Academic literature on the topic 'Optimal liquidation portfolio'

Market making and optimal portfolio liquidation in the context of electronic limit order books are of considerably practical importance for high frequency (HF) market makers as well as more traditional brokerage firms supplying optimal execution services for clients. In general, the two problems are based on probabilistic models defined on certain reference probability spaces. However, due to uncertainty in model parameters or in periods of extreme market turmoil, ambiguity concerning the correct underlying probability measure may appear and an assessment of model risk, as well as the uncertainty on the choice of the model itself, becomes important, as for a market maker or a trader attempting to liquidate large positions, the uncertainty may result in unexpected consequences due to severe mispricing. This paper focuses on the market making and the optimal liquidation problems using limit orders, accounting for model risk or uncertainty. Both are formulated as stochastic optimal control problems, with the controls being the spreads, relative to a reference price, at which orders are placed. The models consider uncertainty in both the drift and volatility of the underlying reference price, for the study of the effect of the uncertainty on the behavior of the market maker, accounting also for inventory restriction, as well as on the optimal liquidation using limit orders.

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 paper, 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 C1,1 regularity while classical methods require additional regularity and cannot be applied to all cases found in practice.

AbstractThis study investigates a combined optimal financing, reinsurance and dividend distribution problem for a big insurance portfolio. A manager can control the surplus by buying proportional reinsurance, paying dividends and raising money dynamically. The transaction costs and liquidation values at bankruptcy are included in the risk model. Under the objective of maximising the insurance company's value, we identify the insurer's joint optimal strategies using stochastic control methods. The results reveal that managers should consider financing if and only if the terminal value and the transaction costs are not too high, less reinsurance is bought when the surplus increases or dividends are always distributed using the barrier strategy.

This thesis studies the Hamilton-Jacobi-Ballman quasivariational inequality (HJBQVI), the corresponding optimal value function, and discrete schemes useful for approximating this value function. Moreover, the structural properties of the optimal policy of particular discrete scheme is studied. The motivation is to find a convergent, approximating scheme for the otherwise complicated HJBQVI that has monotone policy structure that can be exploited in a stochastic gradient estimation scheme to approximate optimal policy function parameters. In order to motivate this approach, we consider the problem of optimal liquidation of a single risky asset portfolio as an impulse control problem. The model is defined over continuous time, state, and compact action sets, and the optimal liquidation value and strategy are found from the viscosity solution of a HJBQVI. It is shown that the optimal strategy is monotone in the number of shares owned and the time remaining to liquidation. This structural result is exploited to estimate the optimal policy via a reinforcement learning method based on the simultaneous perturbation stochastic approximation (SPSA) algorithm. The optimal policy can be estimated without knowledge of the parameters of the underlying model.
Applied Science, Faculty of
Electrical and Computer Engineering, Department of
Graduate

