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Option market-making algorithms in high dimension
In this part of the thesis we leave aside the viewpoint of the exchange to focus on the problem of the trader. Two central topics for traders are optimal market-making and optimal trading. While the latter will be treated in the third and last part of this thesis, the former is the main focus of the second part of this thesis. For more than three decades, the optimal market-making problem on cash markets has been the object of many academic studies, see, for example, Grossman and Miller [135], Ho and Stoll [156]. In Grossman and Miller [135], the authors studied a three-period model representing the interaction between market-makers and market-takers and analyzed its equilibrium state. In Ho and Stoll [156], the authors examined the behavior of a market-maker facing a stochastic demand and an inventory risk. The subject was revived in 2008 in the seminal paper Avellaneda and Stoikov [26], where the authors address the quoting and inventory management problems of market-makers using the stochastic optimal control tools. In this paper, the traded asset follows an arithmetic Brownian motion, and the spread processes are continuous, which makes the model more suited to quote-driven markets. For order-driven markets, the more suitable model is the one developed in Guilbaud and Pham [143]: the market-maker can send limit and market-orders at the first and second best limit of an order book, and the optimal controls (the volumes and limits at which he posts) are obtained through the resolution of a HJBQVI equation. There are numerous extensions of the work of Avellaneda and Stoikov tackling the problem of single-asset market-making. For example, in Guéant, Lehalle, and Fernandez-Tapia [142], the authors show that, in the case of exponential intensity functions, the market-making problem boils down to the reso-lution of a system of linear ordinary differential equations. A significant proportion of the contribution to the market-making literature comes from Cartea and Jaimungal. By considering a risk-adjusted expectation instead of Avellaneda-Stoikov’s CARA objective function, they manage to enrich the ini-tial model by introducing alpha signals, ambiguity aversion, competition with other agents, see, for example, Cartea, Donnelly, and Jaimungal [71, 72], Cartea and Jaimungal [68], Cartea, Jaimungal, and Penalva [70]. Regardless of how rich is the academic literature considering linear markets, the one studying optimal market-making on options is far less extensive. Market-making models on options, or other derivatives, are intrinsically more complicated because they must consider both the market for an underlying asset and the derivatives market. Consequently, one needs to first impose a factorial stochastic volatility model, possibly with jumps, on the underlying asset. Second, option market-makers need to manage several thousands of positions, which leads to very high-dimensional problems that cannot be solved using classical numerical schemes. Even if machine learning techniques are used, involving, for example, deep reinforcement learning methods in the spirit of Guéant and Manziuk [140], Huré, Pham, and Warin [168], E, Han, and Jentzen [103], the computation time can still be an obstacle for a large portfolio of options.
At the time of writing this thesis, the only papers addressing option market-making are El Aoud and Abergel [106], Stoikov and Sağlam [258]. In the former, the authors consider a single-option market driven by a stochastic volatility model and assume that the position is always -hedged. They provide optimal bid and ask quotes for the option and focus on the risk of model misspecification. In the latter, the authors consider three different settings for a market-maker managing a single option and its underlying. The first setting is a complete market with continuous trading in the perfectly liquid underlying. The second is a complete market with an illiquid underlying, where the market-maker sets bid and ask quotes in the option and the stock. The third is an incomplete market with residual risks due to stochastic volatility and overnight jumps in the stock price. These two articles treat only the case of a single option on a single underlying. Thus, we want to solve the following problem. Problem 6: How can we use stochastic control to design option market-making algorithms in high dimension?
We propose two models to build option market-making strategies. In the first one, we consider the case of a market-maker in charge of a book of options (on a single asset) whose prices are driven by a stochastic volatility model. We assume that trading in continuous time can be carried out in the underlying asset so that the residual risk is only that of the Vega associated with the inventory. The dynamics of the underlying asset under a risk-neutral measure Q is given by the following one-factor stochastic volatility model:
High-frequency dynamics of the volatility surface
Modeling the behavior of the implied volatility surface has often been carried out at a coarse time scale, see among others Fengler, Härdle, and Mammen [116], Kamal and Gatheral [181], Skiadopoulos, Hodges, and Clewlow [252] and Cont and Da Fonseca [80]. For example, in the latter, the authors use a time series of option prices on the S&P500 and FTSE indices to study the volatility surface’s deformation. The surface dynamics can be described by fluctuations of a small number of orthogonal random factors:
The “level” factor corresponding to the global level of the whole surface of implied volatilities.
The “calendar” factor corresponding to the skew of the volatility surface.
The “butterfly” factor corresponding to the convexity of the volatility surface.
