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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.
Sommario
Introduction to the Change of Time Methods: History, Finance and Stochastic Volatility.- Change of Time Methods: Definitions and Theory.- Applications of the Change of Time Methods.- Change of Time Method (CTM) and Black-Scholes Formula.- CTM and Variance, Volatility, Covariance and Correlation Swaps for the Classical Heston Model.- CTM and the Delayed Heston Model: Pricing and Hedging of Variance and Volatility Swaps.- CTM and the Explicit Option Pricing Formula for a Mean-reverting Asset in Energy Markets.- CTM and Multi-Factor Levy Models for Pricing Financial and Energy Derivatives.- Epilogue.
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Nikolaos Limnios is a Professor at the Applied Mathematics Laboratory at the Université de Technologie de Compiègne, France. His research interests include stochastic processes, random evolutions, with a focus on semi-Markov processes, and statistics, and applications in reliability, biology, seismology, insurance, and finance. He has published more than 150 journal articles and 10 books on the theory and applications of stochastic processes and statistics. He serves on editorial boards for several research journals.
Anatoliy Swishchuk is a Professor in Mathematical Finance at the Department of Mathematics and Statistics, University of Calgary, Canada. His research areas include financial mathematics, random evolutions and their applications, biomathematics, and stochastic calculus. He serves on editorial boards for several research journals and is the author of more than 180 publications, including 15 books and more than 120 articles in peer-reviewed journals. In 2018 he received a Peak Scholar award.
