Anatoliy Swishchuk - Böcker

The theory and applications of random dynamical systems (RDS) are at the cutting edge of research in mathematics and economics, particularly in modeling the long-run evolution of economic systems subject to exogenous random shocks. Despite this interest, there are no books available that solely focus on RDS in finance and economics. Exploring this emerging area, Random Dynamical Systems in Finance shows how to model RDS in financial applications.Through numerous examples, the book explains how the theory of RDS can describe the asymptotic and qualitative behavior of systems of random and stochastic differential/difference equations in terms of stability, invariant manifolds, and attractors. The authors present many models of RDS and develop techniques for implementing RDS as approximations to financial models and option pricing formulas. For example, they approximate geometric Markov renewal processes in ergodic, merged, double-averaged, diffusion, normal deviation, and Poisson cases and apply the obtained results to option pricing formulas.With references at the end of each chapter, this book provides a variety of RDS for approximating financial models, presents numerous option pricing formulas for these models, and studies the stability and optimal control of RDS. The book is useful for researchers, academics, and graduate students in RDS and mathematical finance as well as practitioners working in the financial industry.

Inhomogeneous Random Evolutions and Their Applications explains how to model various dynamical systems in finance and insurance with non-homogeneous in time characteristics. It includes modeling for: financial underlying and derivatives via Levy processes with time-dependent characteristics; limit order books in the algorithmic and HFT with counting price changes processes having time-dependent intensities; risk processes which count number of claims with time-dependent conditional intensities; multi-asset price impact from distressed selling; regime-switching Levy-driven diffusion-based price dynamics. Initial models for those systems are very complicated, which is why the author’s approach helps to simplified their study. The book uses a very general approach for modeling of those systems via abstract inhomogeneous random evolutions in Banach spaces. To simplify their investigation, it applies the first averaging principle (long-run stability property or law of large numbers [LLN]) to get deterministic function on the long run. To eliminate the rate of convergence in the LLN, it uses secondly the functional central limit theorem (FCLT) such that the associated cumulative process, centered around that deterministic function and suitably scaled in time, may be approximated by an orthogonal martingale measure, in general; and by standard Brownian motion, in particular, if the scale parameter increases. Thus, this approach allows the author to easily link, for example, microscopic activities with macroscopic ones in HFT, connecting the parameters driving the HFT with the daily volatilities. This method also helps to easily calculate ruin and ultimate ruin probabilities for the risk process. All results in the book are new and original, and can be easily implemented in practice.

Stochastic Modelling of Big Data in Finance provides a rigorous overview and exploration of stochastic modelling of big data in finance (BDF). The book describes various stochastic models, including multivariate models, to deal with big data in finance. This includes data in high-frequency and algorithmic trading, specifically in limit order books (LOB), and shows how those models can be applied to different datasets to describe the dynamics of LOB, and to figure out which model is the best with respect to a specific data set. The results of the book may be used to also solve acquisition, liquidation and market making problems, and other optimization problems in finance.FeaturesSelf-contained book suitable for graduate students and post-doctoral fellows in financial mathematics and data science, as well as for practitioners working in the financial industry who deal with big dataAll results are presented visually to aid in understanding of conceptsDr. Anatoliy Swishchuk is a Professor in Mathematical Finance at the Department of Mathematics and Statistics, University of Calgary, Calgary, AB, Canada. He got his B.Sc. and M.Sc. degrees from Kyiv State University, Kyiv, Ukraine. He earned two doctorate degrees in Mathematics and Physics (PhD and DSc) from the prestigious National Academy of Sciences of Ukraine (NASU), Kiev, Ukraine, and is a recipient of NASU award for young scientist with a gold medal for series of research publications in random evolutions and their applications.Dr. Swishchuk is a chair and organizer of finance and energy finance seminar ‘Lunch at the Lab’ at the Department of Mathematics and Statistics. Dr. Swishchuk is a Director of Mathematical and Computational Finance Laboratory at the University of Calgary. He was a steering committee member of the Professional Risk Managers International Association (PRMIA), Canada (2006-2015), and is a steering committee member of Global Association of Risk Professionals (GARP), Canada (since 2015).Dr. Swishchuk is a creator of mathematical finance program at the Department of Mathematics & Statistics. He is also a proponent for a new specialization “Financial and Energy Markets Data Modelling” in the Data Science and Analytics program. His research areas include financial mathematics, random evolutions and their applications, biomathematics, stochastic calculus, and he serves on editorial boards for four research journals. He is the author of more than 200 publications, including 15 books and more than 150 articles in peer-reviewed journals. In 2018 he received a Peak Scholar award.

Inhomogeneous Random Evolutions and Their Applications explains how to model various dynamical systems in finance and insurance with non-homogeneous in time characteristics. It includes modeling for: financial underlying and derivatives via Levy processes with time-dependent characteristics; limit order books in the algorithmic and HFT with counting price changes processes having time-dependent intensities; risk processes which count number of claims with time-dependent conditional intensities; multi-asset price impact from distressed selling; regime-switching Levy-driven diffusion-based price dynamics.Initial models for those systems are very complicated, which is why the author’s approach helps to simplified their study. The book uses a very general approach for modeling of those systems via abstract inhomogeneous random evolutions in Banach spaces. To simplify their investigation, it applies the first averaging principle (long-run stability property or law of large numbers [LLN]) to get deterministic function on the long run. To eliminate the rate of convergence in the LLN, it uses secondly the functional central limit theorem (FCLT) such that the associated cumulative process, centered around that deterministic function and suitably scaled in time, may be approximated by an orthogonal martingale measure, in general; and by standard Brownian motion, in particular, if the scale parameter increases. Thus, this approach allows the author to easily link, for example, microscopic activities with macroscopic ones in HFT, connecting the parameters driving the HFT with the daily volatilities. This method also helps to easily calculate ruin and ultimate ruin probabilities for the risk process. All results in the book are new and original, and can be easily implemented in practice.

