The Microstructure of Stochastic Volatility Models with Self-Exciting Jump Dynamics

Abstract

We provide a general probabilistic framework within which we establish scaling limits for a class of continuoustime stochastic volatility models with self-exciting jump dynamics. In the scaling limit, the joint dynamics of asset returns and volatility is driven by independent Gaussian white noises and two independent Poisson random measures that capture the arrival of exogenous shocks and the arrival of self-excited shocks, respectively. Various well-studied stochastic volatility models with and without self-exciting price/volatility co-jumps are obtained as special cases under different scaling regimes. We analyze the impact of external shocks on the market dynamics, especially their impact on jump cascades and show in a mathematically rigorous manner that many small external shocks may tigger endogenous jump cascades in asset returns and stock price volatility.

Diffusion Approximations for Marked Self-exciting Systems with Applications to General Branching Processes

“…to recount the mean impacts of an event with mark u on the future arrivals of type-i events. An argument similar to the one used in Section 2 in [32] induces the following proposition immediately.…”

“…In addition, the process µ H describes the impacts of events prior to time 0 on the arrivals of future events. For instance, Horst and Xu [32] applied a special class of MHP-measures with immigration and exponential kernel to study the stochastic volatility models with self-exciting jump dynamics. In detail, the process µ H represents the impacts of orders arrived prior to time 0.…”

“…These were extended to a two variate case with a special kernel in [18]. Horst and Xu [32] established a jump-diffusion limit for the rescaled intensity process of a MHP-measure with immigration and exponential kernel. For the general case, our analysis is based on a stochastic Volterra representation for the density process.…”

Portfolio Liquidation Games with Self-Exciting Order Flow

“…Specifically, we assume that the market buy and sell order dynamics follow Hawkes processes whose base intensities depend on the large investors' trading activities. Hawkes processes have recently received considerable attention in the financial mathematics literature as a powerful tool to model self-exciting order flow and its impact on stock price volatility; see [5,6,19,34,32] and references therein. In the context of liquidation models, they have been employed in [1,3,14] albeit in very different settings.…”

Portfolio Liquidation Under Transient Price Impact – Theoretical Solution and Implementation With 100 NASDAQ Stocks

“…Originally introduced in Hawkes (1971) to model the occurrence seismic events, Hawkes processes have recently received considerable attention in the financial mathematics and economics literature as a powerful tool to model financial time series dynamics. Their application range from trade arrivals in high-frequency markets (Aït-Sahalia et al, 2015;Andersen et al, 2015;Bacry et al, 2013;Huang et al, 2015), to volatility modelling (Bates, 2019;El Euch et al, 2018;Horst and Xu, 2019a), and from limit order book modelling (Horst and Xu, 2019b) to market impact and microstructure (Alfonsi and Blanc, 2016;Bacry et al, 2015;Cartea et al, 2014).…”