Author | Roman, Steven. author |
---|---|
Title | Introduction to the Mathematics of Finance [electronic resource] : From Risk Management to Options Pricing / by Steven Roman |
Imprint | New York, NY : Springer New York : Imprint: Springer, 2004 |
Connect to | http://dx.doi.org/10.1007/978-1-4419-9005-1 |
Descript | XV, 356 p. online resource |
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 ยฐt ? 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