Introduction to the Mathematics of Finance

The Mathematics of Finance has become a hot topic in applied mathematics 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 upper division 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.

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.