Background
In ARMA models, we only considered time series that are stationary and driven by white noise processes. We only modeled the mean of given the past values of the time series and did not consider the conditional variability. For better modeling of financial time series, we need to also consider the modeling of conditional variability.
Some of the characteristics of volatility that have been noted in stock price behavior:
- Volatility clustering: most real-life financial time series exhibit non-constant conditional variance. Such non-constant conditional variance also clusters into periods of low and high volatility.
- Volatility is continuous, not a jump process.
- Volatility does not diverge to infinity; it is approximately stationary
- Volatility increases more following big price decreases than after big price increases (the leverage effect)
ARCH model
In a non-parametric regression model, the conditional variability is introduced as
with and .
Note that ARCH is a more structured version of a white noise process.
Let be a martingale difference sequence (MDS), i.e. . Assume . Define process
where
This is called the autoregressive conditional heteroscedastic (ARCH) model of order .
The is a martingale difference sequence, because is only a function of , and we get
This implies that is a martingale difference sequence, which implies that
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