Diese Dissertation analysiert BSDEs und PDEs mit singulären Endbedingungen, welche in Problemen der optimalen Portfolioliquidierung auftreten. In den vergangenen Jahren haben Portfolioliquidierungsprobleme in der Literatur zur Finanzmathematik große Aufmerksamkeit erhalten. Ihre wichtigste Eigenschaft ist die singuläre Endbedingung der durch die Liquidierungsbedingung induzierten Wertfunktion, welche eine singuläre Endbedingung der zugehörigen BSDE oder PDE impliziert. Diese Arbeit besteht aus drei Kapiteln. Das erste Kapitel analysiert ein Portfolioliquidierungsproblem für mehrere Wertpapiere mit sofortigem und anhaltendem Preiseinfluss und stochastischer Resilienz. Wir zeigen, dass die Wertfunktion durch eine mehrdimensionale BSRDE mit singulärer Endbedingung beschrieben werden kann. Wir weisen die Existenz einer Lösung dieser BSRDE nach und zeigen, dass diese durch eine Folge von Lösungen von BSRDEs mit endlicher und wachsender Endbedingung approximiert werden kann. Eine neue a priori-Abschätzung für die approximierenden BSRDEs wird für den Nachweis hergeleitet. Das zweite Kapitel betrachtet ein Portfolioliquidierungsproblem mit unbeschränkten Kostenkoeffizienten. Wir weisen die Existenz einer eindeutigen nichtnegativen Viskositätslösung der HJB-Gleichung nach. Das Existenzresultat basiert auf einem neuartigen Vergleichsprinzip für semi-stetige Viskositätssub-/-superlösungen für singuläre PDEs. Stetigkeit der Viskositätslösung ist hinreichend für das Verifikationsargument. Im dritten Kapitel untersuchen wir ein optimales Liquidierungsproblem unter Mehrdeutigkeit der Parameter des Preiseinflusses. In diesem Fall kann die Wertfunktion durch die Lösung einer semilinearen PDE mit superlinearem Gradienten beschrieben werden. Zuerst zeigen wir die Existenz einer Viskositätslösung indem wir unser Vergleichsprinzip für singuläre PDEs erweitern. Sodann weisen wir die Regularität mit einer asymptotischen Entwicklung der Lösung am Endzeitpunkt nach.
This dissertation analyzes BSDEs and PDEs with singular terminal condition arising in models of optimal portfolio liquidation. Portfolio liquidation problems have received considerable attention in the financial mathematics literature in recent years. Their main characteristic is the singular terminal condition of the value function induced by the liquidation constraint, which translates into a singular terminal state constraint on the associated BSDE or PDE. The dissertation consists of three chapters. The first chapter analyzes a multi-asset portfolio liquidation problem with instantaneous and persistent price impact and stochastic resilience. We show that the value function can be described by a multi-dimensional BSRDE with a singular terminal condition. We prove the existence of a solution to this BSRDE and show that it can be approximated by a sequence of the solutions to BSRDEs with finite increasing terminal condition. A novel a priori estimate for the approximating BSRDEs is established for the verification argument. The second chapter considers a portfolio liquidation problem with unbounded cost coefficients. We establish the existence of a unique nonnegative continuous viscosity solution to the HJB equation. The existence result is based on a novel comparison principle for semi-continuous viscosity sub-/supersolutions for singular PDEs. Continuity of the viscosity solution is enough to carry out the verification argument. The third chapter studies an optimal liquidation problem under ambiguity with respect to price impact parameters. In this case the value function can be characterized by the solution to a semilinear PDE with superlinear gradient. We first prove the existence of a solution in the viscosity sense by extending our comparison principle for singular PDEs. Higher regularity is then established using an asymptotic expansion of the solution at the terminal time.

