Chapter 8 Modelling volatility and correlation
8-1 Chapter 8 Modelling volatility and correlation
8-2 1 An Excursion into Non-linearity Land Motivation: the linear structural (and time series ) models cannot explain a number of important features common to much financial data leptokurtosis:尖峰性,厚尾 volatility clustering or volatility pooling波动性集群 较卖 age effects与价格同幅上升相比,价格大幅下降后,波动性上升 levers Our "traditional"structural model could be something like: y,= Bi+Bx2+.+ Bkrk+u, or y=XB+u We also assumed u,N(0, 0
8-2 1 An Excursion into Non-linearity Land • Motivation: the linear structural (and time series) models cannot explain a number of important features common to much financial data - leptokurtosis：尖峰性，厚尾 - volatility clustering or volatility pooling 波动性集群 - leverage effects 与价格同幅上升相比，价格大幅下降后，波动性上升 较多 • Our “traditional” structural model could be something like: yt = 1 + 2x2t + ... + kxkt + ut, or y = X + u. We also assumed ut N(0, 2 )
A Sample financial asset 8-3 Returns Time series Daily s&p 500 returns for Januar y1990-De ecember 1999 Return 0.06 0.04 0.02 000 wHwwHANwH -0.02 -0.04 -0.06 -0.08 1/01/90 11/01/93 Date 9/0197
8-3 A Sample Financial Asset Returns Time Series Daily S&P 500 Returns for January 1990 – December 1999 -0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 1/01/90 11/01/93 9/01/97 Return Date
8-4 Non-linear models: a Definition Campbell, Lo and macKinlay(1997)define a non-linear data generating process as one that can be written y=f(up1,u12,…) where u, is an iid error term and f is a non-linear function. They also give a slightly more specific definition as y1=g(u1,u12,…)+uJ2(u1,u12,…) where g is a function of past error terms only and ol is variance term Models with nonlinear g( are"non-linear in mean", while those with nonlinear o() are"non-linear in variance Models can be linear in mean and variance(Clrm,arma) or linear in mean but non-linear in variance(GarCh)
8-4 Non-linear Models: A Definition • Campbell, Lo and MacKinlay (1997) define a non-linear data generating process as one that can be written yt = f(ut , ut-1 , ut-2 , …) where ut is an iid error term and f is a non-linear function. • They also give a slightly more specific definition as yt = g(ut-1 , ut-2 , …)+ ut 2 (ut-1 , ut-2 , …) where g is a function of past error terms only and 2 is a variance term. • Models with nonlinear g(•) are “non-linear in mean”, while those with nonlinear 2 (•) are “non-linear in variance”. • Models can be linear in mean and variance(CLRM,ARMA), or linear in mean but non-linear in variance(GARCH)
8-5 1.1 Types of non-linear models The linear paradigm is a useful one. Many apparently non linear relationships can be made linear by a suitable transformation On the other hand, it is likely that many relationships in finance are intrinsically non-linear. There are many types of non-linear models, e.g. ARCH/ GARCH for modelling and forecasting volatility switching models allow the behaviour of a series to follow different processes at different points in time bilinear models
8-5 1.1 Types of non-linear models • The linear paradigm is a useful one. Many apparently nonlinear relationships can be made linear by a suitable transformation. On the other hand, it is likely that many relationships in finance are intrinsically non-linear. • There are many types of non-linear models, e.g. - ARCH / GARCH for modelling and forecasting volatility - switching models : allow the behaviour of a series to follow different processes at different points in time. - bilinear models
8-6 1.2 Testing for Non-linearity The“ traditional” tools of time series analysis(acf’s, spectral analysis)may find no evidence that we could use a linear model, but the data may still not be independent. General test( Portmanteau多用途 tests) for non- - linear dependence have been developed. The simplest is Ramseys RESET test (chapter 4) Many other non-linearity tests are available the bds test(19:检验数据是否是纯随机的。 Eview4提供 The bispectrum test(Hinich, 1982) bicorrelation test(Hsieh, 1993; Hinich, 1996) One particular non-linear model that has proved very useful in finance is the arch model due to Engle 1982. Specific tests to find specific types of non-linear structure
