Long memory patterns in high-frequency trading

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Table of contents

1 Introduction 
1.1 A introduction to the limit order book
1.2 Directional bets and the power of information in market making
1.3 Semi Markov model for market microstructure
1.4 HFT and asymptotics for small risk aversion in a Markov renewal model
1.5 Long memory patterns in high-frequency trading
2 Directional bets and the power of information in market making 
2.1 The market framework and the control problem
2.2 The explicit solution in the no risk aversion case
2.2.1 A nancial interpretation of the value function
2.2.2 A nancial interpretation of the optimal controls
2.3 The perturbation approach for positive risk aversion
2.3.1 A nancial interpretation of the value function
2.3.2 A nancial interpretation of the optimal controls
2.3.3 The special case of small martingale deviation
2.4 Examples
2.4.1 The Ornstein-Uhlenbeck process
2.4.2 The arithmetic Heston model
2.5 Numerical experiments
2.6 Appendix: the multi-asset model
2.6.1 The no risk aversion case
2.6.2 The case of small risk aversion under small martingale deviation
3 Semi Markov model for market microstructure 
3.1 Semi-Markov model
3.1.1 Price return modelling
3.1.2 Tick times modeling
3.1.3 Statistical inference
3.1.4 Price simulation
3.1.5 Semi-Markov property
3.1.6 Comparison with respect to Hawkes processes
3.2 Scaling limit
3.3 Mean Signature plot
3.4 Appendix: the mean signature plot
3.5 Appendix: a comparison to the Eurostoxx50
4 HFT and asymptotics for small risk aversion in a Markov renewal model 
4.1 Stock price in the limit order book
4.1.1 Markov renewal model
4.1.2 The stock price conditional mean and the trend indicator
4.2 Market order ow modeling and adverse selection
4.3 The market making problem
4.4 Value function and optimal controls: a perturbation approach
4.4.1 The no risk aversion case
4.4.2 The small risk aversion case
4.5 Appendix: the trade intensity function
4.6 Appendix: the estimation of the agent execution distribution #(dk;L)
4.7 Appendix: properties of the function T (t; s)
4.7.1 The PDE representation
4.7.2 The probabilistic representation
4.8 Appendix: proof of Theorem 4.3
5 Long memory patterns in high-frequency trading 
5.1 From the mid-price to the fair one
5.2 The dynamic of the fair price
5.2.1 The tick times
5.2.2 The marks of the stock price
5.3 Execution via limit orders
5.4 The optimal trading problem
5.4.1 The agent strategy and the portfolio dynamic
5.4.2 The HJB equation
5.5 Numerical results
5.6 Appendix: proof of Theorem 5.1
5.7 Appendix: proof of Proposition 5.1
Notations