A Haar-Fisz Technique for Locally Stationary Volatility Estimation

We propose a locally stationary model for financial log-returns whereby the returns are independent and the volatility is a piecewise constant function with an unknown number and location of jumps, defined on a compact interval to enable a meaningful estimation theory. We demonstrate that our model explains well the common stylised facts of log-returns. We propose a new wavelet thresholding algorithm for volatility estimation in our model, where Haar wavelets are combined with the variance-stabilizing Fisz transform. The resulting volatility estimator is mean-square consistent with a near-parametric rate, does not require any pre-estimates, is rapidly computable and easy to implement. We show that our modelling and estimation approach both gives an excellent fit to selected currency exchange datasets, and achieves accurate long- and short-term volatility forecasts in comparison to the GARCH(1,1) and moving window techniques.