Limit theorems for nearly unstable Hawkes processes

By Thibault Jaisson, Mathieu Rosenbaum. 2013

181 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/57ca5a736dbac6338bec0d5c0e689566dc019fdf

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Citations in the corpus

Pricing under rough volatility (2015, unknown): “Moreover, the RFSV model has a quantitative market microstructure-based foundation based on the modelling of order flow using Hawkes processes, see Gatheral et al. (2014, Section 6) and Jaisson and Rosenbaum (2015).”
Rough fractional diffusions as scaling limits of nearly unstable heavy tailed Hawkes processes (2015, methodology): “This is exactly the approach used in [26].”
Volatility is rough (2014, background): “ a metaorder whose full execution can take a large amount of time. We believe these two phenomena together lead to a superposition eect inducing this irregular volatility. Indeed, it is explained in [33,34] that the macroscopic scaling limit of Hawkes processes with power law tail and kernel with L1”
A weak solution theory for stochastic Volterra equations of convolution type (2019, methodology): “The following example provides a concrete specification for the special case of fractional powers, extending results in Jaisson and Rosenbaum (2015, 2016) to nonlinear Hawkes processes.”
Hawkes processes and their applications to finance: a review (2018, result): “This result is in contrast to the case of a regression kernel with light tail, for which Jaisson and Rosenbaum (2015) obtained a classical Brownian CIR process in the limit.”
Renewal in Hawkes processes with self-excitation and inhibition (2018, unknown): “, [2, 22, 23, 38]).”
The characteristic function of rough Heston models (2016, background): “…condition ( ∫ ∞ 0 𝑎𝑇 𝜙(𝑠)ds ) < 1, where denotes the spectral radius operator, should almost be saturated and that the intensity of exogenous orders, namely, 𝜇𝑇 , should be small; see El Euch et al. (2016), Hardiman, Bercot, and Bouchaud (2013), and Jaisson and Rosenbaum (2015, 2016).”
Hawkes Processes in Finance (2015, methodology): “For precise formulation of the corresponding theorem, we refer the reader to [42].”
No‐arbitrage implies power‐law market impact and rough volatility (2018, background): “atible with empirical studies showing that markets are highly endogenous. In the Hawkes process context, the degree of endogeneity of the market is measured by the L1 norm of ˚, denoted by k˚k 1, see [13, 17, 22, 21]. Therefore a highly endogenous market corresponds to the case where k˚k 1 is close ”
Hawkes model for price and trades high-frequency dynamics (2013, methodology): “We refer the reader to †Let us also mention that, after completion of this work, Jaisson and Rosembaum (2013) proved that the process converges towards a CoxIngersoll Process in the limit case when the Hawkes process is critical.”