Rough fractional diffusions as scaling limits of nearly unstable heavy tailed Hawkes processes

By Thibault Jaisson, Mathieu Rosenbaum. 2015

145 citations recorded by the source corpus.

We investigate the asymptotic behavior as time goes to infinity of Hawkes processes whose regression kernel has $L^1$ norm close to one and power law tail of the form $x^{-(1+\alpha)}$, with $\alpha\in(0,1)$. We in particular prove that when $\alpha\in(1/2,1)$, after suitable rescaling, their law converges to that of a kind of integrated fractional Cox-Ingersoll-Ross process, with associated Hurst parameter $H=\alpha-1/2$. This result is in contrast to the case of a regression kernel with light tail, where a classical Brownian CIR process is obtained at the limit. Interestingly, it shows that persistence properties in the point process can lead to an irregular behavior of the limiting process. This theoretical result enables us to give an agent-based foundation to some recent findings about the rough nature of volatility in financial markets.

Subfields: Applied Mathematics, Finance, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/d7148378ac1604198be62ee7f6bc6a4b04329d96

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Finance

In Molecular Biology

Citations in the corpus

Volatility is rough (2014, unknown): “Indeed, it is explained in [33, 34] that the macroscopic scaling limit of Hawkes processes with power law tail and kernel with L1 norm close to one can be seen as an integrated fractional process, with Hurst parameter H smaller than 1/2.”
Generalized Feller processes and Markovian lifts of stochastic Volterra processes: the affine case (2018, background): “, [4, 21, 15, 16]), stochastic Volterra processes have been studied extensively, for instance in [24, 3, 17] and the references therein.”
Hawkes processes and their applications to finance: a review (2018, background): “Jaisson and Rosenbaum (2016). investigated the asymptotic behaviour as time goes to infinity of Hawkes processes whose regression kernel has L1 norm close to one and power law tail of the form x−(1 + α), with α ∈ (0, 1).”
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).”
Perfect hedging in rough Heston models (2017, methodology): “Writing the dynamic of V under the following form as in [16]:”
Rough volatility: Evidence from option prices (2017, background): “Other developments about rough volatility models can be found in [2, 3, 8, 9, 10, 12, 14, 18, 21].”
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.”
No‐arbitrage implies power‐law market impact and rough volatility (2018, methodology): “Using the population approach to Hawkes processes, see [3, 13, 21, 22], it is easily seen that in our model such property can hold only provided aT goes to one.”
The Quadratic Rough Heston Model and the Joint S&P 500/VIX Smile Calibration Problem (2020, methodology): “This model arises as limit of natural Hawkes process-based models of price and order flow, see [12, 27, 28].”
The characteristic function of rough Heston models (2016, methodology): “This is a remarkable feature of rough volatility models because this exploding behavior is commonly observed on real data and very important for practical applications; see Bayer et al. (2016), Fukasawa (2011), and Jaisson and Rosenbaum (2016).”
Renewal in Hawkes processes with self-excitation and inhibition (2018, unknown): “, [2, 22, 23, 38]).”