On Bellman equation in the limit order optimization problem for high-frequency trading
M. I. Balakaeva, A. Yu. Veretennikov
TL;DR
The paper addresses high-frequency limit-order optimization by formulating a zero-fee, complete-market setting with a Bachelier mid-price and exponential utility, and then applying a Bellman-equation framework to optimize limit-order quotes. It specializes the order-arrival intensities to $\lambda(\delta)=A\exp(-\kappa\delta)$ and carries out a small-inventory asymptotic expansion, deriving closed-form leading-order results for the optimal quotes and their spread. The main contributions are the explicit leading-order expressions $r(s,q,t)=s-\gamma\sigma^2(T-t)q$ and $\delta^a+\delta^b=\gamma\sigma^2(T-t)+\tfrac{2}{\gamma}\ln\left(1+\tfrac{\gamma}{\kappa}\right)$, together with a corrected derivation for the individual reservation prices $r^a$ and $r^b$, while showing that these corrections do not alter the principal outcome. The work clarifies how inventory sensitivity and order-arrival parameters shape inventory-aware market-making strategies in high-frequency trading and reconciles gaps from earlier work on the Avellaneda–Stoikov framework.
Abstract
An approximation method for construction of optimal strategies in the bid \& ask limit order book in the high-frequency trading (HFT) is studied. The basis is the article by M. Avellaneda \& S. Stoikov 2008, in which certain seemingly serious gaps have been found; in the present paper they are carefully corrected. However, a bit surprisingly, our corrections do not change the main answer in the cited paper, so that, in fact, the gaps turn out to be unimportant. An explanation of this effect is offered.