Book on optimal execution and market making
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Research Papers in Quantitative Finance
Price-Aware Automated Market Makers: Models Beyond Brownian Prices and Static Liquidity, with P. Bergault, L. Bertucci, D. Bouba, J. Guilbert, submitted
Abstract : In this paper, we introduce a suite of models for price-aware automated market making platforms willing to optimize their quotes. These models incorporate advanced price dynamics, including stochastic volatility, jumps, and microstructural price models based on Hawkes processes. Additionally, we address the variability in demand from liquidity takers through models that employ either Hawkes or Markov-modulated Poisson processes. Each model is analyzed with particular emphasis placed on the complexity of the numerical methods required to compute optimal quotes.
Market Making in Spot Precious Metals, with A. Barzykin and P. Bergault, to appear in Risk Magazine (Cutting Edge) in a shortened version
Abstract : The primary challenge of market making in spot precious metals is navigating the liquidity that is mainly provided by futures contracts. The Exchange for Physical (EFP) spread, which is the price difference between futures and spot, plays a pivotal role and exhibits multiple modes of relaxation corresponding to the diverse trading horizons of market participants. In this paper, we introduce a novel framework utilizing a nested Ornstein-Uhlenbeck process to model the EFP spread. We demonstrate the suitability of the framework for maximizing the expected P\&L of a market maker while minimizing inventory risk across both spot and futures. Using a computationally efficient technique to approximate the solution of the Hamilton-Jacobi-Bellman equation associated with the corresponding stochastic optimal control problem, our methodology facilitates strategy optimization on demand in near real-time, paving the way for advanced algorithmic market making that capitalizes on the co-integration properties intrinsic to the precious metals sector.
Dispensing with optimal control: a new approach for the pricing and management of share buyback contracts, with B. Baldacci and P. Bergault, Risk Magazine (Cutting Edge, August 2024) with the title Pricing and managing complex share buy-back contracts: an alternative to optimal control
Abstract : This paper introduces a novel methodology for the pricing and management of share buyback contracts, overcoming the limitations of traditional optimal control methods, which frequently encounter difficulties with high-dimensional state spaces and the intricacies of selecting appropriate risk penalty or risk aversion parameter. Our methodology applies optimized heuristic strategies to maximize the contract's value. The computation of this value utilizes classical methods typically used for pricing path-dependent Bermudan options. Additionally, our approach naturally leads to the formulation of a hedging strategy.
Asset and Factor Risk Budgeting: a balanced approach, with A. R. Cetingoz, to appear in Quantitative Finance
Abstract: Portfolio optimization methods have evolved significantly since Markowitz introduced the mean-variance framework in 1952. While the theoretical appeal of this approach is undeniable, its practical implementation poses important challenges, primarily revolving around the intricate task of estimating expected returns. As a result, practitioners and scholars have explored alternative methods that prioritize risk management and diversification. One such approach is Risk Budgeting, where portfolio risk is allocated among assets according to predefined risk budgets. The effectiveness of Risk Budgeting in achieving true diversification can, however, be questioned, given that asset returns are often influenced by a small number of risk factors. From this perspective, one question arises: is it possible to allocate risk at the factor level using the Risk Budgeting approach? This paper introduces a comprehensive framework to address this question by introducing risk measures directly associated with risk factor exposures and demonstrating the desirable mathematical properties of these risk measures, making them suitable for optimization. We also propose a framework to find the portfolio that effectively balances the risk contributions from both assets and factors. Leveraging standard stochastic algorithms, our framework enables the use of a wide range of risk measures.
Liquidity Dynamics in RFQ Markets and Impact on Pricing, with P. Bergault, submitted
Abstract: To assign a value to a portfolio, it is common to use Mark-to-Market prices. However, how should one proceed when the securities are illiquid? When transaction prices are scarce, how can one use all the available real-time information? In this article, we address these questions for over-the-counter (OTC) markets based on requests for quotes (RFQs). We extend the concept of micro-price, which was recently introduced for assets exchanged through limit order books in the market microstructure literature, and incorporate ideas from the recent literature on OTC market making. To account for liquidity imbalances in RFQ markets, we use an approach based on bidimensional Markov-modulated Poisson processes. Beyond extending the concept of micro-price to RFQ markets, we introduce the new concept of Fair Transfer Price. Our concepts of price can be used to value securities fairly, even when the market is relatively illiquid and/or tends to be one-sided.
