Optimal inventory management and order book modeling

Abstract

We model the behavior of three agent classes acting dynamically in a limit order book of a financial asset. Namely, we consider market makers (MM), high-frequency trading (HFT) firms, and institutional brokers (IB). Given a prior dynamic of the order book, similar to the one considered in the Queue-Reactive models [12,18,19], the MM and the HFT define their trading strategy by optimizing the expected utility of terminal wealth, while the IB has a prescheduled task to sell or buy many shares of the considered asset. We derive the variational partial differential equations that characterize the value functions of the MM and HFT and explain how almost optimal control can be deduced from them. We then provide a first illustration of the interactions that can take place between these different market participants by simulating the dynamic of an order book in which each of them plays his own (optimal) strategy. . D. Evangelista was partially supported by KAUST baseline funds and KAUST OSR-CRG2017-3452. ¶ CMAP, École Polytechnique. othmane.mounjid@polytechnique.edu. arXiv:1802.08135v2 [q-fin.TR] 9 Nov 2018 v(t, z) := sup φ∈C(t,z)Remark 3.3. Note that v is bounded from above by 0 by definition. On the other hand, for all E[U (P t,z,0 T , 0, 0, g, i, 0, 0, 0, 0, j)] = e −η(g− j) min i∈[−I * ,I * ] E[U (P t,z,0 T , 0, 0, 0, i, 0, 0, 0, 0, 0)], where P t,z,0 corresponds to the dynamics in the case that the MM does not act on the order book up to T . Moreover, it follows from (3.3) that E[U (P t,z,0 T , 0, 0, 0, i, 0, 0, 0, 0, 0)] ≥ −e ηI * |p b | E[e ηI * (|P t,z,0,b T −p b |+2d+κ) ]where supby Remark 2.1 and the fact that the price can jump only by d when a market event occurs. Thus, v belongs to the class L exp ∞ of functions ϕ such that ϕ/L is bounded, in which The dynamic programming equationThe derivation of the dynamic programming equation is standard, and is based on the dynamic programming principle. We state below the weak version of Bouchard and Touzi [10], we let v * and v * denote the lower-and upper-semicontinuous envelopes of v. Proposition 3.1. Fix (t, z) ∈ [0, T ] × D Z and a family {θ φ , φ ∈ C(t, z)} such that each θ φ is a [t, T ]-valued F t,z,φ -stopping time and Z t,z,φ θ φ L∞ < ∞. Then, sup φ∈C(t,z) E v * (θ φ , Z t,z,φ θ φ ) ≤ v(t, z) ≤ sup φ∈C(t,z)E v * (θ φ , Z t,z,φ θ φ ) .

Forecasting with Importance-Sampling and Path-Integrals: Applications to COVID-19

“…Although high-frequency trading (HFT) is a relatively new development in financial markets, it has become a primary force in market pricing. A large scientific literature has developed to understand the nature of these forces (Ait-Sahalia & Saglam, 2017;Avellaneda & Stoikov, 2008;Baradely et al, 2018;Cartea et al, 2014;Cont, 2011;Cont et al, 2009;Fodra & Labadie, 2012;Gueant, 2017;Huang et al, 2014). Since HFT by definition implies fast trading, this has generally prevented complex sophisticated algorithms from being applied to trading in real time.…”

Forecasting with Importance-Sampling and Path-Integrals: Applications to COVID-19

“…High-frequency trading (HFT) is a relatively new development in financial markets, where it has become a primary force in market pricing. A large scientific literature has developed to understand the nature of these forces (Ait-Sahalia & Saglam, 2017; Avellaneda & Stoikov, 2008;Baradely et al, 2018;Cartea et al, 2014;Cont, 2011;Cont et al, 2009;Fodra & Labadie, 2012;Gueant, 2017;Huang et al, 2014).…”

Forecasting with Importance-Sampling and Path-Integrals

“…Although high-frequency trading (HFT) is a relatively new development in financial markets, it has become a primary force in market pricing. A large scientific literature has developed to understand the nature of these forces (Ait-Sahalia & Saglam, 2017; Avellaneda & Stoikov, 2008;Baradely et al, 2018;Cartea et al, 2014;Cont, 2011;Cont et al, 2009;Fodra & Labadie, 2012;Gueant, 2017;Huang et al, 2014). Since HFT by definition implies fast trading, this has generally prevented complex sophisticated algorithms from being applied to trading in real time.…”