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Table of contents
Introduction
1 Market making on OTC markets
1.1 The problem of the market maker
1.2 A first model: Avellaneda-Stoikov
1.3 Main existing results
1.4 Our contribution
2 Optimal tick sizes
2.1 The problem
2.2 The model with uncertainty zones
2.3 Our contribution
3 Optimal execution
3.1 The problem
3.2 A first model: Almgren-Chriss
3.3 Our contribution
I Optimal market making on OTC markets
1 Size matters for OTC market makers: general results and dimensionality reduction techniques
1.1 Introduction
1.2 Market making with marked point processes
1.2.1 Modeling framework and notations
1.2.2 Existence and uniqueness of a solution to (1.2)
1.2.3 Verification theorem
1.3 Solving the multi-asset market making problem with factors
1.3.1 A low-dimensional approximation
1.3.2 A Monte-Carlo method to take the residual risk into account
1.4 Numerical results
1.4.1 The case of two assets: one factor vs. two factors
1.4.2 Dealing with 30 assets
1.5 Appendix
1.5.1 A particular case: hedging the market factors
1.5.2 On the construction of the processes Ji;b and Ji;a
2 From passive to active: when market makers are also market takers
2.1 Introduction
2.2 A model for active market makers
2.2.1 Modeling framework and notations
2.2.2 Preliminary results
2.3 Viscosity solution to (HJ)
2.3.1 Existence
2.3.2 Uniqueness
2.4 Numerical results
2.5 Appendix
2.5.1 Proof of Lemma 3
2.5.2 Proof of Proposition 4
3 Algorithmic market making for options
3.1 Introduction
3.2 Description of the problem
3.2.1 The market
3.2.2 The optimization problem of the market maker
3.2.3 Assumptions and approximations
3.3 An approximate solution to the problem
3.3.1 Change of variables: beating the curse of dimensionality
3.3.2 Hamilton-Jacobi-Bellman equation and optimal controls
3.4 Numerical results
3.4.1 Model parameters
3.4.2 Optimal quotes
3.4.3 Conclusion
3.5 Appendix
3.5.1 An alternative to the -hedging assumption
3.5.2 Beyond the constant-vega assumption
4 Closed-form approximations in multi-asset market making
4.1 Introduction
4.2 The multi-asset market making model
4.2.1 Model setup
4.2.2 The optimization problems
4.2.3 The Hamilton-Jacobi-Bellman and Hamilton-Jacobi equations
4.2.4 Existing theoretical results
4.3 A quadratic approximation of the value function and its applications
4.3.1 Introduction
4.3.2 An approximation of the value function in closed form
4.3.3 From value functions to heuristics and quotes
4.4 Beyond the quadratic approximation: a perturbative approach
4.5 A multi-asset market making model with additional features
4.5.1 A more general model
4.5.2 The Hamilton-Jacobi equation
4.5.3 Quadratic approximation
4.5.4 From value functions to heuristics and quotes
4.6 Numerical results
4.6.1 Value function and optimal quotes
4.6.2 Closed-form approximations
II Other problems in mathematical finance
5 On bid and ask side-specific tick sizes
5.1 Introduction
5.2 The model with uncertainty zones
5.3 High frequency market making under side-specific tick values and interaction with the exchange
5.3.1 The market maker’s problem
5.3.2 The platform’s problem
5.4 Numerical results
5.4.1 Similar tick values on both sides
5.4.2 Side-specific tick values: additional opportunities for the market maker
5.4.3 Side-specific tick values: effect of
5.5 Conclusion
5.6 Appendix
5.6.1 Proof of Proposition 7
5.6.2 Proof of Theorem 5
5.6.3 Effects of the uncertainty zones on h
6 Optimal execution and statistical arbitrage under Ornstein-Uhlenbeck dynamics
6.1 Introduction
6.2 The optimal liquidation problem
6.2.1 Modelling framework and notations
6.2.2 Hamilton-Jacobi-Bellman equation
6.2.3 Main mathematical results
6.3 Numerical results
6.3.1 Single-asset case
6.3.2 Multi-asset case
6.4 Appendix – Multi-asset optimal execution with correlated Brownian motions and execution costs
6.5 Appendix – Merton portfolio optimization problem under Ornstein-Uhlenbeck dynamics and exponential utility
6.5.1 Modelling framework
6.5.2 HJB equation
Bibliography