Automated Market Makers: Mean-Variance Analysis of LPs Payoffs and Design of Pricing Functions, with P. Bergault, L. Bertucci and D. Bouba, Digital Finance, Volume 6, 2024
Abstract: With the emergence of decentralized finance, new trading mechanisms called Automated Market Makers have appeared. The most popular Automated Market Makers are Constant Function Market Makers. They have been studied both theoretically and empirically. In particular, the concept of impermanent loss has emerged and explains part of the profit and loss of liquidity providers in Constant Function Market Makers. In this paper, we propose another mechanism in which price discovery does not solely rely on liquidity takers but also on an exchange rate or price oracle. We also propose to compare the different mechanisms from the point of view of liquidity providers by using a mean / variance analysis of their profit and loss compared to that of agents holding assets outside of Automated Market Makers. In particular, inspired by Markowitz' modern portfolio theory, we manage to obtain an efficient frontier for the performance of liquidity providers in the idealized case of a perfect oracle. Beyond that idealized case, we show that even when the oracle is lagged and in the presence of adverse selection by liquidity takers, optimized oracle-based mechanisms perform better than popular Constant Function Market Makers.
Risk Budgeting Portfolios: Existence and Computation, with A. R. Cetingoz and J.-D. Fermanian, Mathematical Finance, Volume 34, Issue 3, 2024
Abstract: Modern portfolio theory has provided for decades the main framework for optimizing portfolios. Because of its sensitivity to small changes in input parameters, especially expected returns, the mean-variance framework proposed by Markowitz (1952) has however been challenged by new construction methods that are purely based on risk. Among risk-based methods, the most popular ones are Minimum Variance, Maximum Diversification, and Risk Budgeting (especially Equal Risk Contribution) portfolios. Despite some drawbacks, Risk Budgeting is particularly attracting because of its versatility: based on Euler's homogeneous function theorem, it can indeed be used with a wide range of risk measures. This paper presents mathematical results regarding the existence and the uniqueness of Risk Budgeting portfolios for a very wide spectrum of risk measures and shows that, for many of them, computing the weights of Risk Budgeting portfolios only requires a standard stochastic algorithm.
Dealing with multi-currency inventory risk in FX cash markets, with A. Barzykin and P. Bergault, Risk Magazine (Cutting Edge, March 2023)
Abstract: In FX cash markets, market makers provide liquidity to clients for a wide variety of currency pairs. Because of flow uncertainty and market volatility, they face inventory risk. To mitigate this risk, they typically skew their prices to attract or divert the flow and trade with their peers on the dealer-to-dealer segment of the market for hedging purposes.
This paper offers a mathematical framework to FX dealers willing to maximize their expected profit while controlling their inventory risk. Approximation techniques are proposed which make the framework scalable to any number of currency pairs.
Market making by an FX dealer: tiers, pricing ladders and hedging rates for optimal risk control, with A. Barzykin and P. Bergault, a short version appeared in Risk Magazine (Cutting Edge, August 2022), with the title Market-making by a foreign exchange dealer
Abstract: Dealers make money by providing liquidity to clients but face flow uncertainty and thus price risk. They can efficiently skew their prices and wait for clients to mitigate risk (internalization), or trade with other dealers in the open market to hedge their position and reduce their inventory (externalization). Of course, the better control associated with externalization comes with transaction costs and market impact. The internalization vs. externalization dilemma has been a topic of recent active discussion within the foreign exchange (FX) community. This paper offers an optimal control framework for market making tackling both pricing and hedging, thus answering a question well known to dealers: `to hedge, or not to hedge?'
Computational methods for market making algorithms, a chapter of the book "Progress in Industrial Mathematics at ECMI 2021"
Abstract: With the rise of electronification and trading automation, the task of quoting assets on many financial markets must be carried out algorithmically by market makers. Market making models and algorithms have therefore been an important research topic in recent years, at the frontier between economics, quantitative finance, scientific computing, and machine learning. The goal of this text is (i) to present a typical multi-asset market making model relevant for most over-the-counter markets, (ii) to show how to use stochastic optimal control tools to derive a theoretical characterization of optimal quotes in that model, and (iii) to discuss the various methods proposed in the literature that could be used in practice in the financial industry for building market making algorithms.
