Limit theorems for nearly unstable Hawkes processes

Abstract

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.Comment: 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

An introduction to econophysics and quantitative finance

“…The relationship between these two sequences will determine the scaling behavior of the sequence of Hawkes processes. Recall that it is shown in [3] that when φ 1 is fixed and smaller than one, after appropriate scaling, the limit of the sequence of Hawkes processes is deterministic, as it is for example the case for Poisson processes, see [29]. In our setting, if 1 − a T tends "slowly" to zero, we can expect the same result.…”

Quadratic Hawkes processes for financial prices

“…The above limiting process is the major result of this section. Although it was derived for a Markovian ZHawkes process, we believe that this is the limiting process for the whole class of non-critical ZHawkes processes with short memory, and is the analogue of the Heston-CIR limiting process for Hawkes, as in [25]. The limiting behaviour corresponding to long-memory/critical ZHawkes processes, in the spirit of [26], is left for future investigations.…”

“…The low-frequency asymptotics of nearly critical Hawkes processes with short-ranged kernels have been investigated in details by Jaisson and Rosenbaum [25,26]. They show that for suitable scaling and convergence to the critical point n H = 1, the short memory Hawkes-based price process of Bacry et al [6] converges towards a Heston process (since the Hawkes intensity converges towards a CIR volatility process).…”

“…Since we fixed the values of n H and n Z , our procedure can be called a "constant endogeneity rescaling", as opposed to the scaling used by Jaisson and Rosenbaum in [25] and [26], where the endogeneity ratio n H of the process needs to converge to unity as T goes to infinity. Our choice is partly motivated by the calibration results of Section 3.2 for intra-day returns, that yield an endogeneity ratio in the range 0.7 − 0.9, close to what is obtained at the daily time scale in [14] and [9], and significantly away from the critical value n H = 1.…”

“…First, one obtains a power-law behavior that emerges naturally from the fact that since the volatility behaves as |Z t | for large values ofZ t , the process describing its dynamics is simply a multiplicative Brownian motion with drift (see 25). This is at variance with the "diagonal" Hawkes counterpart of [25] where the coefficient in front of the Brownian noise is only the square-root of the volatility, which inevitably leads to a process that has a characteristic scale and thin tails. Second, the stationary distribution of V only depends on the Zumbach norm n Z , that can be seen as the endogeneity of the process.…”