How GARCH and Heston Volatility Dynamics Relate to Return Moments
Summary
The document compares a discrete GARCH(1,1) return model with a continuous-time stochastic-volatility model identified as Heston. It describes GARCH returns as driven by independent normal innovations, with variance evolving from a constant term, a lagged shock, and prior variance. It then presents a continuous-time variance process and says it resembles a limit of GARCH, while noting a difference in how volatility enters the stochastic term.
The central issue is whether this difference helps explain the models’ behavior for skewness and kurtosis. The document states that higher return moments are nontrivial in discrete GARCH, whereas an infinitesimal return in the continuous model has zero third and higher moments because the process has finite quadratic variation. It poses, rather than resolves, the question of why the discrete model can capture kurtosis more realistically. The comparison is therefore a conceptual prompt; it does not provide a derivation or empirical evidence, and its moment claims concern infinitesimal continuous-time increments rather than finite-horizon returns.
Key ideas
- GARCH models discrete returns with variance depending on a lagged shock and prior variance.
- The document presents Heston as a continuous-time stochastic-volatility model related to a limit of GARCH.
- It states that discrete GARCH can produce nontrivial higher return moments.
- It attributes zero third and higher moments in infinitesimal continuous-time returns to finite quadratic variation.
- The document raises the difference in kurtosis behavior as an open conceptual question rather than supplying a resolution.
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# Skewness and Kurtosis in GARCH vs Heston
# Skewness and Kurtosis in GARCH vs Heston
### GARCH(1,1)
In discrete time, we can model returns as follows \begin{align} r_t &= \mu + \sigma_t\epsilon_t\\ \sigma_t^2 &= \omega + \alpha \epsilon_{t-1}^2 + \beta\sigma_{t-1}^2 \end{align}
- Returns are driven by white noise ($\epsilon_t\overset{iid}{\sim} N(0,1)$)
- The GARCH process converges to $d\sigma_t^2=\kappa(\theta-\sigma_t^2)dt+\xi\sigma_t^2dW$ (Nelson, 1990)
### Heston
In continuous time, we can model returns as follows \begin{align} \frac{dS}{S} &= \mu dt+\sigma_tdW_t \\ \sigma_t^2 &= \kappa(\theta-\sigma_t^2)dt+\xi\sigma_tdW_t^2 \end{align}
- Returns are driven by white noise ($dW_t\sim N(0,dt)$)
- This is pretty much the limit of the GARCH process with the exception of $\sigma_t$ instead of $\sigma_t^2$ in the $dW_t^2$ part
### Skewness and Kurtosis
- In the discrete GARCH model, higher moments of returns are non-trivial, see this answer by @RichardHardy
- In the continuous model, higher moments of returns are zero: $\mathbb{E}\left(\left(\frac{dS}{S}\right)^3\right)=0$. Essentially, the Ito process $dS$ has finite quadratic variation and therefore all higher moments are zero.
Question: Why does the discrete time volatility model allow us to realistically capture kurtosis whereas the time continuous model does not?
In quant finance, discrete time and continuous time models often fit well (e.g. random walk $\to$ Brownian motion, binomial tree $\to$ Black-Scholes). Why do these models differ? I understand that a GARCH model has predictable volatility whereas the Heston model has true stochastic volatility, but I'm not sure that's the explanation for the zero kurtosis in the time continuous model?
Note: the same ideas (finite quadratic variation $\Rightarrow$ zero skewness and kurtosis) apply if we use GARCH to model log-returns and consider $d\ln(S)$ instead of $\frac{dS}{S}$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.