across implementations; its notation also includes apparent typographical inconsistencies.
Key ideas
- The document treats initial variance and long-run variance as separate sources of volatility sensitivity.
- It defines each vega with respect to the square root of its variance parameter.
- The chain rule converts derivatives with respect to variance into derivatives with respect to volatility.
- The initial-variance vega is expressed through sensitivities of characteristic-function pricing probabilities.
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# Vega in Heston / Bates Model
# 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
## Answer by user16891 (score 4, accepted)
https://quant.stackexchange.com/a/18857
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 thisShown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
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