L46 Stochastic Volatility Modelling

ARIMA models: describe the mean development of time series
Example
- VIX Index
- Daily returns of NYSE Composite Index
Properties
- Volatility clusters exist (higher and low periods)
- Volatility evolves over time in a continuous manner (no jumps)
- Volatility is often stationary and doesn't diverge to infinity (varies within fixed limits)
- Big price drops have a greater impact on volatility. Leverage effect (volatility reacts differently to rise and drops)
Changing variance
- Heteroscedastic \(\implies\) non-constant variance, follows a mixture of normal distribution \(\implies\) it will follow a heavy-tailed or outlier-prone (more outliers) probability distribution
- Homoscedastic

Key distinction is between conditional and unconditional variance

\[ Var(X) = E[X - E(X)]^{2} \]

\[ Cond\ Var(X) = E[X - E(X|\Omega)]^{2} \]

Facts that can be evidently observed everywhere but cannot be proved. "This is how it is"

Stylized Fact	What?
Thick Tails	leptokurtic
Leverage Effects	change in stock prices to be negatively correlated with change in volatility
Non-trading period effects	when market closes, information seems to accumulate at different rate (Information accumulates differently over the weekends)
Forecastable events	When the market opens for a day (the asset price is extremely volatile), or when an announcement happens.
Volatility and serial correlation	There is a suggestion of an inverse relationship between the two
Co-movements in volatility	High volatility is positively correlated across assets of the same class.

Historical Volatility
- Measure of volatility calculated using the past data
Implied Volatility
- derived from option pricing models (Black-Scholes)
- market's expectations to capture future volatility
Volatility clustering
- periods of high volatility followed by periods of low volatility
- ARCH and GARCH
Realized volatility
- actual volatility observed over a past period data
- Can be estimated using high-frequency data