La programmation mathématique fournit un cadre pour l'étude et la résolution des problèmes d'optimisation contraints ou non. Elle constitue une branche active des mathématiques appliquées, depuis la deuxième moitié du XXème siècle.L'objet de cette thèse est la résolution d'un programme mathématique non convexe non linéaire en variables entières, sous contrainte linéaire d'égalité. Le problème proposé, bien qu'abordé dans cette étude uniquement pour le cas déterministe, trouve son origine en finance, sous le nom d'écoulement de larges blocs d'actifs, ou de liquidation optimale de portefeuille. Il consiste à vendre une (très large) quantité M donnée d'un actif financier en temps fini (discrétisé en N instants) en maximisant le produit de cette vente. A chaque instant, le prix de vente est modélisé par une fonction de pénalité qui reflète le comportement antagoniste du marché face à l'écoulement progressif.Du point de vue, de la programmation mathématique, cette classe de problème est NP-difficile résoudre d'après Garey et Johson, car la non-convexité de la fonction objectif impose d'adapter les méthodes classiques de résolutions (Branch and Bound , coupes) en variables entières. De plus, comme on ne connait pas de méthode de résolution générale pour cette classe de problèmes, les méthodes utilisées doivent être adaptées aux spécificités du problème.La première partie de cette thèse est consacrée à la résolution exacte ou approchée utilisant la programmation dynamique. Nous montrons en effet, que l'équation de Bellman s'applique au problème proposé et permet ainsi de résoudre exactement et rapidement les petites instances. Pour les moyennes et grandes instances, où la programmation dynamique n'est plus disponible et/ou performante, nous proposons des bornes inférieures via différentes heuristiques utilisant la programmation dynamique ainsi que des méthodes de recherche locale, dont nous étudions la qualité (optimalité, temps CPU) et la complexité.La seconde partie de la thèse s'intéresse à la reformulation équivalente du problème de thèse sous forme factorisée et à sa relaxation convexe via les inégalités de McCormick. Nous proposons alors deux algorithmes de résolution exacte du type Branch and Bound, qui fournissent l'optimum global ou un encadrement en temps limité.Dans une troisième partie, dédiée aux expérimentations numériques, nous comparons les méthodes de résolutions proposées entre elles et aux solvers de l'état de l'art. Nous observons notamment que les bornes obtenues sont souvent proches et mêmes parfois meilleures que celles des solvers libres ou commerciaux auxquels nous nous comparons (ex : LocalSolver, Scip, Baron, Couenne et Bonmin).De plus, nous montrons que nos méthodes de résolutions peuvent s'appliquer à toute fonction de pénalité suffisamment régulière et croissante, ce qui comprend notamment des fonctions qui ne sont pas actuellement pas prises en charge par certains solvers, bien qu'elles aient un sens économique pour le problème, comme par exemple les fonctions trigonométriques ou la fonction arctangente.Numériquement, la programmation dynamique permet de résoudre à l'optimum, sous la minute, des instances de taille N<100 et M<10 000. Les heuristiques proposées fournissent de très bonnes bornes inférieures, qui atteignent très souvent l'optimum, pour N<1 000 et M<100 000. Par contraste, la résolution du problème factorisé n'est efficace que pour N< 10, M<1 000, mais nous obtenons des bornes supérieures relativement bonnes. Enfin, pour les grandes instances (M>1 000 000), nos heuristiques à base de programmation dynamique, lorsqu'elles sont disponibles, fournissent les meilleures bornes inférieures, mais nous n'avons pas d'encadrement précis de l'optimum car nos bornes supérieures ne sont pas fines
Mathematical programming provides a framework to study and resolve optimization problems, constrained or not. It represents an active domain of Applied Mathematics, for the second half of the 20th century.The aim of this thesis is to solve an non convex, non linear, pure integer, mathematical program, under a linear constraint of equality. This problem, although studied in this dissertation only in the deterministic case, stems from a financial application, known as the large block sale problem, or optimal portfolio liquidation. It consists in selling a (very large) known quantity M of a financial asset in finite time, discretized in N points in time, while maximizing the proceeds of the sale. At each point in time, the sell price is modeled by a penalty function, which reflects the antagonistic behavior of the market in response to our progressive selling flow.From the standpoint of the mathematical programming, this class of problems is NP-hard to solve according to Garey and Johnson, because the non convexity of the objective function imposes on us to adapt classical resolutions methods (Branch and Bound, cuts) for integer variables. In addition, as no general resolution method for this class of problems is known, the methods used for solving must be adapted to the problem specifics.The first part of the thesis is devoted to solve the problem, either exactly or approximately, using Dynamic Programming. We indeed prove that Bellman's equation applies to the problem studied and thus enables to solve it exactly and quickly for small instances. For medium and large instances, for which Dynamic Programming is either not available and/or efficient, we provide lower bounds using different heuristics relying on Dynamic Programming, or local search methods, for which performance (tightness and CPU time) and complexity are studied.The second part of this thesis focuses on the equivalent reformulation of the problem in a factored form, and on its convex relaxation using McCormick's inequalities. We introduce two exact resolution algorithms, which belongs to the Branch and Bound category. They return the global optimum or bound it in limited time.In a third part, dedicated to numerical experiments, we compare our resolution methods between each other and to state of the art solvers. We notice in particular that our bounds are comparable and sometimes even better than solvers' bounds, both free and commercial (e.g LocalSolver, Scip, Baron, Couenne et Bonmin), which we use as benchmark.In addition, we show that our resolution methods may apply to sufficiently regular and increasing penalty functions, especially functions which are currently not handled by some solvers, even though they make economic sense for the problem, as does trigonometric functions or the arctangent function for instance.Numerically, Dynamic Programming does optimally solve the problem, within a minute, for instances of size N<100 and M< 10 000. Our heuristics provide very tight lower bounds, which often reach the optimum, for N<1 000 and M<100 000. By contrast, optimal resolution of the factored problem proves efficient for instances of size N<10, M<1 000, even though we obtain relatively good upper bounds. Lastly, for large instances (M>1 000 000), our heuristics based on Dynamic Programming, when available, return the best lower bounds. However, we are not able to bound the optimum tightly, since our upper bounds are not thin

