Stochastic Volatility Modelling¶
While ARIMA models describe the mean development of a time series, stochastic volatility models focus on the evolution of variance (risk) over time. This is particularly crucial for financial indices like the VIX Index or daily returns of the NYSE Composite Index.
Properties of Volatility¶
- Volatility Clusters: High volatility tends to be followed by high volatility, and low by low.
- Continuity: Volatility usually evolves in a continuous manner without sudden jumps.
- Stationarity: It is often stationary, varying within fixed limits rather than diverging to infinity.
- Leverage Effect: Volatility reacts differently to price rises and drops; big price drops typically have a significantly greater impact on volatility than corresponding rises.
Changing Variance¶
- Heteroscedastic: Non-constant variance. In finance, this often follows a mixture of normal distributions, leading to heavy-tailed or outlier-prone (leptokurtic) probability distributions.
- Homoscedastic: Constant variance over time.
Conditional & Unconditional Variance¶
The key distinction in volatility modelling is between these two types of variance:
Unconditional Variance:
The overall variance of the series without considering time-specific information.
$\(Var(X) = E[X - E(X)]^{2}\)$
Conditional Variance:
The variance at a specific time \(t\), given a model and an information set \(\Omega\) (past observations).
$\(Cond\ Var(X) = E[X - E(X|\Omega)]^{2}\)$
Stylized Facts of Asset Returns¶
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Stylized facts are empirical findings that are so consistently observed across different markets and instruments that they are accepted as truths, even if they cannot be formally "proven."
| Stylized Fact | Description |
|---|---|
| Thick Tails | Returns are leptokurtic, meaning extreme outcomes (outliers) happen more often than a normal distribution would predict. |
| Leverage Effects | Changes in stock prices are negatively correlated with changes in volatility; when prices fall, volatility usually spikes. |
| Non-trading Period Effects | Information accumulates at different rates when markets are closed (e.g., weekends vs. weekdays). |
| Forecastable Events | Volatility spikes predictably during market openings or scheduled economic announcements. |
| Volatility and Serial Correlation | There is a suggested inverse relationship between volatility and the degree of serial correlation in returns. |
| Co-movements in Volatility | High volatility is positively correlated across assets of the same class (e.g., if one tech stock is volatile, others likely are too). |
Types of Volatility¶
- Historical Volatility: A measure calculated using past realized data.
- Implied Volatility: Derived from option pricing models (like Black-Scholes). It represents the market's expectations of future volatility.
- Volatility Clustering: The tendency of large changes to follow large changes. This is the primary motivation for ARCH and GARCH models.
- Realized Volatility: The actual volatility observed over a past period, often estimated using high-frequency (intra-day) data.