However, such a model does not exist at thin time scales, where the implied volatility moves as a discrete process. Models for thin and coarse time scales have only been developed for price processes, which have inspired our study. In Bacry and Muzy [30], the authors develop a Hawkes-based model for the high-frequency dynamics of an asset’s price. Assuming that all jumps are of the same size, the microscopic price of the asset is the difference of the number of upward and downward jumps, that is where (N+; N ) is a bi-dimensional Hawkes process with intensity kernel where is an endogenous source of price moves: for example, + raises the intensity of upward price jumps after a downward price jump, creating a mean-reversion effect, while ++ creates a trending effect. This model accurately reproduces some stylized facts of the high-frequency price behavior, which can be encoded easily in the Hawkes kernel: no statistical arbitrage property, bid-ask asymmetry, long memory property of order flow, and a high degree of endogeneity of financial markets. At the coarse time scale, the price process defined above converges to a stochastic rough volatility process. The rough volatility models are particularly appreciated for their ability to capture key features of the implied volatility surface and its dynamics, see Bayer, Friz, and Gatheral [38], El Euch, Gatheral, and Rosenbaum [108], Horvath, Jacquier, and Tankov [161], Jacquier, Martini, and Muguruza [171], Alòs, León, and Vives [15]. Their properties have raised interest in building microscopic models for market dynamics which reproduce rough volatility at a macroscopic scale, see among others Jaisson and Rosenbaum [174, 175], Jusselin and Rosenbaum [177], Tomas and Rosenbaum [263]. These models focus on stock prices, mainly because it is easy to stylize stock markets’ characteristics with Hawkes kernels. However, one can apply similar ideas to model the microscopic behavior of the volatility surface. Problem 7: How to model the intraday dynamics of the implied volatility surface? To answer this question, we propose a tick-by-tick model for the high-frequency dynamics of the volatility surface. Assuming a set K of strikes and T of maturities, the microscopic volatility surface is modeled as a multi-dimensional process ( t(k; ))(k; )2K T where.
Adaptive trading strategies across liquidity pools
Most of the first studies of optimal trading examined the problem of optimal scheduling. The scheduling consists in finding an optimal plan for a large buy or sell order minimizing the overall trading costs. There is a trade-off between trading fast, which leads to high execution costs and market impact and trading slow, which leads to low market impact but high uncertainty on price change. In Almgren and Chriss [9, 10], the authors propose a framework to address this issue of large order optimal splitting. In their model, the authors assume a Bachelier price dynamics with linear execution costs and discrete periods at which the trader chooses the volume to execute. Since then, the Almgren-Chriss model has been widely used in practice with or without numerous generalizations such as different price dynamics, non-linear or transient market impact, illiquidity of the traded asset, see among others Almgren [8, 7], Forsyth [119], Forsyth, Kennedy, Tse, and Windcliff [120], Gatheral, Schied, and Slynko [127], Lehalle [198], Schied and Schöneborn [248]. In all these models, the market’s interactions are hidden: their aim is only to find the right quantity to buy or sell at a given time for certain market conditions. The first main contribution in this direction, considering aggressive orders only, is Obizhaeva and Wang [224], extended in Alfonsi, Fruth, and Schied [5], Predoiu, Shaikhet, and Shreve [234]. In Bayraktar and Ludkovski [39], the authors introduce an optimal liquidation framework using passive orders. This idea has been used notably in Guéant [136], Guéant and Lehalle [139], Guéant, Lehalle, and Fernandez-Tapia [141] to solve the optimal scheduling and trading problems together. The framework used to solve optimal liquidation using passive orders is directly inspired by the market-making model of Avellaneda and Stoikov [26], and it is more suitable for liquidation on quote-driven markets. Optimal liquidation models in limit order books have been notably developed by Guilbaud and Pham [143] using stochastic control theory. The trader can choose the order’s volume, aggressiveness, and the limit at which to post. One important drawback of these models is the impossibility of providing simple expressions for the trader’s optimal control: they rely either on first order approximations or on numerical approximations for PDEs. Moreover, they do not address the problem of optimal trading in several liquidity platforms.
The problem of optimal splitting of orders across liquidity pools has been treated for instance in Almgren and Harts [11], Cont and Kukanov [81], Laruelle, Lehalle, and Pagès [193, 194]. In Almgren and Harts [11], the authors develop a dynamic estimate of the hidden liquidity present on several venues and use this information to make order splitting decisions (Smart Order Routing). The paper Cont and Kukanov [81] solve a general order placement problem and provide an explicit solution for the optimal split between limit and market orders in different venues. Finally, in Laruelle, Lehalle, and Pagès [193, 194], the authors build a stochastic algorithm to find the optimal splitting between liquidity pools, including dark pools. These models are mostly “static” because they do not combine the optimal scheduling of orders with a long term inventory target. Moreover, an optimal trading model’s quality mainly relies on estimating the market parameters (cross-dependence between the imbalance and spread of each venue and the probability and the proportion of execution of limit orders). Some optimal liquidation models treat online updates of market parameters, see Almgren and Lorenz [12, 13], however, those are usually rather parsimonious models, where the parameters are, for example, only the drift and the volatility of the price process, updated via Bayesian updates. There is still a lack of literature treating optimal trading in a more dynamic fashion, which leads to the following problem.
