# Heston model vs. GARCH

Heston model is a stochastic volatility extension of the Black-Scholes model. On the other hand, there is also closed-form expression for option pricing that uses GARCH stochastic volatility model. Which model to use for option pricing?

I guess there is no clear cut answer, but still, what are the criteria to use? It seems like GARCH is always better as it is a discrete time model.

• I think the following thread might answer your question (or at least be a helpful starting point): quant.stackexchange.com/questions/16951 Jan 20, 2021 at 7:38
• Thank you! But the post is not very helpful. I disagree with the best answer in that thread. Analytical formulas are available for both Heston model and GARCH model (both developed by Heston). I'm also not sure why the author of the best answer discussed physical probabilities: Heston model is by construction an option-pricing model. But the author seems to be discussing the square-root process. Jan 20, 2021 at 13:30
• I cannot talk for the GARCH SV model, however, in the original paper of the Heston model, he derives a semi-analytical solution for option pricing and not a closed-form solution (which by the way was corrected for a better well-behaved characteristic function), since you have to evaluate the probabilities numerically (I believe they are derived by the Fourier inverse transform). Now, when referring to the Heston model, one refers to the SDE which is defined under the physical measure. You can then do a change of measure to the risk-neutral measure and derive an option-pricing formula.
– Pleb
Jan 20, 2021 at 13:54
• Implying that the Heston model is not an option-pricing model per construction.
– Pleb
Jan 20, 2021 at 14:11
• The main difference between the two models is that Heston allows for controllable correlation between the return and volatility, thus allowing us to capture risk reversals in the data. Jan 21, 2021 at 5:50