Nonlinear models in option pricing --- An introduction

Nonlinear Black-Scholes equations have been increasingly attracting interest over the last two decades, since they provide more accurate values by taking into account more realistic assumptions, such as transaction costs, risks from an unprotected portfolio, large investor's preferences or illiquid markets, which may have an impact on the stock price, the volatility, the drift and the option price itself. This book consists of a collection of contributed chapters of well-known outstanding scientists working successfully in this challenging research area. It discusses concisely several models from the most relevant class of nonlinear Black-Scholes equations for European and American options with a volatility depending on different factors, such as the stock price, the time, the option price and its derivatives. We will present in this book both analytical techniques and numerical methods to solve adequately the arising nonlinear equations. The purpose of this book is to give an overview on the current state-of-the-art research on nonlinear option pricing. The intended audience is on the one hand graduate and Ph.D. students of (mathematical) finance and on the other hand lecturer of mathematical finance and and people working in banks and stock markets that are interested in new tools for option pricing

Absorbing boundary conditions for hyperbolic systems

This paper deals with absorbing boundary conditions for hyperbolic systems in one and two space dimensions. We prove the strict well-posedness of the resulting initial boundary value problem in 1D. Afterwards we establish the GKS-stability of the corresponding Lax-Wendroff-type finite difference scheme. Hereby, we have to extend the classical proofs, since the (discretized) absorbing boundary conditions do not fit the standard form of boundary conditions for hyperbolic systems

Nonlinear models in option pricing : an introduction

Nonlinear Black-Scholes equations have been increasingly attracting interest over the last two decades, since they provide more accurate values by taking into account more realistic assumptions, such as transaction costs, risks from an unprotected portfolio, large investor's preferences or illiquid markets, which may have an impact on the stock price, the volatility, the drift and the option price itself. This book consists of a collection of contributed chapters of well-known outstanding scientists working successfully in this challenging research area. It discusses concisely several models from the most relevant class of nonlinear Black-Scholes equations for European and American options with a volatility depending on different factors, such as the stock price, the time, the option price and its derivatives. We will present in this book both analytical techniques and numerical methods to solve adequately the arising nonlinear equations. The purpose of this book is to give an overview on the current state-of-the-art research on nonlinear option pricing. The intended audience is on the one hand graduate and Ph.D. students of (mathematical) finance and on the other hand lecturer of mathematical finance and and people working in banks and stock markets that are interested in new tools for option pricing

Problem perception and public expectations in international institutions: evidence from a German representative survey

"Much of what can be subsumed under the label of 'political cosmopolitanism' argues that some internationalization of political authority is desirable because of the superior capability of international institutions in solving transnational problems. Surprisingly, however, few scholars have asked whether ordinary citizens share this way of thinking. To address this question, falsifiable hypotheses about the quality and scope conditions of such a 'public political cosmopolitanism' are derived from the literature and tested using the results of a representative survey of German citizens. I show that there is significant support for what I call the 'interdependence model' of cosmopolitan politicization: German citizens' perception of transnational interdependencies (in terms of functional sensitivity as well as moral commitments) fosters beliefs in the capability of international institutions to solve problems. This model has significant explanatory power over different levels of education, and thus disproves a common claim that cognitive mobilization is a crucial scope condition of cosmopolitan politicization. Remarkably, however, the relationships specified by the interdependence model are only marginally moderated by German citizens' sense of their own vulnerability, i.e. their beliefs that the national government is incapable of solving such problems." (author's abstract)"Viele Arbeiten zum 'politischen Kosmopolitismus' begründen ihre Präferenz für starke internationale Institutionen mit deren Fähigkeit transnationale Probleme besser zu lösen. Überraschender Weise hat die Frage bislang kaum wissenschaftliche Aufmerksamkeit erfahren inwiefern die Bürger ähnlich denken. Das Papier leitet Thesen zu Struktur und Bedingungen eines solchen 'öffentlichen politischen Kosmopolitismus' ab und testet sie mit Hilfe einer Repräsentativumfrage deutscher Bürger. Es wird gezeigt, dass deren Einstellungen tatsächlich einem 'Interdependenzmodell' kosmopolitisischer Politisierung folgen. Die Wahrnehmung transnationaler Interdependenz (sowohl im Sinne funktionaler wie auch moralischer Bindungen) fördert Überzeugungen, dass internationale Institutionen in der Lage sind daraus resultierende Probleme zu lösen. Das Interdependenzmodell zeigt auch über verschiedene Bildungsniveaus ein hohes Maß an Erklärungskraft und widerspricht damit der herrschenden Lehrmeinung, der zufolge kognitive Mobilisierung eine kritische Bedingung für kosmopolitischer Politisierung darstellt. Bemerkenswerter Weise, sind die im Interdependenzmodell spezifizierten Zusammenhänge zudem kaum von variierenden Graden subjektiver Vulnerabilität abhängig, d.h. der Wahrnehmung mangelnder Problemlösungsfähigkeit des Nationalstaats." (Autorenreferat

