It has been widely reported that volatility of many financial and macroeconomic time series is highly persistent.  Hence there has been a lot of research to come up with models that model this persistent volatility.  Proper estimation of volatility is relevant both to the calculation of value at risk of a portfolio and to the valuation of derivatives.  The objective of this effort is to estimate these persistent volatilities using the following three highly used mathematical models.

1.      Exponentially Weighted Moving Average (EWMA) model

2.      Generalized Auto-Regressive Conditional Heteroscedasticity (GARCH) model

3.      Fractionally Integrated Generalized Auto-Regressive Conditional Heteroscedasticity (FIGARCH) model

EWMA and GARCH models assume that the weights given to the historical volatilities should decrease exponentially (hence a shorter memory) whereas the FIGARCH models assume that the weights should decrease hyperbolically (longer memory) rather than exponentially.  The main characterization of a FIGARCH model is that conditional variances exhibit not only short-run dynamics as in GARCH model, but also the long-run persistence that decays slowly at hyperbolic rates.

 For this project the EWMA, GARCH (1, 1) and FIGARCH (1, d, 1) models have been compared in terms of how well they are able to explain the autocorrelation structure of the financial data.  All three models are applied to two sets of data.  The first set of data is the Japanese yen to US Dollar exchange rate between August 1996 and August 2006.  The second set of data is the S&P 500 index between August 1996 and August 2006.