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.