-----BEGIN PGP SIGNED MESSAGE----- Hash: SHA1 Hello,
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GARCH(1,1)
Generalized Autoregressive Conditional Heteroskedasticity.
The asset and its volatility evolve as follows:
$$\Delta S_t = S_t(\mu\Delta t + \sigma_{t} \sqrt{\Delta t} \varepsilon_t)$$ $$\sigma^2_t = (1 - \alpha -\beta) \sigma^{2} + \sigma^{2}_{t-1}(\beta + \alpha\varepsilon^{2}_{t-1})$$ $$\varepsilon_{t} \sim N(0,1)$$
Autoregressive Term: $\alpha\sigma^{2}_{t-1}\varepsilon^{2}_{t-1}$. This term makes the conditional variance at time $t$ a function of the actual innovation (change) of the previous time period $t-1$. This produces the effect that periods of big moves or small moves are clustered together (volatility clustering). This can lead to a high kurtosis or fat-tailed distribution [Chatterjee 2012].
Generalized Term: $\beta\sigma^{2}_{t-1}$. This produces a richer autocorrelation term structure of squared returns.
Reference:
[Chatterjee 2012] - Rupak Chatterjee - Introduction to Financial Engineering and Risk Management - 2012.
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