Choosing the Filtration for a Risk-Neutral Heston Model

Summary

The document asks what information the filtration represents when defining a risk-neutral measure for the Heston stochastic volatility model. A filtration describes the information available up to each time, and the response gives a market-oriented example: the information generated by the asset price and variance processes observed through time. This provides a concrete way to interpret the sigma-field appearing in the Radon–Nikodym derivative used for the change of measure.

The answer suggests that the Wiener processes are generally not included as directly observable market information; their effects are reflected indirectly in the modeled price and variance paths. This is an illustrative modeling choice rather than a unique definition for every Heston setup. The document does not discuss technical conditions required for the measure change, alternative filtrations, or how latent variance might be inferred from market data. Its contribution is a concise explanation of the information concept and a plausible filtration for the stated context.

Key ideas

• A filtration represents the information available up to each point in time.
• For a Heston model, one possible filtration is generated by the asset price and variance processes observed through time.
• The driving Wiener processes are generally treated as unobservable market quantities.
• The suggested filtration is a modeling example, not a universal specification for every application.

Tags

# Filtration used in the change of measure of Heston model


# Filtration used in the change of measure of Heston model












The risk-neutral measure $\mathbb{Q}$ in changing the measure of the Heston model is defined through the following Radon-Nikodym derivative:

\begin{equation*} \begin{aligned} \frac{d{\mathbb{Q}}}{{d\mathbb{P}}}\Bigg\vert_{\mathcal{F}_{t}}=\exp\biggl[ &- \int_{0}^{t}\gamma_{1,s}d{W_{S,s}}-\int_{0}^{t}\gamma_{2,s}d{W_{v,s}} \\ &+\int_{0}^{t}\rho\gamma_{1,s}\gamma_{2,s}d{s}-\frac{1}{2}\int_{0}^{t}\left(\gamma_{1,s}^{2}+\gamma_{2,s}^{2}\right)d{s}\biggr], \end{aligned} \end{equation*}

where $\gamma_{1,t}$ and $\gamma_{2,t}$ are the market prices of risk given by

\begin{equation*} \gamma_{1,t}=\frac{\mu-r}{\sqrt{v_{t}}},\gamma_{2,t}=\frac{\lambda\sqrt{v_{t}}}{\sigma}. \end{equation*} What is the mathematical definition of the filtration $\mathcal{F}_{t}$ here?

For example, the filtration $\mathcal{F}_{t}^{X}=\sigma(X_{0},X_{1},\ldots,X_{t})$ is the information generated by the stochastic process $X_{t}$ on $[0,t]$.

## Answer by welcra (score 0)

https://quant.stackexchange.com/a/81689

The filtration should include all the observable information until time t. In the context of Heston models, a definition for filtration could be $\mathcal{F}_t = \sigma (\{S_s, v_s : 0 \le s \le t\})$. Typically you do not include the wiener processes in the filtration because they are not directly observable in the market and are implicitly included in the filtration through $v_t$.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.