ARCH Models for Financial Applications

To my husband and my son, a wonderful family Evdokia Xekalaki To the memory of the most important person in my life, my father Antonis, and to my mother and my brother Stavros Degiannakis

[...]... returns that conform to an ARCH process are discussed Chapter 9 introduces the notion of implied volatility and discusses implied volatility indices and their use in ARCH modelling It also discusses techniques for forecasting implied volatility Chapter 10 deals with evaluation and selection of ARCH models for forecasting applications The topics of consistent ranking and of proxy measures for the actual... Higher-order quantities, such as the correlation, absolute correlation and so forth, are much more important tools in the analysis of stochastic process than their paths ARCH Models for Financial Applications Evdokia Xekalaki and Stavros Degiannakis © 2010 John Wiley & Sons Ltd ISBN: 978-0-470-06630-0 2 ARCH MODELS FOR FINANCIAL APPLICATIONS 1140 1120 1100 1080 1060 1040 1020 1000 980 1000 Figure 1.1 N... ARFIMAX models are presented Chapter 8 illustrates applications of volatility forecasting in risk management and options pricing Step-by-step empirical applications provide an insight into obtaining value-at-risk estimates and expected shortfall forecasts An options trading game driven by volatility forecasts produced by various methods of ARCH model selection is illustrated, and option pricing models for. .. the daily e returns are normally distributed 8 ARCH MODELS FOR FINANCIAL APPLICATIONS 1.2 The autoregressive conditionally heteroscedastic process Autoregressive conditional heteroscedasticity (ARCH) models have been widely used in financial time series analysis and particularly in analysing the risk of holding an asset, evaluating the price of an option, forecasting time-varying confidence intervals and... introd duced by Granger (1980) and Granger and Joyeux (1980) There is a plethora of formulations of the conditional mean in the literature For example, Sarkar (2000) illustrated the ARCH model for a regression model in which the dependent variable is Box–Cox transformed He referred to this as the Box–Cox transformed ARCH, or BCARCH, model: & À l Á À1 yt À1 l ; l 6¼ 0; ðBC Þ yt ¼ l ¼ 0; logðyt Þ; À 0 Á ðBC... j s2 ¼ ðrft ; Þð1; Þ0 t 18 ARCH MODELS FOR FINANCIAL APPLICATIONS 1.6.2 Volatility and serial correlation LeBaron (1992) noted a strong inverse relationship between volatility and serial correlation for the returns of the S&P500 index, the CRSP value-weighted market index, the Dow Jones and the IBM returns He introduced the exponential autoregressive GARCH, or EXP-GARCH( p,q), model in which the... conditional realized variance at time t þ 1 based on information available at time t Point in time (i.e days) for out-of-sample forecasting Also days to maturity for options Vector of predetermined variables included in It Functional form of conditional variance in conditional mean in GARCH-M model Confluent hypergeometric function Polynomial of FIGARCH Characteristic function of stable Paretian distribution... Þ ð tÞ Ctðià ;iÞ  ð tÞ Cðià ;iÞ Mean squared error loss function for model i volatility forecasts t days ahead,  2 P 2ðtÞ 2ðtÞ  ðtÞ ~ À1 i.e CðSEÞðiÞ ¼ T ~ t¼1T stþ1jtðiÞ Àstþ1ðiÞ Squared error loss function for models i volatility forecasts t days ahead Average of a loss function for t-days-ahead volatility ~  ðtÞ ~ À1 PT CðtÞ forecasts, i.e C ¼ T t¼1 t Loss function that measures the distance... non-linear models a Applications of SETAR and ARFIMA models can be found in Peel and Speight (1996) and Barkoulas et al (2000), respectively 9 They proposed the estimation of the conditional mean and conditional variance parameters of the SEMIFAR ARCH model separately 10 For example, a two-step estimator of conditional variance with ordinary least squares is a consistent but inefficient estimator 14 ARCH MODELS. .. of one length when in fact they were recorded at time intervals of another, 16 ARCH MODELS FOR FINANCIAL APPLICATIONS not necessarily regular, length is an important factor affecting the return series, an effect known as the non-synchronous trading effect For details, see Campbell et al (1997, p 84) and Tsay (2002, p 176) For example, the daily prices of securities usually analysed are the closing prices