High-frequency trading in a limit order book

Abstract

We study a stock dealer's strategy for submitting bid and ask quotes in a limit order book. The agent faces an inventory risk due to the diffusive nature of the stock's mid-price and a transactions risk due to a Poisson arrival of market buy and sell orders. After setting up the agent's problem in a maximal expected utility framework, we derive the solution in a two step procedure. First, the dealer computes a personal indifference valuation for the stock, given his current inventory. Second, he calibrates his bid and ask quotes to the market's limit order book. We compare this "inventory-based" strategy to a "naive" strategy that is symmetric around the mid-price, by simulating stock price paths and displaying the P&L profiles of both strategies. We find that our strategy yields P&L profiles and final inventories that have significantly less variance than the benchmark strategy.

Stock Returns and Market Making with Inventory

“…Different from our model, Avellaneda and Stoikov (2008) assume the stock price process is an arithmetic Brownian motion without any drift, and also assume the zero risk-free rate. Seemingly, their model focuses on daily (or extremely short term) bidding and asking strategies of a market maker, while our model could be used in both short-term and long-term market making.…”

“…Following Avellaneda and Stoikov (2008), we assume the symmetric order arrival intensities of the following form: I(A t ) = Cexp(-kA t ), I(B t ) = Cexp(-kB t ), where C and k are positive constants.…”

“…Thus, the assumption of a monopolistic dealer in a stock market is not Platonic. Our paper is considered as an extension of the paper by Avellaneda and Stoikov (2008), but we employ asset price processes different from them. Avellaneda and Stoikov (2008) assume that the stock price process follows an arithmetic Brownian motion without any drift and the risk-free rate is zero.…”

“…Our paper is considered as an extension of the paper by Avellaneda and Stoikov (2008), but we employ asset price processes different from them. Avellaneda and Stoikov (2008) assume that the stock price process follows an arithmetic Brownian motion without any drift and the risk-free rate is zero. The stock price process in our paper is also assumed to be an arithmetic Brownian motion but we take a constant expected rate of the stock return and assume that the stock volatility is an inverse function of the stock price level, which could reflect the so-called leverage effect.…”

“…They employ the dynamic programming approach to find the maximal expected utility of the dealer at the terminal time and determine her optimal bid and ask strategies. Following them, Avellaneda and Stoikov (2008) suggest the optimal bid and ask strategies of an agent with constant absolute risk aversion (CARA) type utility with respect to her terminal wealth.…”