**Graduate Studies in Mathematics**

Volume: 31;
2001;
253 pp;
Hardcover

MSC: Primary 62; 91; 93;
Secondary 49; 60

**Print ISBN: 978-0-8218-2123-7
Product Code: GSM/31**

List Price: $51.00

AMS Member Price: $40.80

MAA Member Price: $45.90

**Electronic ISBN: 978-1-4704-2085-7
Product Code: GSM/31.E**

List Price: $48.00

AMS Member Price: $38.40

MAA Member Price: $43.20

# Option Pricing and Portfolio Optimization: Modern Methods of Financial Mathematics

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*Ralf Korn; Elke Korn*

Understanding and working with the current models of
financial markets requires a sound knowledge of the mathematical tools
and ideas from which they are built. Banks and financial houses all
over the world recognize this and are avidly recruiting
mathematicians, physicists, and other scientists with these
skills.

The mathematics involved in modern finance springs from the heart
of probability and analysis: the Itô calculus, stochastic
control, differential equations, martingales, and so on. The authors
give rigorous treatments of these topics, while always keeping the
applications in mind. Thus, the way in which the mathematics is
developed is governed by the way it will be used, rather than by the
goal of optimal generality. Indeed, most of purely mathematical topics
are treated in extended “excursions” from the applications
into the theory. Thus, with the main topic of financial modelling and
optimization in view, the reader also obtains a self-contained and
complete introduction to the underlying mathematics.

This book is specifically designed as a graduate textbook. It could
be used for the second part of a course in probability theory, as it
includes an applied introduction to the basics of stochastic processes
(martingales and Brownian motion) and stochastic calculus. It would
also be suitable for a course in continuous-time finance that assumes
familiarity with stochastic processes.

The prerequisites are basic probability theory and calculus. Some
background in stochastic processes would be useful, but not
essential.

#### Readership

Graduate level and research mathematicians, physicists, financial analysts, and actuarians interested in mathematical finance.

#### Reviews & Endorsements

Especially useful for students seeking a lively introduction to Itô calculus.

-- Short Book Reviews, International Statistical Institute

#### Table of Contents

# Table of Contents

## Option Pricing and Portfolio Optimization: Modern Methods of Financial Mathematics

- Cover Cover11 free
- Title v6 free
- Copyright vi7 free
- Contents vii8 free
- Preface ix10 free
- Frequently Used Notation xiii14 free
- Chapter 1. The Mean-Variance Approach in a One-Period Model 116 free
- Chapter 2. The Continuous-Time Market Model 1126
- § 2.1. Modeling the Security Prices 1126
- Excursion 1: Brownian Motion and Martingales 1530
- 2.1. Continuation: Modeling the Security Prices 2338
- Excursion 2: The Itô Integral 2641
- Excursion 3: The Itô Formula 4257
- §2.2. Trading Strategy and Wealth Process 5671
- §2.3. Properties of the Continuous-Time Market Model 6479
- Excursion 4: The Martingale Representation Theorem 7186
- Exercises 7691

- Chapter 3. Option Pricing 7994
- § 3.1. Introduction 7994
- §3.2. Option Pricing via the Replication Principle 8398
- Excursion 5: Girsanov's Theorem 93108
- 3.2. Continuation: Option Pricing via the Replication Principle 99114
- §3.3. Option Pricing by the Partial Differential Approach 105120
- Excursion 6: The Feynman-Kac Representation 111126
- §3.4. Arbitrage Bounds for American and European Options 122137
- §3.5. Pricing of American Options 129144
- §3.6. Arbitrage, Equivalent Martingale Measures and Option Pricing 134149
- §3.7. Market Numeraire and Numeraire Invariance 143158
- Exercises 148163

- Chapter 4. Pricing of Exotic Options and Numerical Algorithms 153168
- § 4.1. Exotic Options with Explicit Pricing Formulae 155170
- Excursion 7: Weak Convergence of Stochastic Processes 170185
- §4.2. Monte-Carlo Simulation 175190
- § 4.3. Approximation via Binomial Trees 177192
- §4.4. Trinomial Trees and Explicit Finite-Difference Methods 187202
- §4.5. The Pathwise Binomial Approach of Rogers and Stapleton 191206
- Exercises 201216

- Chapter 5. Optimal Portfolios 203218
- Bibliography 247262
- Index 251266
- Back Cover Back Cover1269