arXiv subjects

Olivier Guéant

Publications and source records attributed to Olivier Guéant.

At least 37 records · Page 2Linked to original sources

arXiv2022

Size matters for OTC market makers: general results and dimensionality reduction techniques

In most OTC markets, a small number of market makers provide liquidity to other market participants. More precisely, for a list of assets, they set prices at which they agree to buy and sell. Market makers face therefore an interesting optimization problem: they need to choose bid and ask prices for making money 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 by using a factor model. Moreover, we generalize existing market making models by the addition of an important feature: the existence of different transaction sizes and the possibility for the market makers in OTC markets to answer different prices to requests with different sizes.

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arXiv2022

Multi-asset optimal execution and statistical arbitrage strategies under Ornstein-Uhlenbeck dynamics

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 \textit{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.

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arXiv2020

Algorithmic market making for options

In this article, we tackle the problem of a market maker in charge of a book of options on a single liquid underlying asset. By using an approximation of the portfolio in terms of its vega, we show that the seemingly high-dimensional stochastic optimal control problem of an option market maker is in fact tractable. More precisely, when volatility is modeled using a classical stochastic volatility model -- e.g. the Heston model -- the problem faced by an option market maker is characterized by a low-dimensional functional equation that can be solved numerically using a Euler scheme along with interpolation techniques, even for large portfolios. In order to illustrate our findings, numerical examples are provided.

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arXiv2020

Recipes for hedging exotics with illiquid vanillas

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.

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arXiv2019

Optimal control on graphs: existence, uniqueness, and long-term behavior

The literature on continuous-time stochastic optimal control seldom deals with the case of discrete state spaces. In this paper, we provide a general framework for the optimal control of continuous-time Markov chains on finite graphs. In particular, we provide results on the long-term behavior of value functions and optimal controls, along with results on the associated ergodic Hamilton-Jacobi equation.

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arXiv2019

Accelerated Share Repurchase and other buyback programs: what neural networks can bring

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 contract. 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.

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arXiv2019

Deep reinforcement learning for market making in corporate bonds: beating the curse of dimensionality

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.

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arXiv2019

Mid-price estimation for European corporate bonds: a particle filtering approach

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 / sequential Monte-Carlo literature.

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arXiv2019

Portfolio choice, portfolio liquidation, and portfolio transition under drift uncertainty

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.

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arXiv2017

Optimal market making

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 both 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). Since they seldom buy and sell simultaneously, and therefore hold long and/or short inventories, they also need to mitigate the risk associated with price changes, and subsequently skew their quotes dynamically. 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 we obtain general closed-form approximations for the optimal quotes of a multi-asset market maker, and (v) we show how the model can be used in practice in the specific (and original) case of two credit indices.

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arXiv2017

The behavior of dealers and clients on the European corporate bond market: the case of Multi-Dealer-to-Client platforms

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.

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arXiv2016

Optimal Real-Time Bidding Strategies

The ad-trading desks of media-buying agencies are increasingly relying on complex algorithms for purchasing advertising inventory. In particular, Real-Time Bidding (RTB) algorithms respond to many auctions -- usually Vickrey auctions -- throughout the day for buying ad-inventory with the aim of maximizing one or several key performance indicators (KPI). The optimization problems faced by companies building bidding strategies are new and interesting for the community of applied mathematicians. In this article, we introduce a stochastic optimal control model that addresses the question of the optimal bidding strategy in various realistic contexts: the maximization of the inventory bought with a given amount of cash in the framework of audience strategies, the maximization of the number of conversions/acquisitions with a given amount of cash, etc. In our model, the sequence of auctions is modeled by a Poisson process and the \textit{price to beat} for each auction is modeled by a random variable following almost any probability distribution. We show that the optimal bids are characterized by a Hamilton-Jacobi-Bellman equation, and that almost-closed form solutions can be found by using a fluid limit. Numerical examples are also carried out.

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arXiv2016

Optimal execution of ASR contracts with fixed notional

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.

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arXiv2015

Option pricing and hedging with execution costs and market impact

This article considers the pricing and hedging of a call option when liquidity matters, that is, either for a large nominal or for an illiquid underlying asset. In practice, as opposed to the classical assumptions of a price-taking agent in a frictionless market, traders cannot be perfectly hedged because of execution costs and market impact. They indeed face a trade-off between hedging errors and costs that can be solved by using stochastic optimal control. Our modelling framework, which is inspired by the recent literature on optimal execution, makes it possible to account for both execution costs and the lasting market impact of trades. Prices are obtained through the indifference pricing approach. Numerical examples are provided, along with comparisons to standard methods.

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arXiv2014

Optimal execution and block trade pricing: a general framework

In this article, we develop a general framework to study optimal execution and to price block trades. We prove existence of optimal liquidation strategies and we provide regularity results for optimal strategies under very general hypotheses. We exhibit a Hamiltonian characterization for the optimal strategy that can be used for numerical approximation. We also focus on the important topic of block trade pricing and we propose a methodology to give a price to financial (il)liquidity. In particular, we provide a closed-form formula for the price of a block trade when there is no time constraint to liquidate.

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arXiv2014

A convex duality method for optimal liquidation with participation constraints

In spite of the growing consideration for optimal execution 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.

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arXiv2014

Accelerated Share Repurchase: pricing and execution strategy

In this article, we consider the 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.

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arXiv2014

VWAP execution and guaranteed VWAP

Optimal liquidation using VWAP strategies has been considered in the literature, though never in the presence of permanent market impact and only rarely with execution costs. Moreover, only VWAP strategies have been studied and the pricing of guaranteed VWAP contracts has never been addressed. In this article, we develop a model to price guaranteed VWAP contracts in a general framework for market impact and we highlight the differences between an agency VWAP and a guaranteed VWAP contract. Numerical methods and applications are also provided.

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