1,720,973 research outputs found
Free Boundaries, Functional Expansions, and Occupied Processes
No full textThe thesis covers several topics at the intersection of stochastic calculus, numerical analysis, and partial differential equations. The first part revolves around free boundary problems in finance (American options) and physics (Stefan problems). We start by proving continuity, relaxation, and asymptotic properties of hitting times of boundaries in optimal stopping. These results lead to the convergence of the neural optimal stopping boundary method, which is introduced and illustrated with numerous financial examples. The second chapter combines the level-set method with the recent probabilistic formulation of Stefan problems to reproduce the melting or freezing of a material. The level-set function, capturing the evolution of the solid, is estimated by a time-space neural network whose parameters are trained using the probabilistic Stefan growth condition. The algorithm can successfully incorporate supercooling of the liquid and surface tension effects, as shown in the numerical experiments.
The second part sheds light on path dependence. First, we provide a comprehensive study of expansions of functionals, including the Functional Taylor expansion (FTE). Blending the functional Itô calculus and the path signature, the FTE provides a powerful tool to decompose functionals, particularly in the context of exotic derivatives. The final chapter presents an Itô calculus for stochastic processes enlarged by their occupation flows, termed occupied processes. We prove Itô’s and Feynman-Kac’s formula in this context and study a novel class of occupation-dependent stochastic differential equations. The developed theory enable the analysis of a challenging optimal stopping problem involving local times. We finally illustrate the prevalence of the occupied process among financial derivatives and explore promising directions in volatility modeling
Dynamische Absicherungsstrategien in illiquiden Finanzmärkten
No full textIn this thesis, we address the problem of constructing effective hedging strategies against the financial risk of writing a contingent claim in an illiquid financial market. Mathematically, this amounts to study various stochastic optimal control problems with suitable nonlinear dynamics. We introduce a price impact model which accounts for finite market depth, market tightness and finite resilience whose coupled bid- and ask-price dynamics induce convex liquidity costs. We provide existence of an optimal solution to the classical problem of maximizing expected utility from terminal liquidation wealth at some finite planning horizon. In a specific configuration of our model, it turns out that the resulting singular optimal stochastic control problem reduces to a deterministic singular control problem. Rather than studying the associated Hamilton-Jacobi-Bellmann PDE, we exploit convex analytic and calculus of variations techniques which allow us to construct the solution explicitly and to describe analytically the free boundaries of the action- and non-action regions in the underlying state space. In the second part, we relate the optimal singular stochastic control problem of utility-based hedging in our original illiquid market model to a considerably simpler classical linear quadratic stochastic optimal tracking problem of a frictionless hedging strategy with constant coefficients. We solve this problem explicitly for general predictable target hedging strategies. The consideration of general predictable reference processes is made possible by the use of a convex analytic approach along the lines of Pontryagin's maximum principle instead of the more common dynamic programming methods. From a financial point of view, our results allow for an intuitively appealing interpretation and yield sensible hedging strategies in illiquid markets. In the third part, we provide a probabilistic formulation of and solution to a more general class of linear quadratic stochastic tracking problems with stochastic coefficients and stochastic terminal state constraint. Proposing a suitable time consistent approximation of the optimization problem allows us to tackle the final state constraint which induces singular terminal conditions on the underlying backward stochastic differential equations (BSDEs). Our approach also allows us to provide necessary and sufficient conditions under which the constrained stochastic optimization problem admits a finite value. We show that the optimal policy is given by a similar form to the one obtained in the constant coefficient case.Diese Arbeit beschäftigt sich mit der Konstruktion effektiver Absicherungsstrategien gegen die finanziellen Risiken, die beim Verkauf von Finanzoptionen in illiquiden Finanzmärkten entstehen. Mathematisch bedeutet dies die Analyse verschiedener stochastischer optimaler Kontrollprobleme mit nicht-linearen Dynamiken. Wir führen ein Preiseinflussmodell ein, das sogenannte endliche Markttiefe, -enge sowie -elastizität berücksichtigt und dessen gekoppelten Dynamiken der Geld- und Briefkurse konvexe Liquiditätskosten erzeugen. Wir zeigen die Existenz einer optimalen Lösung für das klassische Nutzenmaximierungsproblem des erwarteten Endvermögens bei endlichem Investitionszeitraum. In einer bestimmten Modellkonfiguration zeigt sich, dass sich das ergebende optimale singuläre stochastische Kontrollproblem auf ein singuläres deterministisches Kontrollproblem zurückführen lässt. Anstelle die dazugehörige Hamilton-Jacobi-Bellmann PDGL zu untersuchen, verfolgen wir einen konvex-analytischen Variationsrechnungsansatz der uns schließlich erlaubt, die Lösung explizit zu bestimmen und die freien Ränder der aktiven und passiven Kontrollregionen im zugrundeliegenden Zustandsraum analytisch zu