Introductory Stochastic Analysis For Finance And Insurance

Introductory Stochastic Analysis for Finance and Insurance......Page 4
CONTENTS......Page 10
List of Figures......Page 14
List of Tables......Page 16
Preface......Page 18
1 Introduction......Page 20
2 Overview of Probability Theory......Page 24
2.1 Probability Spaces and Information Structures......Page 25
2.1. The price of a stock over a two-day period.......Page 30
2.3 Multivariate Distributions......Page 39
2.4 Conditional Probability and Conditional Distributions......Page 43
2.5 Conditional Expectation......Page 53
2.6 The Central Limit Theorem......Page 62
3.1 Stochastic Processes and Information Structures......Page 64
3.2 Random Walks......Page 66
3.3. The binomial tree of the stock price.......Page 74
3.4 Martingales and Change of Probability Measure......Page 79
3.5 Stopping Times......Page 85
3.6 Option Pricing with Binomial Models......Page 91
3.7 Binomial Interest Rate Models......Page 103
4.1 General Description of Continuous-Time Stochastic Processes......Page 116
4.2 Brownian Motion......Page 117
4.2.  A sample path of Brownian motion with μ = 1 and σ = 1.......Page 123
4.4 The Poisson Process and Compound Poisson Process......Page 131
4.5 Martingales......Page 136
4.6 Stopping Times and the Optional Sampling Theorem......Page 141
5.1 Stochastic (Ito) Integration......Page 150
5.2 Stochastic Differential Equations......Page 160
5.3 One-Dimensional Ito’s Lemma......Page 163
5.4 Continuous-Time Interest Rate Models......Page 167
5.5 The Black-Scholes Model and Option Pricing Formula......Page 174
5.6 The Stochastic Version of Integration by Parts......Page 181
5.7 Exponential Martingales......Page 184
5.8 The Martingale Representation Theorem......Page 187
6 Stochastic Calculus: Advanced Topics......Page 192
6.1 The Feynman-Kac Formula......Page 193
6.2 The Black-Scholes Partial Differential Equation......Page 194
6.3 The Girsanov Theorem......Page 196
6.4 The Forward Risk Adjusted Measure and Bond Option Pricing......Page 200
6.5 Barrier Hitting Probabilities Revisited......Page 206
6.6 Two-Dimensional Stochastic Differential Equations......Page 210
7 Applications in Insurance......Page 216
7.1 Deferred Variable Annuities and Equity-Indexed Annuities......Page 217
7.2 Guaranteed Annuity Options......Page 225
7.3 Universal Life......Page 229
References......Page 236
Topic Index......Page 240
2.2. The probability tree of the stock price over a two-day period.......Page 45
2.3. The expectation tree of the stock price over a two-day period.......Page 58
3.1. The tree of a standard random walk.......Page 68
3.2. The binomial model of the stock price.......Page 72
3.4. The returns of a stock and a bond.......Page 92
3.6. The payoff function of a put.......Page 93
3.7. The payoff function of a strangle.......Page 96
3.8. Treasury yield curve, Treasury zero curve, and Treasury forward rate curve based on the quotes in Table 3.1.......Page 107
3.9. Constructing a short rate tree: step one.......Page 111
3.11. The complete short rate tree.......Page 112
4.1. A sample path of standard Brownian motion (μ = 0 and σ = 1).......Page 122
4.3. A sample path of Brownian motion with μ = -1 and σ = 1.......Page 124
4.4. A sample path of Brownian motion with μ = 0 and σ = 2.......Page 125
4.5. A sample path of Brownian motion with μ = 0 and σ = 0.5.......Page 126
4.6. A path of standard Brownian motion reflected after hitting.......Page 128
4.7. A path of standard Brownian motion reflected before hitting.......Page 129
4.8. A sample path of a compound Poisson process.......Page 133
4.9. A sample path of the shifted Poisson process {Xτ(t)}.......Page 135
3.1. A sample of quotes on U.S. Treasuries.......Page 105
3.2. The market term structure.......Page 110
5.1. The product rules in stochastic calculus.......Page 165