第三章 条件异方差模型

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3.3 MODEL BUILDING
• Building a volatility model for an asset return series consists of four steps:
• Specify a mean equation by testing for serial dependence in the data and building an econometric model for the return series to remove any linear dependence. • 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.
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• Volatility evolves over time in a continuous manner—that is, volatility jumps are rare. • Volatility does not diverge to infinity—that is, volatility varies within some fixed range. • Volatility seems to react differently to a big price increase or a big price drop, referred to as the leverage effect.
• The SV model of Melino and Turnbull(1990),
Taylor(1994), Harvey,Ruiz,and Shephard(1994), and Jacquier, Polson, and Rossi(1994).
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3.1 CHARACTERISTICS OF VOLATILITY
for rt where p and q are non-negative integers.
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Combining the above equations, we have
The conditional heteroscedastic models of this chapter are concerned with the evolution of σ2t. The manner under which σ2t evolves over time distinguishes one volatility model from another.
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Volatility is an important factor in financial applications
• Option (derivative) pricing
• Risk management, e.g. value at risk (VaR)
• Asset allocation • Interval forecasts
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Var( rt | Ft 1 )
2 t
E[(rt t ) | Ft 1 ]
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E[ a | Ft 1 ]
2 t
E[ at2 | Ft 1 ] {E[ at | Ft 1 ]}2 Var( at | Ft 1 )
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Conditional heteroscedastic models can be classified into two general categories: • Use an exact function to govern the evolution of σ2t. For example the GARCH model. • Use a stochastic equation to describe σ2t. For example the stochastic volatility model.
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Although volatility is not directly observable, it has some characteristics that are commonly seen in asset returns.
• There exist volatility clusters (i.e., volatility may be high for certain time periods and low for other periods).
• In option markets, if one accepts that the prices are governed by an econometric model such as the Black-Scholes formula, then one can use the price to obtain the “implied” volatility. Yet this approach is often criticized for using a special model, which is based on some assumptions that might not hold in practice.
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To put the volatility models in proper perspective, it is informative to consider the conditional mean and variance of rt given Ft-1; that is, • μt=E(rt|Ft-1)
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3.4.1 Properties of ARCH Model
• Consider an ARCH(1) model
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Properties:
• E(at) = 0 • Var(at) = α0/(1 - α1) if 0 < α1 < 1 • Under normality,
• provided 0 < α21 < 1/3 • The 3rd property implies heavy tails.
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AN EXAMPLE
• Black-Scholes European call option:
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The univariate volatility models discussed in this chapter
• The ARCH model of Engle(1982)
• The GARCH model of Bollerslev(1986) • The EGARCH model of Nelson(1991) • The CHARMA model of Tsay(1987) • The RCA model of Nicholls and Quinn(1982)
Chapter 3
Conditional Heteroscedastic Models 条件异方差模型
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The objective of this chapter
• To study some statistical methods and econometric models available in the literature for modeling the volatility of an asset return. • The models are referred to as conditional heteroscedastic models.
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• Under the ARCH framework, large shocks tend to be followed by another large shock. • The ARCH effect also occurs in other financial time series.
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• The null hypothesis is H0 α1=α2=…=αm=0
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3.4 THE ARCH MODEL
• The basic idea of ARCH models is that:
• the shock at of an asset return is serially uncorrelated, but dependent. • the dependence of at can be described by a simple quadratic function of its lagged values.
• σ2t=Var(rt|Ft-1)=E[(rt-μt)2|Ft-1]
Where Ft-1 denotes the information set available at time t-1.
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We assume that rt follows a simple time series model, we entertain the model:
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ARCH MODEL
• Where {εt} is a sequence of independent and identically distributed random variables with mean zero and variance 1, α0>0, α i>o. • In practice, εt is often assumed to follow the standard normal or a standardized Studentdistribution or a generalized error distribution.
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3.3.1 Testing for ARCH Effect
• Let at be the residuals of the mean equation. • The squared series a2t is then used to check for the ARCH effects.
• Two tests are available:
• Apply the usual Ljung-box statistics Q (m) to the {a2t} series. • The Lagrange multiplier test of Engle(1982).
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The Lagrange multiplier test
• This test is equivalent to the usual F statistic for testing α=0 in the linear regression:
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3.2 STRUCTURE OF A MODEL
• The basic idea behind volatility study is that
the series {rt} is either serially uncorrelated or with minor lower order serial correlations, but it is a dependent series.
• Question: What is stock volatility?
• Answer: conditional standard deviation of stock returns
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A key characteristic: Not directly observable!
• Consider the daily log returns of IBM stock.