Realized wavelet-based estimation of integrated variance and jumps in the presence of noise

Abstract

We introduce wavelet-based methodology for estimation of realized variance allowing its measurement in the time-frequency domain. Using smooth wavelets and maximum overlap discrete wavelet transform, we allow for the decomposition of the realized variance into several investment horizons and jumps. Basing our estimator in the two-scale realized variance framework, we are able to utilize all available data and obtain a feasible estimator in the presence of microstructure noise as well. The estimator is tested in a large numerical study of the finite sample performance and is compared to other popular realized variation estimators. We use different simulation settings with changing noise as well as jump level in different price processes including the long memory fractional stochastic volatility model. The results reveal that our wavelet-based estimator is able to estimate and forecast the realized measures with the greatest precision. Our time-frequency estimators not only produce feasible estimates, but also decompose the realized variation into arbitrarily chosen investment horizons. We apply them to study the volatility of forex futures during the recent crisis at several investment horizons and obtain results which provide us with better understanding of the volatility dynamics.

Keywords:

• Quadratic variation
• Realized variance
• Jumps
• Market microstructure noise
• Wavelets

AMS Subject Classifications:

• C14
• C53
• G17

Acknowledgements

We are indebted to Ionut Florescu, Karel Najzar, David Veredas, anonymous referees and seminar participants at the Modeling High Frequency Data in Finance 4 in New York (July 2012) and Computational and Financial Econometrics in London (December 2011) for many useful comments, suggestions and discussions.

Notes

No potential conflictof interest was reported by the authors.

1 Due to the nature of the MODWT filters, we need to correct the position of the wavelet coefficient to get the precise position of the jump. For more details see Percival and Mofjeld (1997).

2 We have also computed the results for lower number of simulations, up to 1000 generated independent sample paths and we found that the results do not change at all. These results are available upon request from authors.

3 We have also computed the results for lower number of simulations, up to 1000 generated independent sample paths and we found that the results do not change at all. These results are available upon request from authors.

4 http://www.tickdata.com/

5 It should be noted that any investment horizons of interest may be chosen arbitrarily.

Additional information

Funding

The support from the Czech Science Foundation under the [13-32263S] and [13-24313S] projects is gratefully acknowledged. The research leading to these results has received funding from the European Union’s Seventh Framework Programme [FP7/2007-2013] under grant agreement No. FP7-SSH- 612955 (FinMaP).