Table of contents :
Introduction
Part I Market-making regulation activity via Principal-Agent theory
I.1 Market-making and incentives design in the presence of a dark pool: a deep reinforcement learning approach
I.2 Optimal make-take fees in a multi-market-maker environment
I.3 On bid and ask side-specific tick sizes
I.4 Derivatives market design
I.5 Governmental incentives for green bonds investment
Part II Options market-making in high dimension
II.1 Option market-making algorithms in high dimension
II.2 High-frequency dynamics of the volatility surface
Part III Optimal trading in high dimension
III.1 Adaptive trading strategies across liquidity pools
III.2 Optimal trading without optimal control
III.3 Two new developments on optimal execution
Part I Regulation of market-making activity via Principal-Agent theory
1 Market-making and incentives design in the presence of a dark pool: a deep reinforcement learning approach
1.1 Introduction
1.2 The market model
1.2.1 Stochastic basis
1.2.2 Traded volumes, market impact and latency
1.3 Market-making without the intervention of the exchange
1.4 Market-making with the intervention of the exchange
1.4.1 Modified PnL of the market-maker
1.4.2 Objective function of the exchange
1.4.3 Design of an optimal make-take fees policy
1.4.4 Problem of the exchange
1.5 Numerical solution: a deep reinforcement learning approach
1.5.1 Description
1.5.2 Numerical results: influence of the make-take fees policy
1.5.3 Numerical results: sensitivity analysis
1.6 Conclusion
1.A Appendix
1.A.1 Dynamic programming principle and contract representation
1.A.2 Proof of Theorem 1.2
1.A.3 Proof of Theorem 1.3
2 Optimal make-take fees in a multi-market-maker environment
2.1 Introduction
2.2 The model
2.2.1 Framework
2.2.2 Market-makers’ problem
2.2.3 The exchange optimal contracting problem
2.2.4 Stackelberg games in a nutshell
2.3 Solving the market-maker’s problem
2.3.1 Preliminaries
2.3.2 Contract representation
2.3.3 On the shape of compensation proposed and contractible variables
2.4 Solving the principal’s problem
2.4.1 Saturation of utility constraint
2.4.2 The HJB equation for the reduced exchange problem
2.4.3 Change of variable and verification theorem
2.4.4 Discussion
2.5 Impact of the presence of several market-makers
2.5.1 One market-maker
2.5.2 Two market-makers
2.5.3 Five market-makers
2.A Appendix
2.A.1 Dynamic programming principle
2.A.2 Proof of Theorem 2.1
2.A.3 Proof of Lemma 2.1
2.A.4 Exchange’s Hamiltonian maximisation
2.A.5 Proof of Lemma 2.3
2.A.6 Proof of Theorem 2.2
2.A.7 First-best exchange problem
3 On bid and ask side-specific tick sizes
3.1 Introduction
3.2 The model with uncertainty zones
3.3 High frequency market-making under side-specific tick values and interaction with the exchange
3.3.1 The market-maker’s problem
3.3.2 The platform’s problem
3.4 Numerical results
3.4.1 Similar tick values on both sides
3.4.2 Side-specific tick values: additional opportunities for the market-maker
3.4.3 Side-specific tick values: effect of
3.5 Conclusion
3.A Appendix
3.A.1 Proof of Proposition 3.1
3.A.2 Proof of Theorem 3.1
3.A.3 Effects of the uncertainty zones on h
4 How to design a derivatives market?
4.1 Introduction
4.2 Market driven selection of the listed options
4.2.1 How to choose the strikes in order to match market demand?
4.2.2 Solving the quantization problem
4.2.3 Application
4.3 Incentive policy of the exchange
4.3.1 The market
4.3.2 Market maker’s problem and contract representation
4.3.3 Solving the exchange’s problem
4.3.4 Numerical results
4.3.5 Conclusion
4.A Appendix
4.A.1 Proof of the convergence of Lloyd’s algorithm
4.A.2 Stochastic basis
4.A.3 Well-posedness of the optimization problems
4.A.4 Dynamic programming principle
4.A.5 Proof of Lemma 4.1
4.A.6 Proof of Theorem 4.1
4.A.7 Proof of Theorem 4.2
4.A.8 Perfect and imperfect Delta-hedging
5 Governmental incentives for green bonds investment
5.1 Introduction