8-6 1.2 Testing for Non-linearity • The “traditional” tools of time series analysis (acf’s, spectral analysis) may find no evidence that we could use a linear model, but the data may still not be independent. • General test (Portmanteau 多 用 途 tests) for non-linear dependence have been developed. The simplest is Ramsey’s RESET test (chapter 4). • Many other non-linearity tests are available. the BDS test(1996) :检验数据是否是纯随机的。Eview4提供 The bispectrum test (Hinich,1982) bicorrelation test (Hsieh, 1993; Hinich,1996) • One particular non-linear model that has proved very useful in finance is the ARCH model due to Engle (1982). • Specific tests : to find specific types of non-linear structure
8-7 2 Models for volatility ·建模和预测股票市场波动性已经成为过去十年中实证和理论研 究中的一个重要主题 波动性是金融中最重要的概念之一。通常用收益的标准差或方 差来衡量。波动性常常用于金融资产的总体风险的粗略测量 许多测量市场风险的vaR模型需要估计和预测波动性参数,在 Black-Scholes期权定价模型中也需要利用股票市场价格的波动 性 ·描述波动性典型特征的一些模型 Historical volatility历史的波动性:计算过去一段时期的收益方差,并用 于未来的波动性预测。可以作为其他方法的比较基准 benchmark plied volatility models在给定期权价格的条件下,可以计算出基础资 产收益波动性的市场预测值。 指数加权移动平均模型EWMA近期数据对波动性的预测有更大的影响 自回归波动性模型:对代表波动性的序列建立ARMA模型并用于预测
8-7 2 Models for volatility • 建模和预测股票市场波动性已经成为过去十年中实证和理论研 究中的一个重要主题. • 波动性是金融中最重要的概念之一。通常用收益的标准差或方 差来衡量。波动性常常用于金融资产的总体风险的粗略测量。 • 许多测量市场风险的VaR模型需要估计和预测波动性参数，在 Black-Scholes期权定价模型中也需要利用股票市场价格的波动 性。 • 描述波动性典型特征的一些模型 – Historical volatility 历史的波动性：计算过去一段时期的收益方差，并用 于未来的波动性预测。可以作为其他方法的比较基准benchmark – Implied volatility models:在给定期权价格的条件下，可以计算出基础资 产收益波动性的市场预测值。 – 指数加权移动平均模型 EWMA: 近期数据对波动性的预测有更大的影响 – 自回归波动性模型: 对代表波动性的序列建立ARMA模型并用于预测
8-8 Heteroscedasticity revisited An example of a structural model is V=B1+ A2x2t+ B3x3t+ B4x4t+ur with u, n(o, ox). The assumption that the variance of the errors is constant is known as homoscedasticity, i. e Var(u=o What if the variance of the errors is not constant heteroscedasticity would imply that standard error estimates could be wrong. Is the variance of the errors likely to be constant over time? not for financial data
8-8 Heteroscedasticity Revisited • An example of a structural model is with ut N(0, ). The assumption that the variance of the errors is constant is known as homoscedasticity, i.e. Var (ut ) = . • What if the variance of the errors is not constant? - heteroscedasticity - would imply that standard error estimates could be wrong. • Is the variance of the errors likely to be constant over time? Not for financial data. u 2 u 2 yt = 1 + 2 x 2t + 3 x 3t + 4 x 4t + u t
3 Autoregressive Conditionally 8-9 Heteroscedastic (ARCH) Models a model which does not assume that the variance is constant the definition of the conditional variance of u. ariu, u,i u t-1 )=El(u E(uD)2 u,p, ur2 We usually assume that e(up=0 SO var(u2|u1,2…)=E t1,“t2 What could the current value of the variance of the errors plausibly depend upon? Previous squared error terms. This leads to the autoregressive conditionally heteroscedastic model for the variance of the errors. This is known as an arCH(1) model
8-9 3 Autoregressive Conditionally Heteroscedastic (ARCH) Models • a model which does not assume that the variance is constant. • the definition of the conditional variance of ut : = Var(ut ut-1 , ut-2 ,...) = E[(ut -E(ut ))2 ut-1 , ut-2 ,...] We usually assume that E(ut ) = 0 so = Var(ut ut-1 , ut-2 ,...) = E[ut 2 ut-1 , ut-2 ,...]. • What could the current value of the variance of the errors plausibly depend upon? Previous squared error terms. • This leads to the autoregressive conditionally heteroscedastic model for the variance of the errors: = 0 + 1 • This is known as an ARCH(1) model. t 2 t 2 t 2 ut−1 2
8-10 ARCH Models(contd) One example of a full model would be D,=B1+B2x2t+.+Bkk +u, u,N(O, o) where ot We can easily extend this to the general case where the error variance depends on g lags of squared errors: o=a+a,u +au2+... +a This is an arCH(@ model. Instead of calling the variance ot, in the literature it is usually called h. so the model is D,=B1+ Bix2r+.+ BkXkt+ut, u N(o,h) where h,=0+a1 u-1+a2 u 22+. +aa u-q
8-10 ARCH Models (cont’d) • One example of a full model would be yt = 1 + 2x2t + ... + kxkt + ut , ut N(0, ) where = 0 + 1 • We can easily extend this to the general case where the error variance depends on q lags of squared errors: = 0 + 1 +2 +...+q • This is an ARCH(q) model. • Instead of calling the variance , in the literature it is usually called ht , so the model is yt = 1 + 2x2t + ... + kxkt + ut , ut N(0,ht ) where ht = 0 + 1 +2 +...+q t 2 t 2 t 2 ut−1 2 ut −q 2 ut −q 2 t 2 2 t −1 u 2 ut −2 2 t −1 u 2 t −2 u