ARCH and GARCH type models used to estimate volatility are also nonlinear models expressed as a function (linear or not) of past variations in stocks.
ARCH-GARCH models and more recently the range process have generated an extensive amount of research and papers. Just as chaos the Hurst, exponent and memory modeling have been topics of interest in many areas outside finance and economics. ARCH and GARCH models, which are important for modeling and estimating volatility, are an important part of modern finance. Since the value of an option depends essentially on its volatility and volatility studies are assuming an important role in financial modeling.
A primary feature of the autoregressive Conditional heteroscedasticity (ARCH) model as developed by Engle (1982), is that the conditional variances change over time. Following the seminal idea, numerous models incorporating this feature have been proposed. Among these models, Bollerslev's (1986) generalized ARCH (GARCH) model is certainly the most popular and successful because it is easy to estimate and interpret by analogy with the autoregressive moving average (ARMA) time series model. ...
prominent role in the analysis of many aspects of financial econometrics such as the term structure of interest rates, the pricing of options, the presence of time varying risk premia in the foreign exchange market. The quintessence of the ARCH model is to make volatility depend on variability of past observations. An alternative formulation initiated by Taylor (1986) makes volatility be driven by unobserved component, and has come to be known as the stochastic volatility (SV) model. AS for the ARCH models SV models have also been intensively used in the last decade, especially after the progress accomplished in the corresponding estimation techniques, as illustrated in the excellent surveys of Ghysels ET al (1996) and Shepard (1996). Early contributions that aimed at relating changes in volatility of asset returns to economic intuition include Clark (1973) who assumed that a stochastic process of information arrival generates a random number of intraday changes of the asset price.
The Black-Scholes model for instance assumes that the price of the asset underlying the option contract follows a geometric Brownian motion and one of the most successful extensions has been the continuous time SV model. In these models, volatility is not a constant as in the original Black-Scholes model; rather, it is another random process typically driven by a Brownian motion that is imperfectly correlated with the Brownian motion driving the primitive asset price dynamics. In technical terms, the volatility process generated within arch type models converges in distribution towards a well-defined solution of a stochastic differential equation as the sampling frequency increases. One concomitant reason is that the continuous record asymptotics developed for the ARCH models do not deliver