Algorithmic market making in dealer markets with hedging and market impact, with A. Barzykin and P. Bergault, Mathematical Finance, Volume 33, Issue 1, 2023
Abstract: In OTC markets, one of the main tasks of dealers / market makers consists in providing prices at which they agree to buy and sell the assets and securities they have in their scope. With ever increasing trading volume, this quoting task has to be done algorithmically. Over the last ten years, many market making models have been designed that can be the basis of quoting algorithms in OTC markets. Nevertheless, in most (if not all) OTC market making models, the market maker is a pure internalizer, setting quotes and waiting for clients. However, on many markets such as foreign exchange cash markets, market makers have access to liquidity pools where they can hedge part of their inventory. In this paper, we propose a model taking this possibility into account, therefore allowing market makers to externalize part of their risk by trading in a liquidity pool. The model displays an important feature well known to practitioners that within a certain inventory range the market maker internalizes the flow by appropriately adjusting the quotes and externalize outside of that range. The larger the market making franchise, the wider is the inventory range suitable for internalization. The model is illustrated numerically with realistic parameters for USDCNH spot market.
Multi-asset optimal execution and statistical arbitrage strategies under Ornstein-Uhlenbeck dynamics, with P. Bergault and F. Drissi, SIAM Journal on Financial Mathematics, Volume 13, Issue 1, 2022
Abstract: In recent years, academics, regulators, and market practitioners have increasingly addressed liquidity issues. Amongst the numerous problems addressed, the optimal execution of large orders is probably the one that has attracted the most research works, mainly in the case of single-asset portfolios. In practice, however, optimal execution problems often involve large portfolios comprising numerous assets, and models should consequently account for risks at the portfolio level. In this paper, we address multi-asset optimal execution in a model where prices have multivariate Ornstein-Uhlenbeck dynamics and where the agent maximizes the expected (exponential) utility of her PnL. We use the tools of stochastic optimal control and simplify the initial multidimensional Hamilton-Jacobi-Bellman equation into a system of ordinary differential equations (ODEs) involving a Matrix Riccati ODE for which classical existence theorems do not apply. By using a priori estimates obtained thanks to optimal control tools, we nevertheless prove an existence and uniqueness result for the latter ODE, and then deduce a verification theorem that provides a rigorous solution to the execution problem. Using examples based on data from the foreign exchange and stock markets, we eventually illustrate our results and discuss their implications for both optimal execution and statistical arbitrage.
Reinforcement Learning Methods in Algorithmic Trading, in Machine Learning and Data Sciences for Financial Markets: A Guide to Contemporary Practices edited by A. Capponi and C.-A. Lehalle
Abstract: This subchapter is dedicated to the third paradigm of machine learning alongside supervised and unsupervised learning: reinforcement learning (RL). RL methods have recently been successful in solving complex dynamic optimization problems in domains such as robotics, video games, and board games. Being flexible in terms of modelling and scalable to high dimensions, they are often regarded as good candidates to solve many financial problems, especially in the field of algorithmic trading. The goal of this subchapter is multifold: presenting the main ideas and concepts of RL, discussing their relevance for addressing algorithmic trading problems, reviewing the existing applications, and discussing the future. In particular, our view is that the range of problems that could be addressed with RL techniques is narrower than what most people think, but that RL-based trading programs could be competitive in execution and market making if traditional quants, computer scientists, and engineers united forces.
Closed-form approximations in multi-asset market making, with P. Bergault, D. Evangelista, and D. Vieira, Applied Mathematical Finance, Volume 28, Issue 2, 2021
Abstract: A large proportion of market making models derive from the seminal model of Avellaneda and Stoikov. The numerical approximation of the value function and the optimal quotes in these models remains a challenge when the number of assets is large. In this article, we propose closed-form approximations for the value functions of many multi-asset extensions of the Avellaneda-Stoikov model. These approximations or proxies can be used (i) as heuristic evaluation functions, (ii) as initial value functions in reinforcement learning algorithms, and/or (iii) directly to design quoting strategies through a greedy approach. Regarding the latter, our results lead to new and easily interpretable closed-form approximations for the optimal quotes, both in the finite-horizon case and in the asymptotic (ergodic) regime. Furthermore, we propose a perturbative approach to improve our closed-form approximations through Monte-Carlo simulations.