beschreiben. Im zweiten Teil führen wir das optimale singuläre stochastische Kontrollproblem zur Berechnung nutzenbasierter Absicherungsstrategien in unserem ursprünglichen illiquiden Finanzmarktmodell auf ein erheblich einfacheres klassisches linear-quadratisches stochastisches optimales Zielverfolgungsproblem einer friktionslosen optimalen Absicherungsstrategie mit konstanten Koeffizienten zurück. Wir lösen dieses Problem explizit für allgemeine vorhersehbare Absicherungsstrategien als Zielstrategie. Die Betrachtung allgemeiner Referenzstrategien wird anstatt der üblicheren dynamischen Programmierungsmethoden durch einen konvex-analytischen Ansatz ähnlich zu Pontryagins Maximumsprinzip ermöglicht. Aus finanztechnischer Sicht erlauben unsere Resultate eine intuitiv ansprechende Interpretation und beschreiben sinnvolle Absicherungsmöglichkeiten in illiquiden Finanzmärkten. Im dritten Teil stellen wir eine probabilistische Formulierung sowie Lösung einer allgemeineren Klasse stochastischer linear-quadratischer optimaler Zielverfolgungsprobleme mit stochastischen Koeffizienten sowie stochastischer Endbedingung vor. Um die stochastische Endbedingung, die zu singulären Endwerten in den zugrundeliegenden stochastischen Rückwärtsgleichungen führt, mathematisch handhabbarer zu machen, schlagen wir eine geeignete zeitkonsistente Approximation des Optimierungsproblems vor. Unsere Vorgehensweise erlaubt die Bestimmung von notwendigen und hinreichenden Bedingungen unter denen das Optimierungsproblem einen endlichen Lösungswert besitzt. Zudem erweist sich, dass die optimale Lösung von ähnlicher Form ist wie im vorherigen Fall mit konstanten Koeffizienten
Optimal investment strategies with a reallocation constraint
No full textWe study a Merton type optimization problem under a reallocation constraint. Under this restriction, the stock holdings can not be liquidated faster than a certain rate. This is a common restriction in certain type of investment firms. Our main objective is to study the large time optimal growth rate of the expected value of the utility from wealth. We also consider a discounted infinite horizon problem as a step towards understanding the first problem. A numerical study is done by solving the dynamic programming equations. Under the assumption of a power utility function, an appropriate dimension reduction argument is used to reduce the original problem to a two dimensional one in a bounded domain with convenient boundary conditions. Computation of the optimal growth rate introduces additional numerical difficulties as the straightforward approach is unstable. In this direction, new analytical results characterizing the growth rate as the limit of a sequence of finite horizon problems with continuously derived utility are proved
Going Beyond Counting First Authors in Author Co-citation Analysis
Get PDFThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Options hedging under liquidity costs
Get PDFFollowing the framework of Cetin, Jarrow and Protter (CJP) we study the problem of super-replication in presence of liquidity costs under additional restrictions on the gamma of the hedging strategies in a generalized Black-Scholes economy. We find that the minimal super-replication price is different than the one suggested by the Black-Scholes formula and is the unique viscosity solution of the associated dynamic programming equation. This is in contrast with the results of CJP who find that the arbitrage free price of a contingent claim coincides with the Black-Scholes price. However, in CJP a larger class of admissible portfolio processes is used and the replication is achieved in the L^2 approximating
sense
Variations on the Author
Get PDF“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
Liquidity in a binomial market
No full textWe study the binomial version of the illiquid market model introduced by Cetin, Jarrow, and Protter for continuous time and develop efficient numerical methods for its analysis. In particular, we characterize the liquidity premium that results from the model. In Cetin, Jarrow, and Protter, the arbitrage free price of a European option traded in this illiquid market is equal to the classical value. However, the corresponding hedge does not exist and the price is obtained only in L2-approximating sense. Cetin, Soner, and Touzi investigated the super-replication problem using the same supply curve model but under some restrictions on the trading strategies. They showed that the super-replicating cost differs from the Black-Scholes value of the claim, thus proving the existence of liquidity premium. In this paper, we study the super-replication problem in discrete time but with no assumptions on the portfolio process. We recover the same liquidity premium as in the continuous-time limit. This is an independent justification of the restrictions introduced in Cetin, Soner, and Touzi. Moreover, we also propose an algorithm to calculate the options price for a binomial market
Appropriate Similarity Measures for Author Cocitation Analysis
Get PDFWe provide a number of new insights into the methodological discussion about author cocitation analysis. We first argue that the use of the Pearson correlation for measuring the similarity between authors’ cocitation profiles is not very satisfactory. We then discuss what kind of similarity measures may be used as an alternative to the Pearson correlation. We consider three similarity measures in particular. One is the well-known cosine. The other two similarity measures have not been used before in the bibliometric literature. Finally, we show by means of an example that our findings have a high practical relevance.information science;Pearson correlation;cosine;similarity measure;author cocitation analysis
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