0000000000160892
AUTHOR
Ansgar Jüngel
Die Monte-Carlo-Methode
Der Preis einer europ¨aischen (Plain-vanilla) Option kann mit der Black-Scholes-Formel aus Abschnitt 4.2 berechnet werden. Leider existieren zu komplexeren Optionen im allgemeinen keine expliziten Formeln mehr. In diesem Abschnitt stellen wir die Monte-Carlo-Methode zur Integration von stochastischen Differentialgleichungen vor, mit der faire Preise von komplizierten Optionsmodellen numerisch berechnet werden k¨onnen. Zuerst f¨uhren wir in Abschnitt 5.1 in die Thematik ein. Das Monte-Carlo-Verfahren erfordert die Simulation von Realisierungen eines Wiener-Prozesses. Die Simulation wiederum ben¨otigt normalverteilte Zufallszahlen. Die Erzeugung von Zufallszahlen ist Gegenstand von Abschnitt …
Numerical Simulation of Thermal Effects in Coupled Optoelectronic Device-circuit Systems
The control of thermal effects becomes more and more important in modern semiconductor circuits like in the simplified CMOS transceiver representation described by U. Feldmann in the above article Numerical simulation of multiscale models for radio frequency circuits in the time domain. The standard approach for modeling integrated circuits is to replace the semiconductor devices by equivalent circuits consisting of basic elements and resulting in so-called compact models. Parasitic thermal effects, however, require a very large number of basic elements and a careful adjustment of the resulting large number of parameters in order to achieve the needed accuracy.
Multi-Scale Modeling of Quantum Semiconductor Devices
This review is concerned with three classes of quantum semiconductor equations: Schrodinger models, Wigner models, and fluid-type models. For each of these classes, some phenomena on various time and length scales are presented and the connections between micro-scale and macro-scale models are explained. We discuss Schrodinger-Poisson systems for the simulation of quantum waveguides and illustrate the importance of using open boundary conditions. We present Wigner-based semiconductor models and sketch their mathematical analysis. In particular we discuss the Wigner-Poisson-Focker-Planck system, which is the starting point of deriving subsequently the viscous quantum hydrodynamic model. Furt…
Convergence of a high-order compact finite difference scheme for a nonlinear Black–Scholes equation
A high-order compact finite difference scheme for a fully nonlinear parabolic differential equation is analyzed. The equation arises in the modeling of option prices in financial markets with transaction costs. It is shown that the finite difference solution converges locally uniformly to the unique viscosity solution of the continuous equation. The proof is based on a careful study of the discretization matrices and on an abstract convergence result due to Barles and Souganides.
A Sequential Quadratic Programming Method for Volatility Estimation in Option Pricing
Our goal is to identify the volatility function in Dupire's equation from given option prices. Following an optimal control approach in a Lagrangian framework, we propose a globalized sequential quadratic programming (SQP) algorithm with a modified Hessian - to ensure that every SQP step is a descent direction - and implement a line search strategy. In each level of the SQP method a linear-quadratic optimal control problem with box constraints is solved by a primal-dual active set strategy. This guarantees L^1 constraints for the volatility, in particular assuring its positivity. The proposed algorithm is founded on a thorough first- and second-order optimality analysis. We prove the existe…
Eine kleine Einführung in MATLAB
In Abschnitt 9.1 stellen wir knapp die Informationen zusammen, die es Leserinnen und Leser ohne Matlab-Kenntnisse ermoglichen sollen, die in diesem Buch entwickelten Matlab-Programme nachzuvollziehen. Alle weiteren, uber die Kurzeinfuhrung hinausgehenden Matlab-Befehle werden im Text jeweils an der Stelle eingefuhrt und erlautert, an der sie benotigt werden. In Abschnitt 9.2 stellen wir drei Matlab-Toolboxen (d.h. Sammlungen von Prozeduren) vor, die fur Finanzanwendungen sehr hilfreich sind.