Recipes for hedging exotics with illiquid vanillas, with J. Fernandez-Tapia, submitted
Abstract: In this paper, we address the question of the optimal Delta and Vega hedging of a book of exotic options when there are execution costs associated with the trading of vanilla options. In a framework where exotic options are priced using a market model (e.g. a local volatility model recalibrated continuously to vanilla option prices) and vanilla options prices are driven by a stochastic volatility model, we show that, using simple approximations, the optimal dynamic Delta and Vega hedging strategies can be computed easily using variational techniques.
It's all relative: mean field game extensions of Merton's problem, with A. Bismuth (slides only, please cite the work as a talk by O. Guéant at DEA Krakow, 2019)
Abstract: Classical objective functions used in models à la Merton do not take account of the natural propensity of agents to compare their results with those of their peers. In this paper, we propose several extensions of Merton's problem involving a large population of economic agents who get utility not only from their running consumption and terminal wealth but also from the ratio between their running consumption (resp. terminal wealth) and the average running consumption (resp. average terminal wealth) of the population. These extensions take the form of mean field games (MFG) of controls with common noise which can, surprisingly, be solved in closed form.
Algorithmic Market Making for options, with B. Baldacci and P. Bergault, Quantitative Finance, Volume 21, Issue 1, 2021
Abstract: In this article, we tackle the problem of a market maker in charge of a book of equity derivatives on a single liquid underlying asset. By using an approximation of the portfolio in terms of its Greeks, we show that the seemingly high-dimensional stochastic optimal control problem of an equity option market maker is in fact tractable. More precisely, the problem faced by an equity options market maker boils down to solving a system of ordinary differential equations with classical numerical tools, even for large portfolios.
Size matters for OTC market makers: general results and dimensionality reduction technique, with P. Bergault, Mathematical Finance, Volume 31, Issue 1, January 2021
Abstract: In most OTC markets, a small number of market makers provide liquidity to clients from the buy side. More precisely, they set prices at which they agree to buy and sell the assets they cover. Market makers face therefore an interesting optimization problem: they need to choose bid and ask prices for making money out of their bid-ask spread while mitigating the risk associated with holding inventory in a volatile market. Many market making models have been proposed in the academic literature, most of them dealing with single-asset market making whereas market makers are usually in charge of a long list of assets. The rare models tackling multi-asset market making suffer however from the curse of dimensionality when it comes to the numerical approximation of the optimal quotes. The goal of this paper is to propose a dimensionality reduction technique to address multi-asset market making with grid methods. Moreover, we generalize existing market making models by the addition of an important feature for OTC markets: the variability of transaction sizes and the possibility for the market maker to answer different prices to requests with different sizes.
Accelerated Share Repurchase and other buyback programs: what neural networks can bring, with I. Manziuk and J. Pu, Quantitative Finance, Volume 20, Issue 8, 2020
Abstract: When firms want to buy back their own shares, they have a choice between several alternatives. If they often carry out open market repurchase, they also increasingly rely on banks through complex buyback contracts involving option components, e.g. accelerated share repurchase contracts, VWAP-minus profit-sharing contracts, etc. The entanglement between the execution problem and the option hedging problem makes the management of these contracts a difficult task that should not boil down to simple Greek-based risk hedging, contrary to what happens with classical books of options. In this paper, we propose a machine learning method to optimally manage several types of buyback contracts. In particular, we recover strategies similar to those obtained in the literature with partial differential equation and recombinant tree methods and show that our new method, which does not suffer from the curse of dimensionality, enables to address types of contract that could not be addressed with grid or tree methods.
