Financial Time Series

Abstract

Volatility measures the volatility of the price of a financial asset. It is a parameter quantifying the risk of performance and price. The volatility is also used for calculations to optimize portfolio diversification financial assets and assessment of financial derivative contracts such as options. The monetary and financial series are characterized by the clustering of volatility, i.e. periods of high volatility alternate with periods of low volatility. This phenomenon, which we also call the conditional heteroscedasticity, (ARCH) and generalized autoregressive conditional heteroskedasticity (GARCH) models left- particularly common in stock market data, exchange rates or other price determined in financial market. These are widely used in the area of ??Finance, as an investor is interested in forecasting the rate of return and volatility only over the holding period, and is the issuer who is interested in analyzing the expected performance and volatility over life of the financial instrument. During our research we discloser that along with idiosyncratic volatilities moving from single assets to portfolios made of multiple assets, correlations and covariance between assets are also predicable and time varying.

Financial Time Series (ARCH AND GARCH models)

Introduction

Volatility models are used by majority of the macro-econometric and financial time series while traders use trading options as a tool in risk management. The gap representations in ARMA model for monetary and financial problems, Engle (1982) proposes a new class of autoregressive condi- traditionally heteroscedastic (ARCH) and or generalized autoregressive conditional heteroskedasticity (GARCH) adapted to capture the behavior of the volatility over time. The model is formed of two equations. The first equation explains the features relationship and a performance variable while second model the conditional variance of the residuals.

The principle proposed by Engle is to introduce a dynamic in the volatility that determined by assuming that the variance is conditional upon information available to us. He advances a specification ARCH (p) where the squared innovations, that is to say, the variance of the error term in time t, depends on the magnitude of the error terms to the square of p past periods. The model ARCH (p) generates episodes of high volatility followed episodes of lower volatility (Engle, Focardi, Fabozzzi, 2007).

The volatility model

The idea behind this concept is the series { } which defined as

Where:

S is stock price: It is dependent though it comprises of serially uncorrelated or with lower order serial correlation. In order to give a clear picture, we can regard as the conditional variance and mean of given; i.e.

Here, : It represents the set of information available at time t - 1. Usually, is considered as a past returns linear functions.

has been assumed as a simple time series model like a stationary ARMA (p, q) model with some explanatory variables.

(2)

In the above equation, k, p, and q are non-negative integers in , while in are explanatory variables and they are flexible. Moreover, equation (2) () has been termed as the mean equation.

As we join (1) and (2), a new equation will be formed which ...