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