Analysis of a Parabolic Cross-Diffusion Semiconductor Model with Electron-Hole Scattering
The global-in-time existence of non-negative solutions to a parabolic strongly coupled system with mixed Dirichlet–Neumann boundary conditions is shown. The system describes the time evolution of the electron and hole densities in a semiconductor when electron-hole scattering is taken into account. The parabolic equations are coupled to the Poisson equation for the electrostatic potential. Written in the quasi-Fermi potential variables, the diffusion matrix of the parabolic system contains strong cross-diffusion terms and is only positive semi-definite such that the problem is formally of degenerate type. The existence proof is based on the study of a fully discretized version of the system…
A derivation of the isothermal quantum hydrodynamic equations using entropy minimization
Isothermal quantum hydrodynamic equations of order O(h 2 ) using the quantum entropy minimization method recently developed by Degond and Ringhofer are derived. The equations have the form of the usual quantum hydrodynamic model including a correction term of order O(h 2 ) which involves the vorticity. If the initial vorticity is of order 0(h), the standard model is obtained up to order O(h 4 ). The derivation is based on a careful expansion of the quantum equilibrium obtained from the entropy minimization in powers of h 2 .
A logarithmic fourth-order parabolic equation and related logarithmic Sobolev inequalities
A logarithmic fourth-order parabolic equation in one space dimension with periodic boundary conditions is studied. This equation arises in the context of fluctuations of a stationary nonequilibrium interface and in the modeling of quantum semiconductor devices. The existence of global-in-time non-negative weak solutions and some regularity results are shown. Furthermore, we prove that the solution converges exponentially fast to its mean value in the ``entropy norm'' and in the Fisher information, using a new optimal logarithmic Sobolev inequality for higher derivatives. In particular, the rate is independent of the solution and the constant depends only on the initial value of the entropy.
Finite-element Discretizations of Semiconductor Energy-transport Equations
Energy-transport models describe the flow of electrons in a semiconductor crystal. Several formulations of these models, in the primal or dual entropy variables or in the drift-diffusion-type variables, are reviewed. A numerical discretization of the steady-state drift-diffusion-type formulation using mixed-hybrid finite elements introduced by Marini and Pietra is presented. The scheme is first applied to the simulation of a one-dimensional ballistic diode with non-parabolic band diagrams. Then a two-dimensional deep submicron MOSFET device with parabolic bands is simulated, using an adaptively refined mesh.
SIMULATION OF THERMAL EFFECTS IN OPTOELECTRONIC DEVICES USING COUPLED ENERGY-TRANSPORT AND CIRCUIT MODELS
A coupled model with optoelectronic semiconductor devices in electric circuits is proposed. The circuit is modeled by differential-algebraic equations derived from modified nodal analysis. The transport of charge carriers in the semiconductor devices (laser diode and photo diode) is described by the energy-transport equations for the electron density and temperature, the drift-diffusion equations for the hole density, and the Poisson equation for the electric potential. The generation of photons in the laser diode is modeled by spontaneous and stimulated recombination terms appearing in the transport equations. The devices are coupled to the circuit by the semiconductor current entering the…
Analysis of a parabolic cross-diffusion population model without self-diffusion
Abstract The global existence of non-negative weak solutions to a strongly coupled parabolic system arising in population dynamics is shown. The cross-diffusion terms are allowed to be arbitrarily large, whereas the self-diffusion terms are assumed to disappear. The last assumption complicates the analysis since these terms usually provide H 1 estimates of the solutions. The existence proof is based on a positivity-preserving backward Euler–Galerkin approximation, discrete entropy estimates, and L 1 weak compactness arguments. Furthermore, employing the entropy–entropy production method, we show for special stationary solutions that the transient solution converges exponentially fast to its…
Einige weiterführende Themen
In diesem Kapitel stellen wir einige weiterfuhrende Themen vor. In Abschnitt 8.1 erlautern wir Volatilitatsmodelle. Zinsderivate werden in Abschnitt 8.2 diskutiert. Eine Einfuhrung in die Bewertung von Wetter- und Energiederivaten wird in Abschnitt 8.3 gegeben. Schlieslich definieren und bewerten wir in Abschnitt 8.4 Kreditderivate, die in der Finanzkrise ab 2007 eine grose Rolle gespielt haben, namlich Credit Default Obligations (CDO). Wege zur besseren Beschreibung des Korrelationsrisikos, dessen Unterschatzung eine der Grunde der Finanzkrise war, diskutieren wir im abschliesenden Abschnitt 8.5.
