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Generalized Integral Transforms in Mathematical Finance https://doi.org/10.1142/12147
October 2021
Pages: 508
Author: Andrey Itkin, Alexander Lipton, and Dmitry Muravey
This book describes several techniques, first invented in physics for solving problems of heat and mass transfer, and applies them to various problems of mathematical finance defined in domains with moving boundaries. These problems include: (a) semi-closed form pricing of options in the one-factor models with time-dependent barriers (Bachelier, Hull-White, CIR, CEV); (b) analyzing an interconnected banking system in the structural credit risk model with default contagion; (c) finding first hitting time density for a reducible diffusion process; (d) describing the exercise boundary of American options; (e) calculating default boundary for the structured default problem; (f) deriving a semi-closed form solution for optimal mean-reverting trading strategies; to mention but some.
The main methods used in this book are generalized integral transforms and heat potentials. To find a semi-closed form solution, we need to solve a linear or nonlinear Volterra equation of the second kind and then represent the option price as a one-dimensional integral. Our analysis shows that these methods are computationally more efficient than the corresponding finite-difference methods for the backward or forward Kolmogorov PDEs (partial differential equations) while providing better accuracy and stability.
We extend a large number of known results by either providing solutions on complementary or extended domains where the solution is not known yet or modifying these techniques and applying them to new types of equations, such as the Bessel process. The book contains several novel results broadly applicable in physics, mathematics, and engineering.
Dr Andrey Itkin is Director, Senior Quantitative Research Associate at Bank of America, and Adjunct Professor of Computational and Algorithmic Finance at the Department of Finance and Risk Engineering at Tandon School of Engineering, New York University. He holds a PhD in Physics of Liquids, Gases, and Plasma from the Moscow Aviation University and a Doctor of Science degree in Computational and Molecular Physics from St. Petersburg Technical University. For a long time, he worked as a physicist, and then moved to finance as a quantitative researcher in equity derivatives, volatility trading, and market making, and then in risk management. He has authored numerous research papers in both physics and finance, and has authored books including Microscopic Theory of Condensation in Gases and Plasma, Pricing Derivatives Under Levy Models: Modern Finite-Difference and Pseudo-Differential Operators Approach, and Fitting Local Volatility: Analytic and Numerical Approaches in Black-Scholes and Local Variance Gamma Models. He is an Associate Editor of the Journal of Derivatives, the International Journal of Computer Mathematics, and the Journal of Advances in Applied & Computational Mathematics. He was a Guest Editor of various special issues including "Computational Methods in Finance" (International Journal of Computer Mathematics, Vol. 92, Issue 12, 2015), "Computational and Algorithmic Finance" (Journal of Computational Science, Vol. 24, 2018), and "Physics of Financial Derivatives" (Journal of Derivatives, 2019). He is also a member of multiple professional associations in finance and physics.
Alexander Lipton is Co-Founder and Chief Technical Officer of Sila, Partner at Numeraire Financial, Partner at Investimizer, Visiting Professor and Dean's Fellow at the Hebrew University of Jerusalem, and Connection Science Fellow at Massachusetts Institute of Technology (MIT). He sits on the Boards of Directors of Sila, and Zilliqa, and on Advisory Boards of several organizations, including Clearmatics, Endor, Katalysen, Sygnum, and University College London (UCL) Centre for Blockchain Technologies. In 2016 he left Bank of America Merrill Lynch, where he served for 10 years in various senior managerial roles including Quantitative Solutions Executive and Co-Head of the Global Quantitative Group. Earlier, he held senior managerial positions at Citadel Investment Group, Credit Suisse, Deutsche Bank, and Bankers Trust. In parallel, Alex held several prestigious professorial appointments at École Polytechnique Fédérale de Lausanne, New York University, Oxford University, Imperial College London, and the University of Illinois. Before switching to finance, Alex was a Full Professor of Mathematics at the University of Illinois and a Consultant at Los Alamos National Laboratory. In 2000 Alex was awarded the first ever Quant of the Year Award by Risk Magazine. Alex published eight books and more than a hundred scientific papers. His most recent book Financial Engineering: Selected Works of Alexander Lipton was published in May of 2018. He is currently working on his next book (with Adrien Treccani) Blockchain and Distributed Ledgers: Mathematics, Technology, and Economics which will be published in the first half of 2020.
Dmitry Muravey is a Research Fellow at Lomonosov State University, Moscow, Russia. He holds a PhD in Operations from Lomonosov State University. He worked as a quantitative researcher in market making and statistical arbitrage and an algorithms researcher in high-dimensional optimization problems. He has authored numerous research papers in finance, stochastic analysis, and partial differential equations. He has also won awards from Machine Learning and Data Science competitions.
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