Choosing Physical or Risk-Neutral Calibration for Heston–Nandi GARCH

Summary

The document asks which probability measure to use when estimating the Heston–Nandi GARCH model for option pricing. It gives the physical-measure return and conditional-variance dynamics, then describes the stated parameter transformation used to obtain risk-neutral dynamics. The central practical distinction in the response is the data source: historical asset returns are used to estimate physical-measure parameters, while option prices are used to calibrate risk-neutral parameters.

This guidance separates forecasting the real-world return process from pricing derivatives under the risk-neutral measure. The response is brief and does not explain estimation procedures, parameter mapping in detail, or how to combine historical and option data in a joint calibration. It also offers no comparison of accuracy or empirical results, so the appropriate calibration depends on the model’s intended use and the available data.

Key ideas

• Historical returns inform estimation of the physical-measure model parameters.
• Option prices are generally used to calibrate risk-neutral parameters for pricing.
• The document presents a parameter transformation connecting the physical and risk-neutral Heston–Nandi dynamics.
• Calibration choice depends on whether the task is modeling real-world returns or valuing options.
• The brief answer provides no empirical accuracy comparison or detailed joint estimation method.

Tags

# Calibration of Heston-Nandi GARCH Model Using Historical Data: Risk-Neutral vs. Physical Measure


# Calibration of Heston-Nandi GARCH Model Using Historical Data: Risk-Neutral vs. Physical Measure












I am currently working on calibrating the Heston-Nandi GARCH model using historical asset return data and am faced with a decision on whether to use the risk-neutral or physical measure for this purpose. I aim to utilize this model for option pricing, although the model does not inherently require risk-neutral pricing, but on the contrary. The physical measure model is described as follows:

$$ \log(S_{t}) = \log(S_{t-1}) + r - \lambda h_t + \sqrt{h_t} z_{t}, $$ $$ h_{t} = \omega + \beta h_{t-1} + \alpha \left(z_{t-1} - \gamma \sqrt{h_{t-1}}\right)^2 $$ where $S_t$ are stock prices, $r$ is risk free rate, $h_t$ variance and $z_t$ standard normal variable. The risk-neutral dynamics of the Heston-Nandi GARCH model are obtained when we replace $\lambda$ with $-1/2$ and applying a correction of the parameter $ \gamma^* $, defined as:

$$ \gamma^* = \lambda + \gamma + \frac{1}{2}, $$

Given this setup, should I use the risk-neutral or the physical measure for calibration? What are the implications of choosing one over the other in terms of model accuracy and applicability? If you have any literature suggestions, please share them.

## Answer by user76968 (score 0)

https://quant.stackexchange.com/a/80862

We usually estimate the parameters of the physical measure with return data and calibrate the parameters of the risk neutral measure using options data. Your question is not clear

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