Focusing on local volatility and its model, this book addresses unresolved issues in mathematical finance, such as deriving analytical solutions for the Dupire equation and creating efficient calibration methods for local volatility surfaces. It explores no-arbitrage techniques for interpolating and extrapolating volatility surfaces and extends the local volatility concept beyond the Black-Scholes framework. Additionally, it examines the integration of deep learning and neural networks into financial engineering, offering fresh perspectives on classical financial problems.
Focusing on contemporary trends in quantitative finance, this volume features six chapters authored by leading experts. It delves into stochastic and fractional volatility models, equity trading, optimal portfolios, and machine learning applications, including natural language processing. Additionally, it explores economic scenario generation, providing original insights and solutions. This comprehensive work serves as a valuable resource for both researchers and practitioners in the finance sector.
This monograph presents a novel numerical approach to solving partial integro-differential equations arising in asset pricing models with jumps, which greatly exceeds the efficiency of existing approaches. The method, based on pseudo-differential operators and several original contributions to the theory of finite-difference schemes, is new as applied to the Lévy processes in finance, and is herein presented for the first time in a single volume. The results within, developed in a series of research papers, are collected and arranged together with the necessary background material from Lévy processes, the modern theory of finite-difference schemes, the theory of M-matrices and EM-matrices, etc., thus forming a self-contained work that gives the reader a smooth introduction to the subject. For readers with no knowledge of finance, a short explanation of the main financial terms and notions used in the book is given in the glossary. The latter part of the book demonstrates the efficacy of the method by solving some typical problems encountered in computational finance, including structural default models with jumps, and local stochastic volatility models with stochastic interest rates and jumps. The author also adds extra complexity to the traditional statements of these problems by taking into account jumps in each stochastic component while all jumps are fully correlated, and shows how this setting can be efficiently addressed within the framework of the new method. Written for non-mathematicians, this book will appeal to financial engineers and analysts, econophysicists, and researchers in applied numerical analysis. It can also be used as an advance course on modern finite-difference methods or computational finance