Deep reinforcement learning for market making in corporate bonds: beating the curse of dimensionality, with I. Manziuk, Applied Mathematical Finance, Volume 26, Issue 5, 2019
Abstract: In corporate bond markets, which are mainly OTC markets, market makers play a central role by providing bid and ask prices for a large number of bonds to asset managers from all around the globe. Determining the optimal bid and ask quotes that a market maker should set for a given universe of bonds is a complex task. Useful models exist, most of them inspired by that of Avellaneda and Stoikov. These models describe the complex optimization problem faced by market makers: proposing bid and ask prices in an optimal way for making money out of the difference between bid and ask prices while mitigating the market risk associated with holding inventory. While most of the models only tackle one-asset market making, they can often be generalized to a multi-asset framework. However, the problem of solving numerically the equations characterizing the optimal bid and ask quotes is seldom tackled in the literature, especially in high dimension. In this paper, our goal is to propose a numerical method for approximating the optimal bid and ask quotes over a large universe of bonds in a model à la Avellaneda-Stoikov. Because we aim at considering a large universe of bonds, classical finite difference methods as those discussed in the literature cannot be used and we present therefore a discrete-time method inspired by reinforcement learning techniques. More precisely, the approach we propose is a model-based actor-critic-like algorithm involving deep neural networks.
Mid-price estimation for European corporate bonds: a particle filtering approach, with J. Pu, Market Microstructure and Liquidity, Volume 4, Issue 1 and 2, 2018
Abstract: In most illiquid markets, there is no obvious proxy for the market price of an asset. The European corporate bond market is an archetypal example of such an illiquid market where mid-prices can only be estimated with a statistical model. In this OTC market, dealers / market makers only have access, indeed, to partial information about the market. In real-time, they know the price associated with their trades on the dealer-to-dealer (D2D) and dealer-to-client (D2C) markets, they know the result of the requests for quotes (RFQ) they answered, and they have access to composite prices (e.g., Bloomberg CBBT). This paper presents a Bayesian method for estimating the mid-price of corporate bonds by using the real-time information available to a dealer. This method relies on recent ideas coming from the particle filtering (PF) / sequential Monte-Carlo (SMC) literature.
Expected Shortfall and optimal hedging payoff, Comptes Rendus Mathématiques (ex-CRAS), Volume 356, Issue 4, April 2018
Abstract: By using variational techniques, we provide an optimal payoff written on a given random variable for hedging - in the sense of minimizing the Expected Shortfall at a given threshold - a payoff written on another random variable. In numerous financially relevant examples, our result leads to optimal payoffs in closed form. From a theoretical viewpoint, our result is also useful for providing bounds to the classical Expected Shortfall minimization problem with given financial instruments.
Portfolio choice, portfolio liquidation, and portfolio transition under drift uncertainty, with A. Bismuth and J. Pu, Mathematics and Financial Economics, Volume 13, Issue 4, September 2019
Abstract: This paper presents several models addressing optimal portfolio choice, optimal portfolio liquidation, and optimal portfolio transition issues, in which the expected returns
of risky assets are unknown. Our approach is based on a coupling between Bayesian learning and dynamic programming techniques that leads to partial differential equations. It enables to recover the well-known results of Karatzas and Zhao in a framework à la Merton, but also to deal with cases where martingale methods are no longer available. In particular, we address optimal portfolio choice, portfolio liquidation, and portfolio transition problems in a framework à la Almgren-Chriss, and we build therefore a model in which the agent takes into account in his decision process both the liquidity of assets and the uncertainty with respect to their expected return.
Optimal market making, Applied Mathematical Finance, Volume 24, Issue 2, 2017
Abstract: Market makers provide liquidity to other market participants: they propose prices at which they stand ready to buy and sell a wide variety of assets. They face a complex optimization problem with static and dynamic components: they need indeed to propose bid and offer/ask prices in an optimal way for making money out of the difference between these two prices (their bid-ask spread), while mitigating the risk associated with price changes -- because they seldom buy and sell simultaneously, and therefore hold long or short inventories which expose them to market risk. In this paper, (i) we propose a general modeling framework which generalizes (and reconciles) the various modeling approaches proposed in the literature since the publication of the seminal paper ``High-frequency trading in a limit order book'' by Avellaneda and Stoikov, (ii) we prove new general results on the existence and the characterization of optimal market making strategies, (iii) we obtain new closed-form approximations for the optimal quotes, (iv) we extend the modeling framework to the case of multi-asset market making, and (v) we show how the model can be used in practice in the specific case of the corporate bond market and for two credit indices.