Numerische Lösung parabolischer Differentialgleichungen
In Kapitel 5 haben wir die Monte-Carlo-Methode zur Losung stochastischer Differentialgleichungen kennengelernt und gesehen, dass diese Methode im allgemeinen recht zeit- und rechenintensiv ist. Die Preise exotischer Optionen konnen haufig auch durch die Losung einer partiellen Differentialgleichung vom Black-Scholes-Typ bestimmt werden. Diese Differentialgleichungen konnen allerdings im allgemeinen nicht explizit gelost werden. In diesem Kapitel stellen wir einige Techniken vor, mit denen diese Gleichungen numerisch gelost werden konnen.
Analysis of the viscous quantum hydrodynamic equations for semiconductors
The steady-state viscous quantum hydrodynamic model in one space dimension is studied. The model consists of the continuity equations for the particle and current densities, coupled to the Poisson equation for the electrostatic potential. The equations are derived from a Wigner–Fokker–Planck model and they contain a third-order quantum correction term and second-order viscous terms. The existence of classical solutions is proved for “weakly supersonic” quantum flows. This means that a smallness condition on the particle velocity is still needed but the bound is allowed to be larger than for classical subsonic flows. Furthermore, the uniqueness of solutions and various asymptotic limits (sem…
Convergence of a high-order compact finite difference scheme for a nonlinear Black-Scholes equation
A high-order compact finite difference scheme for a fully nonlinear parabolic differential equation is analyzed. The equation arises in the modeling of option prices in financial markets with transaction costs. It is shown that the finite difference solution converges locally uniformly to the unique viscosity solution of the continuous equation. The proof is based on a careful study of the discretization matrices and on an abstract convergence result due to Barles and Souganides.
Existence and uniqueness of solutions to a quasilinear parabolic equation with quadratic gradients in financial markets
A quasilinear parabolic equation with quadratic gradient terms is analyzed. The equation models an optimal portfolio in so-called incomplete financial markets consisting of risky assets and non-tradable state variables. Its solution allows to compute an optimal portfolio strategy. The quadratic gradient terms are essentially connected to the assumption that the so-called relative risk aversion function is not logarithmic. The existence of weak global-in-time solutions in any dimension is shown under natural hypotheses. The proof is based on the monotonicity method of Frehse. Furthermore, the uniqueness of solutions is shown under a smallness condition on the derivatives of the covariance (?…
Numerische Lösung freier Randwertprobleme
In diesem Kapitel widmen wir uns der Aufgabe, den fairen Preis fur amerikanische Optionen zu berechnen. Wie in Kapitel 1 bereits erklart, raumen amerikanische Optionen im Gegensatz zu europaischen Optionen das Recht ein, die Option zu einem beliebigen Zeitpunkt innerhalb der Laufzeit auszuuben. Aufgrund des zusatzlichen Rechts der vorzeitigen Ausubung ist eine amerikanische Option im Allgemeinen teurer als die entsprechende europaische Option. Der Preis einer amerikanischen Option kann also nicht uber die Black-Scholes-Gleichung bestimmt werden. In Abschnitt 7.1 zeigen wir, dass der Optionspreis durch Losen einer Black-Scholes-Ungleichung berechnet werden kann. Diese Ungleichung hangt mit s…
Numerical approximation of the viscous quantum hydrodynamic model for semiconductors
The viscous quantum hydrodynamic equations for semiconductors with constant temperature are numerically studied. The model consists of the one-dimensional Euler equations for the electron density and current density, including a quantum correction and viscous terms, coupled to the Poisson equation for the electrostatic potential. The equations can be derived formally from a Wigner-Fokker-Planck model by a moment method. Two different numerical techniques are used: a hyperbolic relaxation scheme and a central finite-difference method. By simulating a ballistic diode and a resonant tunneling diode, it is shown that numerical or physical viscosity changes significantly the behavior of the solu…
ANALYSIS OF A SPHERICAL HARMONICS EXPANSION MODEL OF PLASMA PHYSICS
