Limit theorems for nearly unstable Hawkes processes
Because of their tractability and their natural interpretations in term of market quantities, Hawkes processes are nowadays widely used in high-frequency finance. However, in practice, the statistical estimation results seem to show that very often, only nearly unstable Hawkes processes are able to fit the data properly. By nearly unstable, we mean that the $L^1$ norm of their kernel is close to unity. We study in this work such processes for which the stability condition is almost violated. Our main result states that after suitable rescaling, they asymptotically behave like integrated Cox-Ingersoll-Ross models. Thus, modeling financial order flows as nearly unstable Hawkes processes may be a good way to reproduce both their high and low frequency stylized facts. We then extend this result to the Hawkes-based price model introduced by Bacry et al. [Quant. Finance 13 (2013) 65-77]. We show that under a similar criticality condition, this process converges to a Heston model. Again, we recover well-known stylized facts of prices, both at the microstructure level and at the macroscopic scale.
|Date of creation:||Oct 2013|
|Date of revision:||Mar 2015|
|Publication status:||Published in Annals of Applied Probability 2015, Vol. 25, No. 2, 600-631|
|Contact details of provider:|| Web page: http://arxiv.org/|
References listed on IDEAS
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- Bowsher, Clive G., 2007.
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- Clive G. Bowsher, 2003. "Modelling Security Market Events in Continuous Time: Intensity Based, Multivariate Point Process Models," Economics Papers 2003-W03, Economics Group, Nuffield College, University of Oxford.
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- John C. Cox & Jonathan E. Ingersoll Jr. & Stephen A. Ross, 2005. "A Theory Of The Term Structure Of Interest Rates," World Scientific Book Chapters, in: Theory Of Valuation, chapter 5, pages 129-164 World Scientific Publishing Co. Pte. Ltd..
- Cox, John C & Ingersoll, Jonathan E, Jr & Ross, Stephen A, 1985. "A Theory of the Term Structure of Interest Rates," Econometrica, Econometric Society, vol. 53(2), pages 385-407, March.
- E. Bacry & S. Delattre & M. Hoffmann & J. F. Muzy, 2013. "Modelling microstructure noise with mutually exciting point processes," Quantitative Finance, Taylor & Francis Journals, vol. 13(1), pages 65-77, January.
- E. Bacry & S. Delattre & M. Hoffmann & J. F. Muzy, 2011. "Modeling microstructure noise with mutually exciting point processes," Papers 1101.3422, arXiv.org.
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