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Author: Dmitry Smelov Publisher: ISBN: Category : Languages : en Pages :
Book Description
This thesis treats the problems of exact simulation and parameter inference for jump-diffusion processes. It has two parts. The first part develops a method for the exact simulation of a skeleton, a hitting time and other functionals of a one-dimensional jump-diffusion with state-dependent drift, volatility, jump intensity and jump size. The method requires the drift function to be C1, the volatility function to be C2, and the jump intensity function to be locally bounded. No further structure is imposed on these functions. The method leads to unbiased simulation estimators of security prices, transition densities, hitting probabilities, and other quantities. Numerical results illustrate its features. The second part develops and analyzes likelihood estimators for the parameters of a discretely-observed jump diffusion. We consider the case when the transition density of the process admits an expansion in terms of an infinite series. A randomization technique leads to an unbiased Monte Carlo estimator of the transition density and the likelihood function. We provide conditions under which resulting likelihood estimators are consistent and asymptotically normal. The method avoids the second-order bias of conventional discretization-based estimators. Unlike the estimators based directly on the density expansion, we do not require high-frequency observations. Numerical results confirm the method's properties.
Author: Dmitry Smelov Publisher: ISBN: Category : Languages : en Pages :
Book Description
This thesis treats the problems of exact simulation and parameter inference for jump-diffusion processes. It has two parts. The first part develops a method for the exact simulation of a skeleton, a hitting time and other functionals of a one-dimensional jump-diffusion with state-dependent drift, volatility, jump intensity and jump size. The method requires the drift function to be C1, the volatility function to be C2, and the jump intensity function to be locally bounded. No further structure is imposed on these functions. The method leads to unbiased simulation estimators of security prices, transition densities, hitting probabilities, and other quantities. Numerical results illustrate its features. The second part develops and analyzes likelihood estimators for the parameters of a discretely-observed jump diffusion. We consider the case when the transition density of the process admits an expansion in terms of an infinite series. A randomization technique leads to an unbiased Monte Carlo estimator of the transition density and the likelihood function. We provide conditions under which resulting likelihood estimators are consistent and asymptotically normal. The method avoids the second-order bias of conventional discretization-based estimators. Unlike the estimators based directly on the density expansion, we do not require high-frequency observations. Numerical results confirm the method's properties.
Author: Floyd B. Hanson Publisher: SIAM ISBN: 9780898718638 Category : Mathematics Languages : en Pages : 472
Book Description
This self-contained, practical, entry-level text integrates the basic principles of applied mathematics, applied probability, and computational science for a clear presentation of stochastic processes and control for jump diffusions in continuous time. The author covers the important problem of controlling these systems and, through the use of a jump calculus construction, discusses the strong role of discontinuous and nonsmooth properties versus random properties in stochastic systems.
Author: Eckhard Platen Publisher: Springer Science & Business Media ISBN: 364213694X Category : Mathematics Languages : en Pages : 868
Book Description
In financial and actuarial modeling and other areas of application, stochastic differential equations with jumps have been employed to describe the dynamics of various state variables. The numerical solution of such equations is more complex than that of those only driven by Wiener processes, described in Kloeden & Platen: Numerical Solution of Stochastic Differential Equations (1992). The present monograph builds on the above-mentioned work and provides an introduction to stochastic differential equations with jumps, in both theory and application, emphasizing the numerical methods needed to solve such equations. It presents many new results on higher-order methods for scenario and Monte Carlo simulation, including implicit, predictor corrector, extrapolation, Markov chain and variance reduction methods, stressing the importance of their numerical stability. Furthermore, it includes chapters on exact simulation, estimation and filtering. Besides serving as a basic text on quantitative methods, it offers ready access to a large number of potential research problems in an area that is widely applicable and rapidly expanding. Finance is chosen as the area of application because much of the recent research on stochastic numerical methods has been driven by challenges in quantitative finance. Moreover, the volume introduces readers to the modern benchmark approach that provides a general framework for modeling in finance and insurance beyond the standard risk-neutral approach. It requires undergraduate background in mathematical or quantitative methods, is accessible to a broad readership, including those who are only seeking numerical recipes, and includes exercises that help the reader develop a deeper understanding of the underlying mathematics.
Author: Aurélien Alfonsi Publisher: Springer ISBN: 3319052217 Category : Mathematics Languages : en Pages : 264
Book Description
This book gives an overview of affine diffusions, from Ornstein-Uhlenbeck processes to Wishart processes and it considers some related diffusions such as Wright-Fisher processes. It focuses on different simulation schemes for these processes, especially second-order schemes for the weak error. It also presents some models, mostly in the field of finance, where these methods are relevant and provides some numerical experiments. The book explains the mathematical background to understand affine diffusions and analyze the accuracy of the schemes.
