Converting Heston Variance Parameters from Physical to Risk-Neutral Measure
Summary
The document addresses which probability measure applies to the Heston option-pricing equation and how its variance-process parameters change between the physical measure and the risk-neutral measure. It shows that the variance drift can be rewritten in mean-reverting form after incorporating a volatility risk-premium term. Under the parameterization given, the risk-neutral mean-reversion speed equals the physical speed plus the premium parameter, while the risk-neutral long-run variance equals the physical speed times the physical long-run variance divided by that adjusted speed.
This algebra explains how the stated drift coefficients map across measures and why option valuation uses risk-neutral dynamics. The result depends on the particular specification of the variance risk premium used in the question; other conventions or models may define that premium differently. The document does not discuss estimation, calibration, or empirical validation of the parameters.
Key ideas
- Option pricing in the Heston framework uses variance dynamics under the risk-neutral measure.
- A variance risk-premium term changes the mean-reversion parameters when moving from the physical measure.
- The adjusted risk-neutral mean-reversion speed is the physical speed plus the premium parameter.
- The adjusted long-run variance is rescaled so the variance drift retains mean-reverting form.
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# Heston Equation Parameters
# Heston Equation Parameters
My question might sound dumb, but I hope someone hears me out. I've been learning Heston's stochastic volatility model for a while, but it heavily relies on "The Heston Model and Its Extensions in Matlab and C#." For European call options, the Heston equation is
\begin{multline} \frac{\partial C}{\partial t}+\frac{1}{2}S_{t}^{2}V_{t}\frac{\partial^{2}C}{\partial S_{t}^{2}}+\frac{1}{2}\sigma^{2}V_{t}\frac{\partial^{2}C}{\partial V_{t}^{2}}+\sigma\rho S_{t}V_{t}\frac{\partial^{2}C}{\partial S_{t}\partial V_{t}} \\ -rC+rS_{t}\frac{\partial C}{\partial S_{t}}+[\kappa(\theta-V_{t})-\lambda V_{t}]\frac{\partial C}{\partial V_{t}}=0, \end{multline}
where $\lambda$ is a constant in the volatility risk premium. Here, the underlying asset dynamics, i.e., its price $S_{t}$ and variance $V_{t}$, follow the Heston model. I've understood the derivation of this Heston equation via the risk-neutral argument, but which measure do we use here? The physical measure $\mathbb{P}$ or the risk-neutral measure $\mathbb{Q}$? In the Heston equation, are the parameters $\kappa$ and $\theta$ risk-neutral, i.e., $\theta=\kappa_{\mathbb{P}}\theta_{\mathbb{P}}/(\kappa_{\mathbb{P}}+\lambda)$ and $\kappa=\kappa_{\mathbb{P}}+\lambda$?
## Answer by Kevin (score 2, accepted)
https://quant.stackexchange.com/a/83773
You can move from $\mathbb{P}$ to $\mathbb{Q}$ parameters easily. Your PDE states numbers under $\mathbb{P}$ and the next steps would be \begin{align} \kappa^\mathbb{P}(\theta^\mathbb{P}-V_t)-\lambda V_t &= \kappa^\mathbb{P}\left(\theta^\mathbb{P}-\frac{\kappa^\mathbb{P}+\lambda^\mathbb{P}}{\kappa^\mathbb{P}} V_t\right) \\ &= (\kappa^\mathbb{P}+\lambda)\left(\frac{\kappa^\mathbb{P}\theta^\mathbb{P}}{\kappa^\mathbb{P}+\lambda}- V_t\right) \\ &= \kappa^\mathbb{Q}(\theta^\mathbb{Q}-V_t), \end{align} where $\kappa^\mathbb{Q}=\kappa^\mathbb{P}+\lambda$ and $\theta^\mathbb{Q}=\frac{\kappa^\mathbb{P}\theta^\mathbb{P}}{\kappa^\mathbb{P}+\lambda}$.
Effectively, this parameterisation ensures that $V_t$ remains of the same probability distribution under $\mathbb{P}$ and $\mathbb{Q}$ subject to the change of the parameters.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.