Dynamic Equilibrium of Market Making with Price Competition
- PDF / 1,145,462 Bytes
- 24 Pages / 439.37 x 666.142 pts Page_size
- 23 Downloads / 236 Views
Dynamic Equilibrium of Market Making with Price Competition Jialiang Luo1 · Harry Zheng1 Accepted: 6 November 2020 © The Author(s) 2020
Abstract In this paper, we discuss the dynamic equilibrium of market making with price competition and incomplete information. The arrival of market sell/buy orders follows a pure jump process with intensity depending on bid/ask spreads among market makers and having a looping countermonotonic structure. We solve the problem with the nonzero-sum stochastic differential game approach and characterize the equilibrium value function with a coupled system of Hamilton–Jacobi nonlinear ordinary differential equations. We prove, do not assume a priori, that the generalized Issac’s condition is satisfied, which ensures the existence and uniqueness of Nash equilibrium. We also perform some numerical tests that show our model produces tighter bid/ask spreads than those derived using a benchmark model without price competition, which indicates the market liquidity would be enhanced in the presence of price competition of market makers. Keywords Dynamic equilibrium · Market making · Price competition · Nonzero-sum stochastic differential game · Generalized Issac’s condition Mathematics Subject Classification 93E20 · 90C39 JEL Classification C7 · G1
1 Introduction Market makers play an important role in providing liquidity for other market participants. They keep quoting bid and ask prices at which they stand ready to buy and sell for a wide variety of assets simultaneously. One of the key challenges faced by market makers is to manage inventory risk. Market makers need to decide bid/ask prices which influence both their profit margins and accumulation of inventory. Many market makers compete for market
B
Harry Zheng [email protected] Jialiang Luo [email protected]
1
Department of Mathematics, Imperial College, London SW7 2AZ, UK
Dynamic Games and Applications
order flows as their profits come from the bid/ask spread of each transaction. Traders choose to buy/sell at the most competitive prices offered in the market. Hence, market makers face a complex optimization problem. In this paper, we model market making for a single asset with price competition as a nonzero-sum stochastic differential game. There has been active research on optimal market making in the literature with focus on inventory risk management. Stochastic control and Hamilton–Jacobi–Bellman (HJB) equation, a nonlinear partial differential equation (PDE), are used to derive the optimal bid/ask spread. Ho and Stoll [13] give the first prototype model for the market making problem. Avellaneda and Stoikov [2] propose a basic trading model in which the asset mid-price follows a Brownian motion, market buy/sell order arrivals follow a Poisson process with exponentially decreasing intensity function of bid/ask spread, and market makers optimally set the bid/ask spread to maximize the expected utility of the terminal wealth. Guéant et al. [11] discuss a quote-driven market and include the inventory penalty for
Data Loading...