Roman Introduction to the Mathematics of Finance

From Risk Management to Options Pricing
Erscheinungsjahr 2013
ISBN: 978-1-4419-9005-1
Verlag: Springer US
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 356 Seiten, eBook

Reihe: Undergraduate Texts in Mathematics

The Mathematics of Finance has become a hot topic ever since the discovery of the Black-Scholes option pricing formulas in 1973. Unfortunately, there are very few undergraduate textbooks in this area. This book is specifically written for advanced undergraduate or beginning graduate students in mathematics, finance or economics. With the exception of an optional chapter on the Capital Asset Pricing Model, the book concentrates on discrete derivative pricing models, culminating in a careful and complete derivation of the Black-Scholes option pricing formulas as a limiting case of the Cox-Ross-Rubinstein discrete model. The final chapter is devoted to American options.The mathematics is not watered down, but is appropriate for the intended audience. No measure theory is used, and only a small amount of linear algebra is required. All necessary probability theory is developed throughout the book on a "need-to-know" basis. No background in finance is required, since the book also contains a chapter on options.

Zielgruppe

Graduate

Autoren/Hrsg.

Roman, Steven

Inhaltsverzeichnis

Portfolio Risk Management.- Option Pricing Models.- Assumptions.- Arbitrage.- Probability I: An Introduction to Discrete Probability.- 1.1 Overview.- 1.2 Probability Spaces.- 1.3 Independence.- 1.4 Binomial Probabilities.- 1.5 Random Variables.- 1.6 Expectation.- 1.7 Variance and Standard Deviation.- 1.8 Covariance and Correlation; Best Linear Predictor.- Exercises.- Portfolio Management and the Capital Asset Pricing Model.- 2.1 Portfolios, Returns and Risk.- 2.2 Two-Asset Portfolios.- 2.3 Multi-Asset Portfolios.- Exercises.- Background on Options.- 3.1 Stock Options.- 3.2 The Purpose of Options.- 3.3 Profit and Payoff Curves.- 3.4 Selling Short.- Exercises.- An Aperitif on Arbitrage.- 4.1 Background on Forward Contracts.- 4.2 The Pricing of Forward Contracts.- 4.3 The Put-Call Option Parity Formula.- 4.4 Option Prices.- Exercises.- Probability II: More Discrete Probability.- 5.1 Conditional Probability.- 5.2 Partitions and Measurability.- 5.3 Algebras.- 5.4 Conditional Expectation.- 5.5 Stochastic Processes.- 5.6 Filtrations and Martingales.- Exercises.- Discrete-Time Pricing Models.- 6.1 Assumptions.- 6.2 Positive Random Variables.- 6.3 The Basic Model by Example.- 6.4 The Basic Model.- 6.5 Portfolios and Trading Strategies.- 6.6 The Pricing Problem: Alternatives and Replication.- 6.7 Arbitrage Trading Strategies.- 6.8 Admissible Arbitrage Trading Strategies.- 6.9 Characterizing Arbitrage.- 6.10 Computing Martingale Measures.- Exercises.- The Cox-Ross-Rubinstein Model.- 7.1 The Model.- 7.2 Martingale Measures in the CRR model.- 7.3 Pricing in the CRR Model.- 7.4 Another Look at the CRR Model via Random Walks.- Exercises.- Probability III: Continuous Probability.- 8.1 General Probability Spaces.- 8.2 Probability Measures on ?.- 8.3 Distribution Functions.- 8.4 Density Functions.- 8.5 Types of Probability Measures on ?.- 8.6 Random Variables.- 8.7 The Normal Distribution.- 8.8 Convergence in Distribution.- 8.9 The Central Limit Theorem.- Exercises.- The Black-Scholes Option Pricing Formula.- 9.1 Stock Prices and Brownian Motion.- 9.2 The CRR Model in the Limit: Brownian Motion.- 9.3 Taking the Limit as °t ? 0.- 9.4 The Natural CRR Model.- 9.5 The Martingale Measure CRR Model.- 9.6 More on the Model From a Different Perspective: Ito's Lemma.- 9.7 Are the Assumptions Realistic?.- 9.8 The Black-Scholes Option Pricing Formula.- 9.9 How Black-Scholes is Used in Practice: Volatility Smiles and Surfaces.- 9.10 How Dividends Affect the Use of Black-Scholes.- Exercises.- Optimal Stopping and American Options.- 10.1 An Example.- 10.2 The Model.- 10.3 The Payoffs.- 10.4 Stopping Times.- 10.5 Stopping the Payoff Process.- 10.6 The Stopped Value of an American Option.- 10.7 The Initial Value of an American Option, or What to Do At Time to.- 10.8 What to Do At Time tk.- 10.9 Optimal Stopping Times and the Snell Envelop.- 10.10 Existence of Optimal Stopping Times.- 10.11 Characterizing the Snell Envelop.- 10.12 Additional Facts About Martingales.- 10.13 Characterizing Optimal Stopping Times.- 10.14 Optimal Stopping Times and the Doob Decomposition.- 10.15 The Smallest Optimal Stopping Time.- 10.16 The Largest Optimal Stopping Time.- Exercises.- Appendix A: Pricing Nonattainable Alternatives in an Incomplete Market.- A. 1 Fair Value in an Incomplete Market.- A.2 Mathematical Background.- A.3 Pricing Nonattainable Alternatives.- Exercises.- Appendix B: Convexity and the Separation Theorem.- B. 1 Convex, Closed and Compact Sets.- B.2 Convex Hulls.- B.3 Linear and Affine Hyperplanes.- B.4 Separation.- Selected Solutions.- References.

Über Autor(innen)

Dr. Roman has authored 32 books, including a number of books on mathematics, such as Coding and Information Theory, Advanced Linear Algebra, and Field Theory, published by Springer-Verlag. He has also written Modules in Mathematics, a series of 15 small books designed for the general college-level liberal arts student. Besides his books for O'Reilly, Dr. Roman has written two other computer books, both published by Springer-Verlag.