How Heston and GARCH Serve Different Volatility Modeling Tasks
Summary
The document contrasts Heston stochastic volatility and GARCH through their typical uses. Heston is a continuous-time model with a characteristic function that supports option pricing. This makes it practical to calibrate risk-neutral parameters to vanilla options across strikes and maturities, then use them in pricing other derivatives. Estimating its physical-measure parameters from historical data is harder because volatility is latent and must be inferred.
GARCH uses a discrete-time variance recursion that supports likelihood estimation from historical returns. Its parameters therefore describe volatility under the physical measure, making it useful for time-series analysis and risk work. The answer presents GARCH as less suited to direct static option calibration because it does not provide the same convenient option-pricing framework. These are broad industry-use distinctions, not absolute rules: the response acknowledges that research can extend either model beyond these typical applications, and it does not establish that one model is universally more accurate.
Key ideas
- Heston's characteristic function enables option valuation and calibration to vanilla option prices.
- Heston calibration to historical returns is complicated by unobserved volatility.
- GARCH's variance recursion supports likelihood estimation using historical time series.
- GARCH estimates generally reflect the physical probability measure, while option calibration in Heston targets risk-neutral parameters.
- The models have different strengths, and their typical applications do not make either universally superior.
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Full text
# Difference between GARCH and Heston Volatility model # Difference between GARCH and Heston Volatility model I know that the difference between the GARCH and the Heston model is volatility vs variance in the stochastic part of the volatility sde. However,from my solutions, there is only ever a 2 - 10 cent difference at most in most evaluations of the different models. Therefore why is heston more commonly used than GARCH and what makes one model better than the other? ## Answer by Kiwiakos (score 17, accepted) https://quant.stackexchange.com/a/16965 Heston gives an expression for the characteristic function, from which option prices can be computed. Therefore it can be calibrated (statically) on a set of vanilla option prices with different strikes and maturities. Hence this produces risk neutral parameters that can be used to price other more exotic products. However, it is a pain to estimate the physical measure parameters using time series of the underlying. Hence not popular for econometric applications, used e.g. for risk analytics. This is because volatility is latent (not directly observable) and has to be filtered out. Also it is set in continuous time. GARCH is roughly the opposite. It gives a nice variance recursion that facilitates maximum likelihood estimation, based on historical time series. It captures historical volatility fluctuations nicely. Hence estimates are under the actual (physical) probability measure, and this makes it popular for risk applications. Not useful option pricing formulas, as it is set in discrete time therefore no-arbitrage arguments not applicable. Therefore not really used for calibration to options in a static way, like Heston. This is a high level view that covers the industry practice. For every sentence above, one can raise her hand and shout "there is this and that paper that do it".
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