We present three chapters on theoretical issues of banking. These deal with bank runs, risk sharing, lending and profitability. In the first chapter, we examine the agency problem in the bank-depositor relationship. Depositors are the principals and banks are the agents. Banks choose investment portfolios and are subject to moral hazard in that they have incentive to take on more risk than desirable to depositors because they are residual claimants. We study an incentive-compatible mechanism that prompts banks to follow a safe investment policy. This mechanism leaves the bank a profit margin in a similar manner to a CEO being paid a bonus by a company. In the second chapter, we extend Allen and Gale (1998) by adding a long-term riskless investment opportunity to the original portfolio of a short-term liquid asset and a long-term risky illiquid asset. Through portfolio diversification, we identify the risk-sharing deposit contract in a three-period model that maximizes the ex-ante expected utility of depositors. Unlike Allen and Gale, there are no information-based bank runs in equilibrium. In addition, our model can improve consumers' welfare over the Allen and Gale model. I also show that the bank will choose to liquidate the cheaper investments, in terms of the gain-loss ratios for the two types of existing long-term assets, when there is liquidity shortage in some cases. Such a policy reduces the liquidation cost and enables the bank to meet the outstanding liability to depositors without large liquidation losses. In the third chapter, we study the role of banks in providing loans to borrower firms. This paper extends the theory of designing optimal loan contracts (for profits) in the Bolton and Scharfstein (1996) model to a setting where asymmetry of information exists. Based on the verifiability of information structure, we analyze complete and incomplete contracts. Through this analysis, optimal, incentive-compatible loan contracts that maximize the expected profit of the bank are characterized. Our analysis suggests that a bank could be induced to liquidate a borrower's project under specific conditions. Furthermore, we identify implementable mechanisms for the renegotiation game given the bargaining power between a borrower and a bank.

This chapter examines the performance of liquidity-adjusted risk modeling in obtaining optimum and coherent economic-capital structures, subject to meaningful operational and financial constraints as specified by the portfolio manager. Specifically, the chapter proposes a robust approach to optimum economic-capital allocation in a liquidity-adjusted value at risk (L-VaR) framework. This chapter expands previous approaches by explicitly modeling the liquidation of trading portfolios, over the holding period, with the aid of an appropriate scaling of the multiple-assets' L-VaR matrix along with GARCH-M technique to forecast conditional volatility and expected return. Moreover, in this chapter, the authors develop a dynamic nonlinear portfolio selection model and an optimization algorithm, which allocates both economic-capital and trading assets by minimizing L-VaR objective function. The empirical results strongly confirm the importance of enforcing financially and operationally meaningful nonlinear and dynamic constraints, when they are available, on the L-VaR optimization procedure.