2 = 3[α2 0 + 2α0 α1 Var (at ) + α1 m4 ] α1 ) + 3α2 = 3α2 1 m4 0 (1 + 2 1 − α1
Consequently, m4 =
the conditional variance of a return.
Chapter 3 Conditional Heteroscedastic Models
(3)
Two general categories of conditional heteroscedastic models:
- exact function - stochastic equation
Chapter 3 Conditional Heteroscedastic Models
November 30, 2014
Introduction
The objective of this chapter is to study some econometric methods for modeling the volatility of an asset return. Volatility is an important factor in options trading.
Chapter 3 Conditional Heteroscedastic Models
3.3 Model Building
Building a volatility model consists of four steps:
1
Specify a mean equation for the return series to remove any linear dependence. (e.g., removing the sample mean, or an ARMA model) Use the residuals of the mean equation to test for ARCH effects. Specify a volatility model if ARCH effects are statistically significant, and perform a joint estimation of the mean and volatility equations. Check the fitted model carefully and refine it if necessary.
Therefore,
2 E (at4 ) = E [E (at4 |Ft −1 )] = 3E (α0 + α1 at2 −1 ) 2 2 4 = 3E (α2 0 + 2α0 α1 at −1 + α1 at −1 )
If at is fourth-order stationary with m4 = E (at4 ), then we have m4
p q
rt = µt + at , µt = φ0 +
i =1
φi rt −i −
i =1
θi at −i ,
(2)
Volatility models are concerned with time-evolution of
σ2 t = Var (rt |Ft −1 ) = Var (at |Ft −1 )
m model implication:large past squared shocks {at2 −i }i =1 imply a large conditional variance σ2 t for the innovation at
— large shocks tend to be followed by another large shock.
Chapter 3 Conditional Heteroscedastic Models
3.4.1 Properties of ARCH Models
ARCH(1) model:
2 at = σt t , σ2 t = α0 + α1 at −1
where α0 > 0 and α1 ≥ 0. E (at ) = E [E (at |Ft −1 )] = E [σt E ( t )] = 0 Since Var (at ) = E (at2 ) = E [E (at2 |Ft −1 )]
- see the ACFs of {rt }, {rt2 } and {|rt |}
Volatility models attempt to capture such dependence in the return series.
Chapter 3 Conditional Heteroscedastic Models
Chapter 3 Conditional Heteroscedastic Models
3.1 Characteristics of Volatility
not directly observable volatility clusters (i.e., volatility may be high for certain time periods and low for other periods) volatility evolves over time in a continuous manner — volatility jumps are rare. volatility does not diverge to infinity (i.e. volatility varies within some fixed range). — volatility is often stationary. volatility react differently to a big price increase or a big price drop — leverage effect.
- Black-Scholes option pricing formula
Volatility has many other financial applications:
- calculating value at risk in risk management; - asset allocation under the mean-variance framework; - improving the efficiency in parameter estimation and the accuracy in interval forecast.
Chapter 3 Conditional Heteroscedastic Models
Tail behavior: Under the normality assumption of
t
in |Ft −1 ) = 3[E (at2 |Ft −1 )]2 = 3(α0 + α1 at2 −1 )
Consider the conditional mean and variance of rt given Ft −1 ; that is,
2 µt = E (rt |Ft −1 ), σ2 t = Var (rt |Ft −1 ) = E [(rt − µt ) |Ft −1 ] (1)
where Ft −1 denotes the information set available at time t − 1. the equation for µt in (1) should be simple, and we assume that rt follows a simple time series model such as a stationary ARMA (p , q) model.
2 2 σ2 t = α0 + α1 at −1 + · · · + αm at −m
(4)
{ t } is i.i.d. with mean 0 and variance 1 α0 > 0, and αi ≥ 0 for i > 0.
distribution of t : standard normal, or a standardized Student-t , or a generalized error distribution.
Some notations:
- at is the shock or innovation at time t . - σt is the positive square root of σ2 t. - µt in Eq.(2) is the mean equation. - the model for σ2 t is the volatility equation.
- {rt } is either serially uncorrelated or with minor lower order serial correlations; - but, it is a dependent series.
e.g. Monthly log stock returns of Intel Corporation.
Chapter 3 Conditional Heteroscedastic Models
3.2 Structure of a Model
Let rt be the log return of an asset at time index t . The basic idea behind volatility study is:
Chapter 3 Conditional Heteroscedastic Models
3.4 The ARCH Model
Basic idea: the dependence of at can be described by a simple quadratic function of its lagged values. An ARCH(m) model assumes that at = σt t ,