Гарантированный детерминистский подход к суперхеджированию: свойства бинарного европейского опциона

Для задачи суперрепликации с дискретным временем рассматривается гарантированная детерминистская постановка: задача состоит в гаран- тированном покрытии обусловленного обязательства по опциону при всех допустимых сценариях. Эти сценарии задаются при помощи апри- орно заданных компактов, зависящих от предыстории цен: прираще- ния цены в каждый момент времени должны лежать в соответству- ющих компактах. В общем случае рассматривается рынок с торговы- ми ограничениями и предполагается отсутствие транзакционных издер- жек. Постановка задачи носит теоретико-игровой характер и приводит к уравнениям Беллмана – Айзекса. В настоящей статье анализируется решение этих уравнений для конкретной задачи ценообразования — для бинарного опциона европейского типа, в рамках мультипликатив- ной модели рынка, при отсутствии торговых ограничений. Получен ряд свойств решения и алгоритм численного решения уравнений Беллма- на. Интерес к этой задаче, с математической точки зрения, связан с разрывностью функции выплат по опцион

Idempotent structures in optimization

Consider the set A = R ∪ {+∞} with the binary operations o1 = max and o2 = + and denote by An the set of vectors v = (v1,...,vn) with entries in A. Let the generalised sum u o1 v of two vectors denote the vector with entries uj o1 vj , and the product a o2 v of an element a ∈ A and a vector v ∈ An denote the vector with the entries a o2 vj . With these operations, the set An provides the simplest example of an idempotent semimodule. The study of idempotent semimodules and their morphisms is the subject of idempotent linear algebra, which has been developing for about 40 years already as a useful tool in a number of problems of discrete optimisation. Idempotent analysis studies infinite dimensional idempotent semimodules and is aimed at the applications to the optimisations problems with general (not necessarily finite) state spaces. We review here the main facts of idempotent analysis and its major areas of applications in optimisation theory, namely in multicriteria optimisation, in turnpike theory and mathematical economics, in the theory of generalised solutions of the Hamilton-Jacobi Bellman (HJB) equation, in the theory of games and controlled Marcov processes, in financial mathematics

Trajectory Based Market Models: Evaluation of Minmax Price Bounds

The paper studies sub and super-replication price bounds for contingent claims defined on general trajectory based market models. No prior probabilistic or topological assumptions are placed on the trajectory space which is of unrestricted cardinality. For a given option, there exists an interval bounding the set of possible fair prices; such interval exists under more general conditions than the usual no-arbitrage requirement. The paper develops a backward recursive method to evaluate the option bounds together with the associated hedging strategies; the global minmax optimization, defining the price interval, is reduced to a local minmax optimization via dynamic programming. Trajectory sets are introduced for which existing probabilistic and non-probabilistic market models are nested as particular cases. Several examples are presented, the effect of the presence of arbitrage on the price bounds is illustrated.Fil: Degano, Iván Leonardo. Universidad Nacional de Mar del Plata. Facultad de Ciencias Exactas y Naturales; ArgentinaFil: Sebastián E. Ferrando. Ryerson University; CanadáFil: Alfredo L, González. Universidad Nacional de Mar del Plata. Facultad de Ciencias Exactas y Naturales; Argentin

The largest eigenvalue of nonnegative tensors

In this thesis we study the methods for finding the largest eigenvalue of tensors. In particular, we study the convergence of the methods and show that the method for rectangular tensors is Q-linear convergence under weak irreducibility condition. We further generalise the method to nonnegative polynomial eigenvalue problems. We prove that this method is convergent for nonhomogeneous irreducible nonnegative polynomials. Then, we apply this method to solve nonnegative polynomial optimization problems with spherical constraints

Arbitraje y cubrimiento. Enfoque no probabilístico

La presente tesis estudia el intervalo de precios justos para un derivado nanciero a partir de un conjunto general de trayectorias de precios de los activos que componen una cartera de inversión. Sobre este conjunto no se asume ninguna hipótesis probabilística ni topológica. El enfoque permite que las negociaciones se produzcan un número nito de veces, no necesariamente jo ni igualmente espaciados en el tiempo. Se de - ne a los mercados con estas características como mercados trayectoriales, permitiendo que una gran variedad de modelos no probabilísticos existentes sean vistos como casos particulares de los mismos. El trabajo provee resultados sobre ausencia de arbitraje, basados únicamente en las propiedades del conjunto de trayectorias, y estudia la invarianza de esta condición bajo transformaciones sobre el mercado. Se de ne el concepto de mercado 0-neutral y se relaciona con las condiciones del mercado conocidas. Para un derivado dado, existe un intervalo de posibles precios justos bajo la hipótesis de mercado 0-neutral, que es más general que el requerimiento usual de no arbitraje. Las cotas de este intervalo quedan dadas por una optimización minimax global. Varias propiedades de estas cotas son presentadas. La tesis desarrolla un método recursivo para evaluarlas, el cual consiste en reducir la optimización global a optimizaciones minimax locales por medio de programación dinámica. El trabajo incluye varios ejemplos y varias salidas numéricas que ilustran su aplicabilidad. Como particularidad se muestra numéricamente el efecto que tiene la presencia de oportunidades de arbitraje en las cotas del intervalo de precios.Fil: Degano, Iván Leonardo. Universidad Nacional de Mar del Plata. Facultad de Ciencias Exactas y Naturales; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Mar del Plata; Argentin

Nonexpansive maps and option pricing theory

summary:The famous Black–Sholes (BS) and Cox–Ross–Rubinstein (CRR) formulas are basic results in the modern theory of option pricing in financial mathematics. They are usually deduced by means of stochastic analysis; various generalisations of these formulas were proposed using more sophisticated stochastic models for common stocks pricing evolution. In this paper we develop systematically a deterministic approach to the option pricing that leads to a different type of generalisations of BS and CRR formulas characterised by more rough assumptions on common stocks evolution (which are therefore easier to verify). On the other hand, this approach is more elementary, because it uses neither martingales nor stochastic equations

Game theoretic analysis of incomplete markets : emergence of probabilities, nonlinear and fractional Black–Scholes equations

Expanding the ideas of the author's paper “Nonexpansive maps and option pricing theory” [Kibernetica 34(6) (1998), 713–724] we develop a pure game-theoretic approach to option pricing, by-passing stochastic modeling. Risk neutral probabilities emerge automatically from the robust control evaluation. This approach seems to be especially appealing for incomplete markets encompassing extensive, so to say untamed, randomness, when the coexistence of infinite number of risk neutral measures precludes one from unified pricing of derivative securities. Our method is robust enough to be able to accommodate various markets rules and settings including path dependent payoffs, American options and transaction costs. On the other hand, it leads to rather simple numerical algorithms. Continuous time limit is described by nonlinear and/or fractional Black–Scholes type equations