Author: Helmut Neunzert Publisher: Springer ISBN: 3662482584 Category : Mathematics Languages : en Pages : 440
Book Description
This book offers an insider's view of how industrial problems are translated into mathematics and how solving the mathematics leads to convincing industrial solutions as well. In 6 technical chapters, a wide range of industrial problems is modeled, simulated, and optimized; 4 others describe the modeling, computing, optimization, and data analysis concepts shaping the work of the Fraunhofer ITWM. Each technical chapter illustrates how the relevant mathematics has been adapted or extended for the specific application and details the underlying practical problem and resulting software. The final chapter shows how the use of mathematical modeling in the classroom can change the image of this subject, making it exciting and fun.
Author: Yuliya Mishura Publisher: Walter de Gruyter GmbH & Co KG ISBN: 3110654245 Category : Mathematics Languages : en Pages : 390
Book Description
Financial market modeling is a prime example of a real-life application of probability theory and stochastics. This authoritative book discusses the discrete-time approximation and other qualitative properties of models of financial markets, like the Black-Scholes model and its generalizations, offering in this way rigorous insights on one of the most interesting applications of mathematics nowadays.
Author: M. A. H. Dempster Publisher: CRC Press ISBN: 1315354691 Category : Computers Languages : en Pages : 648
Book Description
High-Performance Computing (HPC) delivers higher computational performance to solve problems in science, engineering and finance. There are various HPC resources available for different needs, ranging from cloud computing– that can be used without much expertise and expense – to more tailored hardware, such as Field-Programmable Gate Arrays (FPGAs) or D-Wave’s quantum computer systems. High-Performance Computing in Finance is the first book that provides a state-of-the-art introduction to HPC for finance, capturing both academically and practically relevant problems.
Author: Joerg Kienitz Publisher: John Wiley & Sons ISBN: 0470744898 Category : Business & Economics Languages : en Pages : 736
Book Description
Financial modelling Theory, Implementation and Practice with MATLAB Source Jörg Kienitz and Daniel Wetterau Financial Modelling - Theory, Implementation and Practice with MATLAB Source is a unique combination of quantitative techniques, the application to financial problems and programming using Matlab. The book enables the reader to model, design and implement a wide range of financial models for derivatives pricing and asset allocation, providing practitioners with complete financial modelling workflow, from model choice, deriving prices and Greeks using (semi-) analytic and simulation techniques, and calibration even for exotic options. The book is split into three parts. The first part considers financial markets in general and looks at the complex models needed to handle observed structures, reviewing models based on diffusions including stochastic-local volatility models and (pure) jump processes. It shows the possible risk-neutral densities, implied volatility surfaces, option pricing and typical paths for a variety of models including SABR, Heston, Bates, Bates-Hull-White, Displaced-Heston, or stochastic volatility versions of Variance Gamma, respectively Normal Inverse Gaussian models and finally, multi-dimensional models. The stochastic-local-volatility Libor market model with time-dependent parameters is considered and as an application how to price and risk-manage CMS spread products is demonstrated. The second part of the book deals with numerical methods which enables the reader to use the models of the first part for pricing and risk management, covering methods based on direct integration and Fourier transforms, and detailing the implementation of the COS, CONV, Carr-Madan method or Fourier-Space-Time Stepping. This is applied to pricing of European, Bermudan and exotic options as well as the calculation of the Greeks. The Monte Carlo simulation technique is outlined and bridge sampling is discussed in a Gaussian setting and for Lévy processes. Computation of Greeks is covered using likelihood ratio methods and adjoint techniques. A chapter on state-of-the-art optimization algorithms rounds up the toolkit for applying advanced mathematical models to financial problems and the last chapter in this section of the book also serves as an introduction to model risk. The third part is devoted to the usage of Matlab, introducing the software package by describing the basic functions applied for financial engineering. The programming is approached from an object-oriented perspective with examples to propose a framework for calibration, hedging and the adjoint method for calculating Greeks in a Libor market model. Source code used for producing the results and analysing the models is provided on the author's dedicated website, http://www.mathworks.de/matlabcentral/fileexchange/authors/246981.
Author: Daniel J. Duffy Publisher: John Wiley & Sons ISBN: 0470684062 Category : Business & Economics Languages : en Pages : 775
Book Description
This is one of the first books that describe all the steps that are needed in order to analyze, design and implement Monte Carlo applications. It discusses the financial theory as well as the mathematical and numerical background that is needed to write flexible and efficient C++ code using state-of-the art design and system patterns, object-oriented and generic programming models in combination with standard libraries and tools. Includes a CD containing the source code for all examples. It is strongly advised that you experiment with the code by compiling it and extending it to suit your needs. Support is offered via a user forum on www.datasimfinancial.com where you can post queries and communicate with other purchasers of the book. This book is for those professionals who design and develop models in computational finance. This book assumes that you have a working knowledge of C ++.
Author: Dmitry Smelov
Author: Dmitry Smelov
Author: Lynda Khalaf
Author: Floyd B. Hanson
Author: Eckhard Platen
Author: Aurélien Alfonsi
Author: Helmut Neunzert
Author: Yuliya Mishura
Author: M. A. H. Dempster
Author: Joerg Kienitz
Author: Daniel J. Duffy