A balanced introduction to the theoretical foundations and
real-world applications of mathematical finance
The ever-growing use of derivative products makes it essential
for financial industry practitioners to have a solid understanding
of derivative pricing. To cope with the growing complexity,
narrowing margins, and shortening life-cycle of the individual
derivative product, an efficient, yet modular, implementation of
the pricing algorithms is necessary. Mathematical Finance is
the first book to harmonize the theory, modeling, and
implementation of today's most prevalent pricing models under one
convenient cover. Building a bridge from academia to practice, this
self-contained text applies theoretical concepts to real-world
examples and introduces state-of-the-art, object-oriented
programming techniques that equip the reader with the conceptual
and illustrative tools needed to understand and develop successful
derivative pricing models.
Utilizing almost twenty years of academic and industry
experience, the author discusses the mathematical concepts that are
the foundation of commonly used derivative pricing models, and
insightful Motivation and Interpretation sections for each concept
are presented to further illustrate the relationship between theory
and practice. In-depth coverage of the common characteristics found
amongst successful pricing models are provided in addition to key
techniques and tips for the construction of these models. The
opportunity to interactively explore the book's principal ideas and
methodologies is made possible via a related Web site that features
interactive Java experiments and exercises.
While a high standard of mathematical precision is retained,
Mathematical Finance emphasizes practical motivations,
interpretations, and results and is an excellent textbook for
students in mathematical finance, computational finance, and
derivative pricing courses at the upper undergraduate or beginning
graduate level. It also serves as a valuable reference for
professionals in the banking, insurance, and asset management
industries.
Autorentext
Christian Fries, PhD, is Lecturer of Mathematical Finance at the University of Frankfurt and head of financial model development at DZ Bank AG Frankfurt, both located in Germany. With extensive knowledge in various programming languages, Dr. Fries has conducted quantitative analysis and overseen the implementation of mathematical modeling platforms at numerous financial institutions. His research interests within the field of mathematical finance include the LIBOR Market Model, Efficient Calculation of Risk Measures with Monte-Carlo Methods, Pricing of Bermudan Options with Monte-Carlo Methods, and Markov Functional Models.
Klappentext
A balanced introduction to the theoretical foundations and real-world applications of mathematical finance
The ever-growing use of derivative products makes it essential for financial industry practitioners to have a solid understanding of derivative pricing. To cope with the growing complexity, narrowing margins, and shortening life-cycle of the individual derivative product, an efficient, yet modular, implementation of the pricing algorithms is necessary. Mathematical Finance is the first book to harmonize the theory, modeling, and implementation of today's most prevalent pricing models under one convenient cover. Building a bridge from academia to practice, this self-contained text applies theoretical concepts to real-world examples and introduces state-of-the-art, object-oriented programming techniques that equip the reader with the conceptual and illustrative tools needed to understand and develop successful derivative pricing models.
Utilizing almost twenty years of academic and industry experience, the author discusses the mathematical concepts that are the foundation of commonly used derivative pricing models, and insightful Motivation and Interpretation sections for each concept are presented to further illustrate the relationship between theory and practice. In-depth coverage of the common characteristics found amongst successful pricing models are provided in addition to key techniques and tips for the construction of these models. The opportunity to interactively explore the book's principal ideas and methodologies is made possible via a related Web site that features interactive Java experiments and exercises.
While a high standard of mathematical precision is retained, Mathematical Finance emphasizes practical motivations, interpretations, and results and is an excellent textbook for students in mathematical finance, computational finance, and derivative pricing courses at the upper undergraduate or beginning graduate level. It also serves as a valuable reference for professionals in the banking, insurance, and asset management industries.
Inhalt
1. Introduction.
1.1 Theory, Modeling and Implementation.
1.2 Interest Rate Models and Interest Rate Derivatives.
1.3 How to Read this Book.
1.3.1 Abridged Versions.
1.3.2 Special Sections.
1.3.3 Notation.
I: FOUNDATIONS.
2. Foundations.
2.1 Probability Theory.
2.2 Stochastic Processes.
2.3 Filtration.
2.4 Brownian Motion.
2.5 Wiener Measure, Canonical Setup.
2.6 Itô Calculus.
2.6.1 Itô Integral.
2.6.2 Itô Process.
2.6.3 Itô Lemma and Product Rule.
2.7 Brownian Motion with Instantaneous Correlation.
2.8 Martingales.
2.8.1 Martingale Representation Theorem.
2.9 Change of Measure (Girsanov, Cameron, Martin).
2.10 Stochastic Integration.
2.11 Partial Differential Equations (PDE).
2.11.1 Feynman-Kac Theorem .
2.12 List of Symbols.
3. Replication.
3.1 Replication Strategies.
3.1.1 Introduction.
3.1.2 Replication in a discrete Model.
3.2 Foundations: Equivalent Martingale Measure.
3.2.1 Challenge and Solution Outline.
3.2.2 Steps towards the Universal Pricing Theorem.
3.3 Excursus: Relative Prices and Risk Neutral Measures.
3.3.1 Why relative prices?
3.3.2 Risk Neutral Measure.
II: FIRST APPLICATIONS.
4. Pricing of a European Stock Option under the Black-Scholes Model.
5. Excursus: The Density of the Underlying of a European Call Option.
6. Excursus: Interpolation of European Option Prices.
6.1 No-Arbitrage Conditions for Interpolated Prices.
6.2 Arbitrage Violations through Interpolation.
6.2.1 Example (1): Interpolation of four Prices.
6.2.2 Example (2): Interpolation of two Prices.
6.3 Arbitrage-Free Interpolation of European Option Prices.
7. Hedging in Continuous and Discrete Time and the Greeks.
7.1 Introduction.
7.2 Deriving the Replications Strategy from Pricing Theory.
7.2.1 Deriving the Replication Strategy under the Assumption of a Locally Riskless Product.
7.2.2 The Black-Scholes Differential Equation.
7.2.3 The Derivative V(t) as a Function of its Underlyings S i(t).
7.2.4 Example: Replication Portfolio and PDE under a Black-Scholes Model.
7.3 Greeks.
7.3.1 Greeks of a European Call-Option under the Black-Scholes model.
7.4 Hedging in Discrete Time: Delta and Delta-Gamma Hedging.
7.4.1 Delta Hedging.
7.4.2 Error Propagation.
7.4.3 Delta-Gamma Hedging.
7.4.4 Vega Hedging.
7.5 Hedging in Discrete Time: Minimizing the Residual Error (Bouchaud-Sornette Method).
7.5.1 Minimizing the Residual Error at Maturity T.
7.5.2 Minimizing the Residual Error in each Time Step.
…