Marchés Financiers

We propose a framework for studying optimal market making policies in a limit

order book (LOB). The bid-ask spread of the LOB is modelled by a Markov chain with

nite values, multiple of the tick size, and subordinated by the Poisson process of the

tick-time clock. We consider a small agent who continuously submits limit buy/sell

orders at best bid/ask quotes, and may also set limit orders at best bid (resp. ask) plus

(resp. minus) a tick for getting the execution order priority, which is a crucial issue in

high frequency trading. By trading with limit orders, the agent faces an execution risk

since her orders are executed only when they meet counterpart market orders, which

are modelled by Cox processes with intensities depending on the spread and on her

limit prices. By holding non-zero positions on the risky asset, the agent is also subject

to the inventory risk related to price volatility. Then the agent can also choose to trade

with market orders, and therefore get immediate execution, but at a least favorable

price because she has to cross the bid-ask spread.

The objective of the market maker is to maximize her expected utility from revenue

over a short term horizon by a tradeo between limit and market orders, while

controlling her inventory position. This is formulated as a mixed regime switching regular/impulse control problem that we characterize in terms of quasi-variational system

by dynamic programming methods. In the case of a mean-variance criterion with

martingale reference price or when the asset price follows a Levy process and with

exponential utility criterion, the dynamic programming system can be reduced to a

system of simple equations involving only the inventory and spread variables.

Calibration procedures are derived for estimating the transition matrix and intensity

parameters for the spread and for Cox processes modelling the execution of limit

orders. Several computational tests are performed both on simulated and real data,

and illustrate the impact and pro t when considering execution priority in

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