A diffusion approximation for limit order book models

Horst, Ulrich; Kreher, Dörte

doi:10.1016/j.spa.2018.11.023

Cited by 8 publications

(20 citation statements)

References 20 publications

(57 reference statements)

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“…While modelling liquid stock markets, prices are typically approximated in the scaling limit by diffusion processes, cf. for example [1,7,16,18]. At the same time, there is a broad consensus that (large) jumps may occur as responses to highly unexpected news, cf.…”

Section: Introductionmentioning

confidence: 99%

“…In [14] a weak law of large numbers is established for a limit order book model with Markovian dynamics depending on prices and standing volumes simultaneously. In contrast, a diffusion limit for the order book dynamics can be found in Cont and de Larrard [7], or Horst and Kreher [16]. While [7] only analyses the volumes standing on top of the book, [16] takes again the prices and whole standing volumes into account leading to diffusive price and volume approximations.…”

Section: Introductionmentioning

confidence: 99%

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Jump diffusion approximation for the price dynamics of a fully state dependent limit order book model

Kreher¹,

Milbradt²

2020

Preprint

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We study a one-sided discrete-time limit order book model, in which the order dynamics depend on the current best available price and the current volume density function, simultaneously. In order to take extreme price movements into account, we include price changes to our model which do not scale with the tick size. We derive a functional convergence result, which states that the limit order book model converges to a continuous-time limit when the size of an individual order as well as the tick size tend to zero and the order arrival rate tends to infinity. In the high frequency limit the order book dynamics are described by a one-dimensional jump diffusion characterizing the price process coupled with an infinite-dimensional fluid process characterizing the standing volumes.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Jump diffusion approximation for the price dynamics of a fully state dependent limit order book model

Kreher¹,

Milbradt²

2020

Preprint

Self Cite

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“…Bayer et al [5] extends the models in [21,22] by introducing additional noise terms in the pre-limit in which case the dynamics can then be approximated by an SPDE in the scaling limit. With a different choice of scaling an SPDE limit for LOB models has recently been established in [19]. Macroscopic SPDE models of limit order markets were studied in [24,28].…”

Section: Introductionmentioning

confidence: 99%

A Scaling Limit for Limit Order Books Driven by Hawkes Processes

Horst¹,

Xu²

2019

SIAM J. Finan. Math.

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In this paper we derive a scaling limit for an infinite dimensional limit order book model driven by Hawkes random measures. The dynamics of the incoming order flow is allowed to depend on the current market price as well as on a volume indicator. With our choice of scaling the dynamics converges to a coupled SDE-ODE system where limiting best bid and ask price processes follows a diffusion dynamics, the limiting volume density functions follows an ODE in a Hilbert space and the limiting order arrival and cancellation intensities follow a Volterra-Fredholm integral equation.

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“…Instead, we use general convergence results for infinite dimensional stochastic integrals established in Kurtz and Protter [48]. Their methods have previously been applied to prove diffusion approximations for limit order book models in [34]. We prove that the rescaled sequence of market models converges in distribution to the unique solution of a stochastic differential equation driven by two independent Gaussiam white noise processes, a Poisson random measure that describes the arrivals of large exogenous shocks and an independent Poisosn random measure that describes the dynamics of endogenously induced self-excited jumps.…”

Section: Introductionmentioning

confidence: 99%

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

Horst¹,

Xu²

2019

Preprint

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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.

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A diffusion approximation for limit order book models

Cited by 8 publications

References 20 publications

Jump diffusion approximation for the price dynamics of a fully state dependent limit order book model

Jump diffusion approximation for the price dynamics of a fully state dependent limit order book model

A Scaling Limit for Limit Order Books Driven by Hawkes Processes

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

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