Eckhard Platen Knihy

In recent years, financial derivatives have become essential tools for risk managers and investors, with insurance products now integral to both personal and business portfolios. The management of mutual and pension funds has gained significance for individuals, while banks, insurance companies, and corporations increasingly utilize financial and insurance instruments for active risk management. A wider range of securities allows for tailored hedging strategies to meet the specific needs of various investors and companies. Mastering modern quantitative methods is crucial for distinguishing market participants in finance and insurance. Consequently, financial institutions, insurance firms, and corporations must develop expertise in quantitative finance, where many associated methods and technologies originate. This book serves as an introduction to quantitative finance, focusing on the mathematical framework used in financial modeling, derivative pricing, portfolio selection, and risk management. It presents a unified approach to risk and performance management through the benchmark approach, which differs from the prevailing paradigm. This systematic and rigorous method employs the growth optimal portfolio as numeraire and the real-world probability measure as the pricing measure.

In financial and actuarial modeling, stochastic differential equations with jumps are used to describe the dynamics of various state variables. Solving these equations numerically is more complex than those driven solely by Wiener processes. This monograph builds on previous work and introduces stochastic differential equations with jumps, focusing on the necessary numerical methods for their solution. It presents new results on higher-order methods for scenario and Monte Carlo simulation, including implicit, predictor-corrector, extrapolation, Markov chain, and variance reduction methods, highlighting their numerical stability. Additionally, it covers exact simulation, estimation, and filtering. Serving as a foundational text on quantitative methods, it also provides access to numerous research problems in a rapidly expanding field. The focus on finance is due to recent research challenges in quantitative finance driving advancements in stochastic numerical methods. The volume introduces a modern benchmark approach for modeling in finance and insurance, extending beyond the standard risk-neutral framework. It requires an undergraduate background in mathematical or quantitative methods, making it accessible to a wide audience, including those seeking numerical recipes. Exercises included help readers deepen their understanding of the underlying mathematics.

In financial and actuarial modeling, stochastic differential equations with jumps are used to describe the dynamics of various state variables. Their numerical solutions are more complex than those driven solely by Wiener processes, as detailed in Kloeden & Platen's earlier work. This monograph serves as an introduction to these equations, focusing on both theory and application, particularly the numerical methods necessary for their solution. It presents new results on higher-order methods for scenario and Monte Carlo simulation, including implicit, predictor-corrector, extrapolation, Markov chain, and variance reduction methods, with an emphasis on numerical stability. Additionally, it covers exact simulation, estimation, and filtering. This text not only serves as a foundational resource on quantitative methods but also highlights numerous potential research problems in a rapidly expanding field. Finance is chosen as the primary application area, reflecting recent research challenges in quantitative finance. The volume introduces a modern benchmark approach for modeling in finance and insurance, extending beyond the traditional risk-neutral framework. It requires an undergraduate background in mathematical or quantitative methods and is accessible to a wide audience, including those seeking numerical recipes. Exercises are included to enhance understanding of the underlying mathematics.