Fractional Stochastic Processes: A New Application to Volatility Estimation

Abstract

It is commonly accepted that some financial data may exhibit long-range dependence, while other financial data exhibit intermediate--range or short--range dependence. These behaviours may be fitted to a continuous--time fractional stochastic model. The estimation procedure proposed in this seminar is based on a continuous--time version of the Gauss--Whittle objective function to find the parameter estimates that minimise the discrepancy between the spectral density and the data periodogram. As a special case, the proposed estimation procedure is applied to a class of fractional stochastic volatility models. The estimation of the volatility process is one of the most difficult problems in econometrics. The authors propose a technique to estimate the drift, standard deviation and memory properties of the volatility process from a transformation of the returns. As an application, the volatility of the Dow Jones, S\&P 500, CAC 40, DAX 30, FTSE 100 and NIKKEI 225 is estimated.