Limit theorems for nearly unstable Hawkes processes
| Autor: | Mathieu Rosenbaum, Thibault Jaisson |
|---|---|
| Přispěvatelé: | Laboratoire de Probabilités et Modèles Aléatoires (LPMA), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS) |
| Jazyk: | angličtina |
| Rok vydání: | 2015 |
| Předmět: |
microstructure modeling
Statistics and Probability Point processes Heston model Point process 62P05 FOS: Economics and business 60F05 Cox–Ingersoll–Ross model FOS: Mathematics Statistical physics order flows ComputingMilieux_MISCELLANEOUS Mathematics Stylized fact Statistical Finance (q-fin.ST) Probability (math.PR) limit theorems Quantitative Finance - Statistical Finance Price model high-frequency data [MATH.MATH-PR]Mathematics [math]/Probability [math.PR] Criticality 60F17 Macroscopic scale Norm (mathematics) 60G55 Statistics Probability and Uncertainty Hawkes processes Mathematics - Probability |
| Zdroj: | Annals of Applied Probability Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2015, 25 (2), pp.600-631 Annals of Applied Probability, 2015, 25 (2), pp.600-631. ⟨10.1214/14-AAP1005⟩ Ann. Appl. Probab. 25, no. 2 (2015), 600-631 |
| ISSN: | 1050-5164 2168-8737 |
| DOI: | 10.1214/14-AAP1005⟩ |
| Popis: | 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. Published in at http://dx.doi.org/10.1214/14-AAP1005 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org) |
| Databáze: | OpenAIRE |
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