A spherical harmonics expansion model arising in plasma and semiconductor physics is analyzed. The model describes the distribution of particles in the position-energy space subject to a (given) electric potential and consists of a parabolic degenerate equation. The existence and uniqueness of global-in-time solutions is shown by semigroup theory if the particles are moving in a one-dimensional interval with Dirichlet boundary conditions. The degeneracy allows to show that there is no transport of particles across the boundary corresponding to zero energy. Furthermore, under certain conditions on the potential, it is proved that the solution converges in the long-time limit exponentially f…
Die Black-Scholes-Gleichung
In diesem Kapitel leiten wir die Black-Scholes-Gleichung her. Dafur benotigen wir den Begriff der stochastischen Differentialgleichung von Ito, den wir in Abschnitt 4.1 einfuhren. Abschnitt 4.2 befasst sich dann mit der Herleitung der Black-Scholes-Gleichung und deren Losung, den sogenannten Black-Scholes-Formeln. Auf die effiziente numerische Auswertung dieser Formeln gehen wir in Abschnitt 4.3 ein. In Abschnitt 4.4 definieren wir dynamische Kennzahlen und erortern, wie die Volatilitat bestimmt werden kann. Erweiterungen der Black-Scholes-Gleichung stellen wir schlieslich in Abschnitt 4.5 vor.
An algorithmic construction of entropies in higher-order nonlinear PDEs
A new approach to the construction of entropies and entropy productions for a large class of nonlinear evolutionary PDEs of even order in one space dimension is presented. The task of proving entropy dissipation is reformulated as a decision problem for polynomial systems. The method is successfully applied to the porous medium equation, the thin film equation and the quantum drift–diffusion model. In all cases, an infinite number of entropy functionals together with the associated entropy productions is derived. Our technique can be extended to higher-order entropies, containing derivatives of the solution, and to several space dimensions. Furthermore, logarithmic Sobolev inequalities can …
A Quasilinear Parabolic Equation with Quadratic Growth of the Gradient modeling Incomplete Financial Markets
We consider a quasilinear parabolic equation with quadratic gradient terms. It arises in the modeling of an optimal portfolio which maximizes the expected utility from terminal wealth in incomplete markets consisting of risky assets and non-tradable state variables. The existence of solutions is shown by extending the monotonicity method of Frehse. Furthermore, we prove the uniqueness of weak solutions under a smallness condition on the derivatives of the covariance matrices with respect to the solution. The in influence of the non-tradable state variables on the optimal value function is illustrated by a numerical example.
The relaxation-time limit in the quantum hydrodynamic equations for semiconductors
Abstract The relaxation-time limit from the quantum hydrodynamic model to the quantum drift–diffusion equations in R 3 is shown for solutions which are small perturbations of the steady state. The quantum hydrodynamic equations consist of the isentropic Euler equations for the particle density and current density including the quantum Bohm potential and a momentum relaxation term. The momentum equation is highly nonlinear and contains a dispersive term with third-order derivatives. The equations are self-consistently coupled to the Poisson equation for the electrostatic potential. The relaxation-time limit is performed both in the stationary and the transient model. The main assumptions are…
Entropies and Equilibria of Many-Particle Systems: An Essay on Recent Research
International audience; .This essay is intended to present a fruitful collaboration which has developed among a group of people whose names are listed above: entropy methods have proved over the last years to be an efficient tool for the understanding of the qualitative properties of physically sound models, for accurate numerics and for a more mathematical understanding of nonlinear PDEs. The goal of this essay is to sketch the historical development of the concept of entropy in connection with PDEs of continuum mechanics, to present recent results which have been obtained by the members of the group and to emphasize the most striking achievements of this research. The presentation is by n…
High Order Compact Finite Difference Schemes for A Nonlinear Black-Scholes Equation
A nonlinear Black-Scholes equation which models transaction costs arising in the hedging of portfolios is discretized semi-implicitly using high order compact finite difference schemes. A new compact scheme, generalizing the compact schemes of Rigal [29], is derived and proved to be unconditionally stable and non-oscillatory. The numerical results are compared to standard finite difference schemes. It turns out that the compact schemes have very satisfying stability and non-oscillatory properties and are generally more efficient than the considered classical schemes.