Public Legitimation by "Going Personal"? The Ambiguous Role of International Organization Officials on Social Media

International organizations increasingly use social media to target citizens with an abundance of content, which tends to stylize officials across ranks as the "personal face" of institutional processes. Such practices suggest a new degree of access to the every day of multilateralism that has traditionally taken place on camera and with the aid of diplomatic discretion. What is more, in these practices the intuitive truth of images on social media often blends with a more credible expression of emotional states - such as enthusiasm, sympathy, anger, or shame - which facilitates the legitimation of international organizations as credible agents of shared values and norms. At the same time, however, such personalization arguably suggests a problematic dependency on the credible conduct of international organization officials as it might undermine institutional claims to depersonalized "rational-legal" authority in international politics and local arenas of implementation alike. Also, it aggravates existing problems of decoupling action in global governance from its political symbolism, because international organizations use social media by and large to communicate "top-down," despite claiming a more personal mode of communication among peers. To illustrate this argument, the article takes on content shared by leading officials of the UN, the IMF, the WHO, and the WTO on Twitter

Implementing exact absorbing boundary condition for the linear one-dimensional Schrödinger problem with variable potential by Titchmarsh-Weyl theory

A new approach for simulating the solution of the time-dependent Schrödinger equation with a general variable potential will be proposed. The key idea is to approximate the Titchmarsh-Weyl m-function (exact Dirichlet-to-Neumann operator) by a rational function with respect to a suitable spectral parameter. With the proposed method we can overcome the usual high-frequency restriction for absorbing boundary conditions of general variable potential problems. We end up with a fast computational algorithm for absorbing boundary conditions that are accurate for the full frequency ban

Fast numerical methods for waves in periodic media

Periodic media problems widely exist in many modern application areas like semiconductor nanostructures (e.g.\ quantum dots and nanocrystals), semi-conductor superlattices, photonic crystals (PC) structures, meta materials or Bragg gratings of surface plasmon polariton (SPP) waveguides, etc. Often these application problems are modeled by partial differential equations with periodic coefficients and/or periodic geometries. In order to numerically solve these periodic structure problems efficiently one usually confines the spatial domain to a bounded computational domain (i.e.\ in a neighborhood of the region of physical interest). Hereby, the usual strategy is to introduce so-called \emph{artificial boundaries} and impose suitable boundary conditions. For wave-like equations, the ideal boundary conditions should not only lead to well-posed problems, but also mimic the perfect absorption of waves traveling out of the computational domain through the artificial boundaries. In the first part of this chapter we present a novel analytical impedance expression for general second order ODE problems with periodic coefficients. This new expression for the kernel of the Dirichlet-to-Neumann mapping of the artificial boundary conditions is then used for computing the bound states of the Schr\"odinger operator with periodic potentials at infinity. Other potential applications are associated with the exact artificial boundary conditions for some time-dependent problems with periodic structures. As an example, a two-dimensional hyperbolic equation modeling the TM polarization of the electromagnetic field with a periodic dielectric permittivity is considered. In the second part of this chapter we present a new numerical technique for solving periodic structure problems. This novel approach possesses several advantages. First, it allows for a fast evaluation of the Sommerfeld-to-Sommerfeld operator for periodic array problems. Secondly, this computational method can also be used for bi-periodic structure problems with local defects. In the sequel we consider several problems, such as the exterior elliptic problems with strong coercivity, the time-dependent Schr\"odinger equation and the Helmholtz equation with damping. Finally, in the third part we consider periodic arrays that are structures consisting of geometrically identical subdomains, usually called periodic cells. We use the Helmholtz equation as a model equation and consider the definition and evaluation of the exact boundary mappings for general semi-infinite arrays that are periodic in one direction for any real wavenumber. The well-posedness of the Helmholtz equation is established via the \emph{limiting absorption principle} (LABP). An algorithm based on the doubling procedure of the second part of this chapter and an extrapolation method is proposed to construct the exact Sommerfeld-to-Sommerfeld boundary mapping. This new algorithm benefits from its robustness and the simplicity of implementation. But it also suffers from the high computational cost and the resonance wave numbers. To overcome these shortcomings, we propose another algorithm based on a conjecture about the asymptotic behaviour of limiting absorption principle solutions. The price we have to pay is the resolution of some generalized eigenvalue problem, but still the overall computational cost is significantly reduced. Numerical evidences show that this algorithm presents theoretically the same results as the first algorithm. Moreover, some quantitative comparisons between these two algorithms are given