Long memory, volatility modelling and pricing of derivative securities

This dissertation studies a model of long memory in volatility and its effects on derivative prices.; The first part undertakes a rigorous study of the Fractionally Integrated Exponential GARCH model introduced by Bollerslev and Mikkelsen [1996]. This model incorporates a fractional differencing operator to induce long memory in volatility. Several important time-series properties are derived: unconditional moments of the returns, the auto-covariance function and the spectral density of the log-squared returns. The Conditional Sum of Squares Estimation is examined. The fractional differencing operator present in the volatility process has a binomial expansion with infinite terms, and needs to be truncated at some finite lag length in this estimation. Simulation results on estimation of ARFIMA and FIEGARCH models with truncation at various lag lengths are presented. The results show that estimation of FIEGARCH is sensitive to this, and the biasedness largely depends on the parameters. A spectral likelihood estimator (using the spectral density derived) is introduced as an alternative to circumvent this problem of truncation. Finally, applications of these results to nominal exchange rate appreciation series for five major currencies are presented.; The second part derives semi closed-form solution of European option prices when the asset returns have long memory in variance, and examines its effects on option prices across moneyness and maturity. To this end, a new long memory in variance model, the Fractionally Integrated Non-linear Asymmetric GARCH (FINGARCH), is introduced. Elliot-Madan Extended Girsanov Principle is used to derive the Equivalent Martingale process. The option prices are calculated using Fourier inversion of characteristic functions. Calibrating the model to Deutschemark foreign currency options, it is shown that the long memory option prices are uniformly higher than their short memory counterpart across maturity and moneyness.