Introduction to the Mathematics of Finance: From Risk Management to Options Pricing

Notation Key and Greek Alphabet Introduction Portfolio Risk Management Option Pricing Models Assumptions Arbitrage 1 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 2 Portfolio Management and the Capital Asset Pricing Model 2.1 Portfolios, Returns and Risk 2.2 Two-Asset Portfolios 2.3 Multi-Asset Portfolios 3 Background on Options 3.1 Stock Options 3.2 The Purpose of Options 3.3 Profit and Payoff Curves 3.4 Selling Short 4 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 5 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 6 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 7 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 8 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 9 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 9.4 The Natural CRR Model 9.5 The Martingale Measure CRR Model 9.6 More on the Model From a Different Perspective: Itô'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 10 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 10.8 What to Do At Time 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 Index