Introduction to Mathematical Finance: Discrete Time Models


This book is designed to serve as a textbook for advanced undergraduate and beginning graduate students who seek a rigorous yet accessible introduction to the modern financial theory of security markets. This is a subject that is taught in both business schools and mathematical science departments. The full theory of security markets requires knowledge of continuous time stochastic process models, measure theory, mathematical economics, and similar prerequisites which are generally not learned before the advanced graduate level. Hence a proper study of the full theory of security markets requires several years of graduate study. However, by restricting attention to discrete time models of security prices it is possible to acquire mathematics. In particular, while living in a discrete time world it is possible to learn virtually all of the important financial concepts. The purpose of this book is to provide such an introductory study.

There is still a lot of mathematics in this book. The reader should be comfortable with calculus, linear algebra, and probability theory that is based on calculus, (but not necessarily measure theory). Random variables and expected values will be playing important roles. The book will develop important notions concerning discrete time stochastic processes; prior knowledge here will be useful but is not required. Presumably the reader will be interested in finance and thus will come with some rudimentary knowledge of stocks, bonds, options, and financial decision making. The last topic involves utility theory, of course; hopefully the reader will be familiar with this and related topics of introductory microeconomic theory. Some exposure to linear programming would be advantageous, but not necessary.

The aim of this book is to provide a rigorous treatment of the financial theory while maintaining a casual style. Readers seeking institutional knowledge about securities, derivatives, and portfolio management should look elsewhere, but those seeking a careful introduction to financial engineering will find that this is a useful and comprehensive introduction to the subject.

Stanley Pliska is the founding editor of the scholarly journal Mathematical Finance. He is noted for his fundamental research on the mathematical and economic theory of security prices, especially his development of important bridges between stochastic calculus and arbitrage pricing theory as well as his discovery of the risk neutral computational approach for portfolio optimization problems. He is currently teaching and researching in the areas of interest rate derivatives and dynamic asset allocation.
Part I: Single Period Securities Markets:.

Model Specifications.

Arbitrage and Other Economic Consideration.

Risk Neutral Probability Measures.

Valuation of Contingent Claims.

Complete and Incomplete Markets.

Risk and Return.

Part II: Single Period Consumption and Investment:.

Optimal Portfolios and Viability.

Risk Neutral Computational Approach.

Consumption Investment Problems.

Mean-Variance Portfolio Analysis.

Portfolio Management with Short Sales Constraints and Similar Restrictions.

Optimal Portfolios in Incomplete Markets.

Equilibrium Models.

Part III: Multiperiod Securities Markets:.

Model Specifications, Filtrations, and Stochastic Processes.

Information Structures.

Stochastic Process Models of Security Prices.

Trading Strategies.

Value Processes and Gains Processes.

Self-Financing Trading Strategies.

Discounted Prices.

Return and Dividend Processes.

Conditional Expectation and Martingales.

Economic Considerations.

The Binomial Model.

Markov Models.

Part IV: Options, Futures, and Other Derivatives:.

Contingent Claims.

European Options Under the Binomial Model.

American Options.

Complete and Incomplete Markets.

Forward Prices and Cash Stream Valuation.


Part V: Optimal Consumption and Investment Problems:.

Optimal Portfolios and Dynamic Programming.

Optimal Portfolios and Martingals Methods.

Consumption-Investment and Dynamic Programming.

Consumption-Investment and Martingale Methods.

Maximum Utility from Consumption and Terminal Wealth.

Optimal Portfolios with Constraints.

Optimal Consumption-Investment with Constraints.

Portfolio Optimization in Incomplete Markets.

Part VI: Bonds and Interest Rate Derivatives:.

The Basic Term Structure Model.

Lattice, Markov Chain Models.

Yield Curve Models.

Forward Risk Adjusted Probability Measures.

Coupon Bonds and Bond Options.

Swaps and Swaptions.

Caps and Floors.

Part VII: Models with Infinite Sample Spaces.

Finite Horizon Models.

Infinite Horizon Models.