**Translations of Mathematical Monographs**

1999;
133 pp;
Hardcover

MSC: Primary 90;
Secondary 60

**Print ISBN: 978-0-8218-1082-8
Product Code: MMONO/184**

List Price: $83.00

AMS Member Price: $66.40

MAA Member Price: $74.70

**Electronic ISBN: 978-1-4704-4598-0
Product Code: MMONO/184.E**

List Price: $78.00

AMS Member Price: $62.40

MAA Member Price: $70.20

# Financial Markets: Stochastic Analysis and the Pricing of Derivative Securities

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*A. V. Mel′nikov*

Financial mathematics is going through a period of intensive
development, particularly in the area of stochastic analysis. This
timely work presents a comprehensive, self-contained introduction to
stochastic financial mathematics. It is based on lectures given at
Moscow State University, “Stochastic Analysis in Finance”,
and comprises the basic methods and key results of the theory of
derivative securities pricing in discrete financial markets.

The following elements: martingales, semimartingales, stochastic
exponents, Itô's formula, Girsanov's theorem, and more, are used to
characterize notions such as arbitrage and completeness of financial
markets, fair price and hedging strategies for options, forward and
futures pricing, and utility maximization. Limiting transition from a
discrete to continuous model with derivation of the famous Black-Scholes
formula is shown.

The book contains a wide spectrum of material and can serve as a
bridge to continuous models. It is suitable as a text for graduate and
advanced graduate students studying economics and/or financial
mathematics.

#### Readership

Graduate students and researchers working in probability theory, stochastic processes, and financial mathematics.

#### Reviews & Endorsements

The book provides a rigorous, self-contained and concise introduction to the rapidly developing field of mathematical finance. It may serve very well as a textbook for graduate students in mathematics or finance.

-- Mathematical Reviews

The book is carefully written and contains a short but clear introduction to the theory of stochastic modelling of financial markets. It could be highly recommended as an introductory course to the topic for graduate students and specialists interested in financial mathematic problems.

-- Zentralblatt MATH

#### Table of Contents

# Table of Contents

## Financial Markets: Stochastic Analysis and the Pricing of Derivative Securities

- Cover Cover11
- Title page v6
- Contents vii8
- Foreword xi12
- Notation xiii14
- Basic concepts and objects of a financial market 116
- The elements of discrete stochastic analysis 722
- A stochastic model for a financial market. Arbitrage and completeness 2136
- Pricing European options in complete markets. The binomial model and the Cox-Ross-Rubinstein formula 3146
- Pricing and hedging American options in complete markets 4762
- Financial computations on a complete market with the use of nonself-financing strategies 5570
- Incomplete markets. Pricing of options and problems of minimizing risk 6378
- The structure of prices of other instruments of a financial market. Forwards, futures, bonds 7388
- The problem of optimal investment 85100
- The concept of continuous models. Limiting transitions from a discrete market to a continuous one. The Black-Scholes formula 93108
- Appendix 1 99114
- Appendix 2 105120
- Appendix 3 109124
- Hints for solving the problems 121136
- Bibliography 129144
- Subject index 131146
- Back Cover Back Cover1153