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This book presents a short introduction to continuous-time financial models. An overview of the basics of stochastic analysis precedes a focus on the Black–Scholes and interest rate models. Other topics covered include self-financing strategies, option pricing, exotic options and risk-neutral probabilities. Vasicek, Cox−Ingersoll−Ross, and Heath–Jarrow–Morton interest rate models are also explored.
The author presents practitioners with a basic introduction, with more rigorous information provided for mathematicians. The reader is assumed to be familiar with the basics of probability theory. Some basic knowledge of stochastic integration and differential equations theory is preferable, although all preliminary information is given in the first part of the book. Some relatively simple theoretical exercises are also provided.
- About continuous-time stochastic models of financial mathematics
- Black-Sholes model and interest rate models
- Requiring a minimum knowledge of stochastic integration and stochastic differential equations
Masters students of mathematics, business mathematics, or financial mathematics; Lecturers in financial mathematics
- 1: Overview of the Basics of Stochastic Analysis
- 1.1 Brownian motion
- 1.2 Stochastic integrals
- 1.3 Martingales, Itô processes and general Itô’s formula
- 1.4 Stochastic differential equations
- 1.5 Change of probability: the Girsanov theorem
- 2: The Black–Scholes Model
- 2.1 Introduction: what is an option?
- 2.2 Self-financing strategies
- 2.3 Option pricing problem: the Black–Scholes model
- 2.4 The Black–Scholes formula
- 2.5 Risk-neutral probabilities: alternative derivation of the Black–Scholes formula
- 2.6 American options in the Black–Scholes model
- 2.7 Exotic options
- 3: Models of Interest Rates
- 3.1 Modeling principles
- 3.2 The Vašíček model
- 3.3 The Cox–Ingersoll–Ross model
- 3.4 The Heath–Jarrow–Morton model
- No. of pages:
- © ISTE Press - Elsevier 2017
- 12th October 2016
- ISTE Press - Elsevier
- Hardcover ISBN:
- eBook ISBN:
Vigirdas Mackevičius is Professor of the Department of Mathematical Analysis in the Faculty of Mathematics and Informatics of Vilnius University in Lithuania. His research interests include stochastic processes, stochastic analysis, and stochastic numerics.
VU MIF Professor, Department of Mathematical Analysis, Vilnius University, Lithuania
"The book is written at a high mathematical level, however very clearly for the reader, and will be useful both for undergraduate and post graduate students, practitioners and everybody who wants to study the basic properties of financial markets with continuous time." --Zentralblatt MATH