The theory and applications of random dynamical systems (RDS) are at the cutting edge of research in mathematics and economics, particularly in modeling the long-run evolution of economic systems subject to exogenous random shocks. Despite this interest, there are no books available that solely focus on RDS in finance and economics. Exploring this emerging area, Random Dynamical Systems in Finance shows how to model RDS in financial applications.Through numerous examples, the book explains how the theory of RDS can describe the asymptotic and qualitative behavior of systems of random and stochastic differential/difference equations in terms of stability, invariant manifolds, and attractors. The authors present many models of RDS and develop techniques for implementing RDS as approximations to financial models and option pricing formulas. For example, they approximate geometric Markov renewal processes in ergodic, merged, double-averaged, diffusion, normal deviation, and Poisson cases and apply the obtained results to option pricing formulas.With references at the end of each chapter, this book provides a variety of RDS for approximating financial models, presents numerous option pricing formulas for these models, and studies the stability and optimal control of RDS. The book is useful for researchers, academics, and graduate students in RDS and mathematical finance as well as practitioners working in the financial industry.

This book is the first of two volumes on random motions in Markov and semi-Markov random environments. This first volume focuses on homogenous random motions. This volume consists of two parts, the first describing the basic concepts and methods that have been developed for random evolutions. These methods are the foundational tools used in both volumes, and this description includes many results in potential operators. Some techniques to find closed-form expressions in relevant applications are also presented. The second part deals with asymptotic results and presents a variety of applications, including random motion with different types of boundaries, the reliability of storage systems and solutions of partial differential equations with constant coefficients, using commutative algebra techniques. It also presents an alternative formulation to the Black-Scholes formula in finance, fading evolutions and telegraph processes, including jump telegraph processes and the estimation of the number of level crossings for telegraph processes.

This book is the second of two volumes on random motions in Markov and semi-Markov random environments. This second volume focuses on high-dimensional random motions. This volume consists of two parts. The first expands many of the results found in Volume 1 to higher dimensions. It presents new results on the random motion of the realistic three-dimensional case, which has so far been barely mentioned in the literature, and deals with the interaction of particles in Markov and semi-Markov media, which has, in contrast, been a topic of intense study. The second part contains applications of Markov and semi-Markov motions in mathematical finance. It includes applications of telegraph processes in modeling stock price dynamics and investigates the pricing of variance, volatility, covariance and correlation swaps with Markov volatility and the same pricing swaps with semi-Markov volatilities.

This book extends the theory and applications of random evolutions to semi-Markov random media in discrete time, essentially focusing on semi-Markov chains as switching or driving processes.

This book extends the theory and applications of random evolutions to semi-Markov random media in discrete time, essentially focusing on semi-Markov chains as switching or driving processes.

This book is devoted to the history of Change of Time Methods (CTM), the connections of CTM to stochastic volatilities and finance, fundamental aspects of the theory of CTM, basic concepts, and its properties. An emphasis is given on many applications of CTM in financial and energy markets, and the presented numerical examples are based on real data. The change of time method is applied to derive the well-known Black-Scholes formula for European call options, and to derive an explicit option pricing formula for a European call option for a mean-reverting model for commodity prices. Explicit formulas are also derived for variance and volatility swaps for financial markets with a stochastic volatility following a classical and delayed Heston model. The CTM is applied to price financial and energy derivatives for one-factor and multi-factor alpha-stable Levy-based models. Readers should have a basic knowledge of probability and statistics, and some familiarity with stochastic processes, such as Brownian motion, Levy process and martingale.

Modeling and Pricing of Swaps for Financial and Energy Markets with Stochastic Volatilities is devoted to the modeling and pricing of various kinds of swaps, such as those for variance, volatility, covariance, correlation, for financial and energy markets with different stochastic volatilities, which include CIR process, regime-switching, delayed, mean-reverting, multi-factor, fractional, Levy-based, semi-Markov and COGARCH(1,1). One of the main methods used in this book is change of time method. The book outlines how the change of time method works for different kinds of models and problems arising in financial and energy markets and the associated problems in modeling and pricing of a variety of swaps. The book also contains a study of a new model, the delayed Heston model, which improves the volatility surface fitting as compared with the classical Heston model. The author calculates variance and volatility swaps for this model and provides hedging techniques. The book considers content on the pricing of variance and volatility swaps and option pricing formula for mean-reverting models in energy markets. Some topics such as forward and futures in energy markets priced by multi-factor Levy models and generalization of Black-76 formula with Markov-modulated volatility are part of the book as well, and it includes many numerical examples such as S&P60 Canada Index, S&P500 Index and AECO Natural Gas Index.