5.2 Framework
5.3 Solving the optimisation problem
5.3.1 The optimal contract
5.3.2 Discussion
5.4 Numerical results
5.4.1 Data, key results and remarks for the policy-maker
5.4.2 Reference case
5.4.3 Influence of the green target
5.4.4 Sensitivity analysis
5.A Appendix
5.A.1 Weak formulation of the problem
5.A.2 Proof of Theorem 5.1
5.A.3 Green investments with stochastic interest rates
Part II Options market-making in high dimension
6 Algorithmic market-making for options
6.1 Introduction
6.2 Description of the problem
6.2.1 The market
6.2.2 The optimization problem of the market-maker
6.2.3 Assumptions and approximations
6.3 An approximate solution to the problem
6.3.1 Change of variables: beating the curse of dimensionality
6.3.2 Hamilton-Jacobi-Bellman equation and optimal controls
6.4 Numerical results
6.4.1 Model parameters
6.4.2 Optimal quotes
6.4.3 Conclusion
6.A Appendix
6.A.1 An alternative to the -hedging assumption
6.A.2 Beyond the constant-Vega assumption
6.A.3 On the construction of the processes Ni;b and Ni;a
7 An approximate solution for options market-making in high dimension
7.1 Introduction
7.2 Framework
7.2.1 The option book
7.2.2 The market-maker’s problem on OTC markets
7.3 Solving the market-maker’s problem with a system of non-linear PDEs
7.4 Numerical results
7.A Appendix
7.A.1 The market-maker’s problem for large number of underlyings
8 High-frequency dynamics of the volatility surface
8.1 Introduction
8.2 Microscopic modelling of the volatility surface
8.2.1 Framework and no-arbitrage conditions
8.2.2 Parametrization of the volatility surface
8.3 Macroscopic limit of the volatility surface
8.3.1 Limiting processes in the separable, one-dimensional case
8.3.2 Limiting process in the semi-separable, factor case
8.3.3 General case
8.4 Applications
8.4.1 Backtesting of option market-making strategies
8.4.2 Market impact curves
8.5 Conclusion
Part III Optimal trading in high dimension
9 Adaptive trading strategies across liquidity pools
9.1 Introduction
9.2 Optimal trading on several liquidity pools
9.2.1 Framework
9.2.2 The Hamilton-Jacobi-Bellman quasi-variational inequality
9.3 Adaptive trading strategies with Bayesian update
9.3.1 Bayesian update of the model parameters
9.3.2 Algorithm description
9.4 Model extensions
9.4.1 Extension 1: incorporation of signals in the price process
9.4.2 Extension 2: market impact
9.4.3 Extension 3: hidden liquidity
9.5 Numerical results
9.5.1 Global parameters
9.5.2 Numerical methods
9.5.3 Two identical venues
9.5.4 Two different venues
9.5.5 Bayesian update
9.A Appendix
9.A.1 Proof of Theorem 9.3
9.A.2 Application to OTC market-making
9.A.3 Bayesian update for OTC market-makers
10 Optimal trading without optimal control
10.1 Introduction
10.2 The long-term trading curve
10.3 From smooth relaxation to microstructure decision
10.4 A general microstructure trading model with long-term trading schedule
10.4.1 Optimal trading
10.4.2 Market-making
10.5 Numerical results
10.5.1 Transaction cost model and microstructure simulation
10.5.2 Results
10.6 Conclusion
11 A note on Almgren-Chriss optimal execution problem with geometric Brownian motion
11.1 Introduction
11.2 The model
11.2.1 Almgren-Chriss framework in continuous time
11.2.2 Reformulation in terms of cash
11.3 Solving explicitly the Almgren-Chriss problem with GBM
11.4 Numerical results
11.5 Extensions of the model
11.5.1 Stochastic drift
11.5.2 Multi-dimensional case
11.6 Conclusion
11.A Appendix
11.A.1 Proof of Lemma 11.1
11.A.2 Gâteaux differentiability in (11.4)
11.A.3 Proof of Lemma 11.2
11.A.4 Proof of Theorem 11.2
12 Liquidity stress testing using optimal portfolio liquidation
12.1 Introduction
12.2 Framework
12.2.1 The locally linear order book model
12.2.2 Single asset liquidation
12.2.3 Portfolio liquidation
12.3 Numerical results
12.3.1 Long-short portfolio with two correlated bonds
12.3.2 Long-short portfolio of 20 bonds
12.4 Conclusion
