Heston Vega Hedging Across Initial and Long-Run Variance
Summary
The document discusses how to hedge an option under the Heston stochastic-volatility model using another option and the underlying stock. Its starting point is to offset the target option’s sensitivity to variance with a volatility hedge, then choose the stock position to address the remaining price sensitivity. It raises the difficulty that Heston volatility depends on both initial variance and the long-run mean level, so a single vega based on an initial-variance bump may not capture exposure across maturities.
The answer suggests considering sensitivities to both parameters when approximating a parallel volatility-surface shift. It explains that changing initial variance mainly affects shorter maturities, while the long-run level can matter for longer maturities, and describes combining the two shifts as an approximation. This is a conceptual response, not a complete formula for constructing a scalar hedge ratio or a tested hedging procedure. The meaning and implementation of the cited cash-vega expression remain only partly explained.
Key ideas
- A Heston hedge can use a second option to offset volatility sensitivity and stock to adjust price sensitivity.
- Heston volatility exposure can depend on both initial variance and its long-run mean level.
- Bumping only initial variance may understate exposure for positions sensitive to longer maturities.
- Combining parameter sensitivities can approximate a parallel volatility shift, subject to assumptions.
Tags
Full text
# Vega in the Heston model
# Vega in the Heston model
I'm trying to calculate the hedging quantities of the Heston model. I undestand that the replicating portfolio consist of one option, $V = V(S,v,t)$, $\Delta$ stocks and $\phi$ units of the option to hedge volatility, $U(S,v,t)$. The quantities are found by: \begin{align} \phi = - \frac{\partial V}{\partial v} / \frac{\partial U}{\partial v} = - \nu_V / \nu_U \quad \text{and} \quad \Delta = - \phi \frac{\partial U}{\partial S} - \frac{\partial V}{\partial S}. \end{align} Next, I need to calculate these quantities. As pointed out by Zhu(2010), the dynamics of the volatility in the Heston model is given by two parameters, the mean reversion level, $\theta$, and the initial level of the variance, $v_0$. He therefore suggest to base the calculation of vega on both parameters by defining vega as a gradient of two partial differentials: \begin{align*} \nu & = (\nu_1, \nu_2) = \left( \frac{\partial C}{\partial v}, \frac{\partial C}{\partial \omega} \right) = \left( \frac{\partial C}{\partial v_0} 2 \sqrt{v_0}, \frac{\partial C}{\partial \theta} 2 \sqrt{\theta} \right), \end{align*} where $\omega = \sqrt{\theta}$ and $v = \sqrt{v_0}$.
Zhu(2010) further states that "The cash amount of mean Vega labeled as mean cash Vega is the total differential: $$ \nu_{cash} = 2\frac{\partial C}{\partial V_0}v_0 \Delta v_0 + 2\frac{\partial C}{\partial V_0}\theta\Delta \theta$$"
My questions:
- as the we now has that vega is a gradients, how do I calculate $\phi$? I'm implementing this hedging procedure, so I need to return a number - not a gradient?
- I don't understand what Zhu means with $\nu_{cash}$? Is this the quantities that I to use for calculating $\phi$? If so, what is $\Delta$ here?
Thank you in advance!
## Answer by jherek (score 4)
https://quant.stackexchange.com/a/60019
Zhu makes sense to me.
The vega cash in Black-Scholes corresponds to a shift of the vol surface by 1%.
If you bump only $v_0$ in Heston, you bump only the short maturities, and if your structure is also dependent on the long maturities, the vega will be vastly underestimated. So you need to bump the $\theta$ as well. I think it implicitly assumes that the other Heston parameters have little relation with a parallel shift, and that a parallel shift can be approximated by the sum of the two independent shifts.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.