The behavior of dealers and clients on the European corporate bond market: the case of Multi-Dealer-to-Client platforms, with J.-D. Fermanian and J. Pu, Market Microstructure and Liquidity, Volume 2, Issue 3 & 4, December 2016
Abstract: For the last two decades, most financial markets have undergone an evolution toward electronification. The market for corporate bonds is one of the last major financial markets to follow this unavoidable path. Traditionally quote-driven (i.e., dealer-driven) rather than order-driven, the market for corporate bonds is still mainly dominated by voice trading, but a lot of electronic platforms have emerged. These electronic platforms make it possible for buy-side agents to simultaneously request several dealers for quotes, or even directly trade with other buy-siders. The research presented in this article is based on a large proprietary database of requests for quotes (RFQ) sent, through the multi-dealer-to-client (MD2C) platform operated by Bloomberg Fixed Income Trading, to one of the major liquidity providers in European corporate bonds. Our goal is (i) to model the RFQ process on these platforms and the resulting competition between dealers, and (ii) to use our model in order to implicit from the RFQ database the behavior of both dealers and clients on MD2C platforms.
Optimal execution of ASR contracts with fixed notional, Journal of Risk, Volume 19, Issue 5, p. 77-99, June 2017
Abstract: Be it for taking advantage of stock undervaluation or in order to distribute part of their profits to shareholders, firms may buy back their own shares. One of the way they proceed is by including Accelerated Share Repurchases (ASR) as part of their repurchase programs. In this article, we study the pricing and optimal execution strategy of an ASR contract with fixed notional. In such a contract the firm pays a fixed notional F to the bank and receives, in exchange, a number of shares corresponding to the ratio between F and the average stock price over the purchase period, the duration of this period being decided upon by the bank. From a mathematical point of view, the problem is related to both optimal execution and exotic option pricing.
A convex duality method for optimal liquidation with participation constraints, with J.-M. Lasry and J. Pu, Market Microstructure and Liquidity, Volume 1, Issue 1, 2015
Abstract: In spite of the growing consideration for optimal execution issues in the financial mathematics literature, numerical approximations of optimal trading curves are almost never discussed. In this article, we present a numerical method to approximate the optimal strategy of a trader willing to unwind a large portfolio. The method we propose is very general as it can be applied to multi-asset portfolios with any form of execution costs, including a bid-ask spread component, even when participation constraints are imposed. Our method, based on convex duality, only requires Hamiltonian functions to have C^{1,1} regularity while classical methods require additional regularity and cannot be applied to all cases found in practice.
Accelerated Share Repurchase: pricing and execution strategy, with J. Pu and G. Royer, International Journal of Theoretical and Applied Finance (IJTAF), Volume 18, Issue 3, May 2015
Abstract: In this article, we consider a specific optimal execution problem associated to accelerated share repurchase contracts. When firms want to repurchase their own shares, they often enter such a contract with a bank. The bank buys the shares for the firm and is paid the average market price over the execution period, the length of the period being decided upon by the bank during the buying process. Mathematically, the problem is new and related to both option pricing (Asian and Bermudan options) and optimal execution. We provide a model, along with associated numerical methods, to determine the optimal stopping time and the optimal buying strategy of the bank.
Option pricing and hedging with execution costs and market impact, with J. Pu, Mathematical Finance, Volume 27, Issue 3, July 2017
Abstract: In this article we consider the pricing and (partial) hedging of a call option when liquidity matters, that is either for a large nominal or for an illiquid underlying. In practice, as opposed to the classical assumptions of a price-taker agent in a frictionless market, traders cannot be perfectly hedged because of execution costs and market impact. They face indeed a trade-off between mishedge errors and hedging costs that can be solved using stochastic optimal control. Our framework is inspired from the recent literature on optimal execution and permits to account for both execution costs and the lasting market impact of our trades. Prices are obtained through the indifference pricing approach and not through super-replication. Numerical examples are provided using PDEs, along with comparison with the Black model.
VWAP execution and guaranteed VWAP, with G. Royer, SIAM Journal of Financial Mathematics, Volume 5, Issue 1, p. 445-471, 2014
Abstract: If optimal liquidation using VWAP strategies has been considered in the literature, it has never been considered in the presence of permanent market impact and only rarely with execution costs. Moreover, only VWAP strategies have been studied and no pricing of guaranteed VWAP contract is provided. In this article, we develop a model to price guaranteed VWAP contracts in the most general framework for market impact. Numerical applications are also provided.
Execution and block trade pricing with optimal constant rate of participation, Journal of Mathematical Finance, Volume 4, Issue 4, 2014
Abstract: In this article, we develop a liquidation model in which the trader is constrained to liquidate a portfolio at a constant participation rate. Considering the functional forms usually used by practitioners, we obtain a closed-form expression for the optimal participation rate and for the liquidity premium a trader should quote to buy a large block. We also show that the difference in terms of liquidity premium between the constant participation rate case and the usual Almgren-Chriss-like case never exceeds 15%.
Optimal execution and block trade pricing: a general framework, Applied Mathematical Finance, Volume 22, Issue 4, 2015
Abstract: In this article, we develop a general CARA framework to study optimal execution and to price block trades. We prove existence and regularity results for optimal liquidation strategies and we provide several differential characterizations. We also give two different proofs that the usual restriction to deterministic liquidation strategies is optimal. In addition, we focus on the important topic of block trade pricing and we therefore give a price to financial (il)liquidity. In particular, we provide a closed-form formula for the price a block trade when there is no time constraint to liquidate, and a differential characterization in the time-constrained case.
General Intensity Shapes in Optimal Liquidation, with C.-A. Lehalle, Mathematical Finance, Volume 25, Issue 3, p. 457-495, July 2015
Abstract: We study the optimal liquidation problem using limit orders. If the seminal literature on optimal liquidation, rooted to Almgren-Chriss models, tackles the optimal liquidation problem using a trade-off between market impact and price risk, it only answers the general question of the liquidation rhythm. The very question of the actual way to proceed is indeed rarely dealt with since most classical models use market orders only. Our model, that incorporates both price risk and non-execution risk, answers this question using optimal posting of limit orders. The very general framework we propose regarding the shape of the intensity generalizes both the risk-neutral model presented of Bayraktar and Ludkovski and the model developed in Guéant, Lehalle and Fernandez-Tapia, restricted to exponential intensity.
Optimal Portfolio Liquidation with Limit Orders, with C.-A. Lehalle and J. Fernandez-Tapia, SIAM Journal on Financial Mathematics, Volume 3, Number 1, p. 740-764, 2012
Abstract: This paper addresses the optimal scheduling of the liquidation of a portfolio using a new angle. Instead of focusing only on the scheduling aspect like Almgren and Chriss, or only on the liquidity-consuming orders like Obizhaeva and Wang, we link the optimal trade-schedule to the price of the limit orders that have to be sent to the limit order book to optimally liquidate a portfolio. Most practitioners address these two issues separately: they compute an optimal trading curve and they then send orders to the markets to try to follow it. The results obtained here solve simultaneously the two problems. As in a previous paper that solved the "intra-day market making problem", the interactions of limit orders with the market are modeled via a Poisson process pegged to a diffusive "fair price" and a Hamilton-Jacobi-Bellman equation is used to solve the problem involving both non-execution risk and price risk. Backtests are carried out to exemplify the use of our results, both on long periods of time (for the entire liquidation process) and on slices of 5 minutes (to follow a given trading curve).
Dealing with the Inventory Risk. A solution to the market making problem, with C.-A. Lehalle and J. Fernandez-Tapia, Mathematics and Financial Economics, Volume 7, Issue 4, September 2013.
Abstract: Market makers continuously set bid and ask quotes for the stocks they have under consideration. Hence they face a complex optimization problem in which their return, based on the bid-ask spread they quote and the frequency they indeed provide liquidity, is challenged by the price risk they bear due to their inventory. In this paper, we consider a stochastic control problem similar to the one introduced by Ho and Stoll and formalized mathematically by Avellaneda and Stoikov. The market is modeled using a reference price S_t following a Brownian motion, arrival rates of buy or sell liquidity-consuming orders depend on the distance to the reference price S_t and a market maker maximizes the expected utility of its PnL over a short time horizon. We show that the Hamilton-Jacobi-Bellman equations can be transformed into a system of linear ordinary differential equations and we solve the market making problem under inventory constraints. We also provide a spectral characterization of the asymptotic behavior of the optimal quotes and propose closed-form approximations.