Stochastic processes with applications to finance / Masaaki Kijima.

Format:

Book

Author/Creator:

Kijima, Masaaki, 1957-

Contributor:

Alumni and Friends Memorial Book Fund.

Language:

English

Subjects (All):

Stochastic processes.

Business mathematics.

Physical Description:

xi, 274 pages : illustrations ; 25 cm

Place of Publication:

Boca Raton, Fla. : Chapman & Hall/CRC, [2003]

Contents:

1 Elementary Calculus: Towards Ito's Formula 1

1.1 Exponential and Logarithmic Functions 1

1.2 Differentiation 4

1.3 Taylor's Expansion 8

1.4 Ito's Formula 10

1.5 Integration 11

2 Elements in Probability 19

2.1 The Sample Space and Probability 19

2.2 Discrete Random Variables 21

2.3 Continuous Random Variables 23

2.4 Multivariate Random Variables 25

2.5 Expectation 28

2.6 Conditional Expectation 32

2.7 Moment Generating Functions 35

3 Useful Distributions in Finance 41

3.1 Binomial Distributions 41

3.2 Other Discrete Distributions 43

3.3 Normal and Log-Normal Distributions 46

3.4 Other Continuous Distributions 50

3.5 Multivariate Normal Distributions 53

4 Derivative Securities 61

4.1 The Money-Market Account 61

4.2 Various Interest Rates 62

4.3 Forward and Futures Contracts 66

4.4 Options 68

4.5 Interest-Rate Derivatives 70

5 A Discrete-Time Model for Securities Market 75

5.1 Price Processes 75

5.2 The Portfolio Value and Stochastic Integral 78

5.3 No-Arbitrage and Replicating Portfolios 80

5.4 Martingales and the Asset Pricing Theorem 84

5.5 American Options 88

5.6 Change of Measure 90

6 Random Walks 95

6.1 The Mathematical Definition 95

6.2 Transition Probabilities 96

6.3 The Reflection Principle 99

6.4 The Change of Measure Revisited 102

6.5 The Binomial Securities Market Model 105

7 The Binomial Model 111

7.1 The Single-Period Model 111

7.2 The Multi-Period Model 114

7.3 The Binomial Model for American Options 118

7.4 The Trinomial Model 119

7.5 The Binomial Model for Interest-Rate Claims 121

8 A Discrete-Time Model for Defaultable Securities 127

8.1 The Hazard Rate 127

8.2 A Discrete Hazard Model 129

8.3 Pricing of Defaultable Securities 131

8.4 Correlated Defaults 135

9 Markov Chains 141

9.1 Markov and Strong Markov Properties 141

9.2 Transition Probabilities 142

9.3 Absorbing Markov Chains 145

9.4 Applications to Finance 148

10 Monte Carlo Simulation 157

10.1 Mathematical Backgrounds 157

10.2 The Idea of Monte Carlo 159

10.3 Generation of Random Numbers 162

10.4 Some Examples from Financial Engineering 165

10.5 Variance Reduction Methods 169

11 From Discrete to Continuous: Towards the Black-Scholes 175

11.1 Brownian Motions 175

11.2 The Central Limit Theorem Revisited 178

11.3 The Black-Scholes Formula 181

11.4 More on Brownian Motions 183

11.5 Poisson Processes 187

12 Basic Stochastic Processes in Continuous Time 193

12.1 Diffusion Processes 193

12.2 Sample Paths of Brownian Motions 197

12.3 Martingales 199

12.4 Stochastic Integrals 202

12.5 Stochastic Differential Equations 205

12.6 Ito's Formula Revisited 208

13 A Continuous-Time Model for Securities Market 215

13.1 Self-Financing Portfolio and No-Arbitrage 215

13.2 Price Process Models 217

13.3 The Black-Scholes Model 222

13.4 The Risk-Neutral Method 225

13.5 The Forward-Neutral Method 231

13.6 The Interest-Rate Term Structure 234

13.7 Pricing of Interest-Rate Derivatives 241

13.8 Pricing of Corporate Debts 245.

Notes:

Includes bibliographical references (pages 261-264) and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Alumni and Friends Memorial Book Fund.

ISBN:

1584882247

OCLC:

49679440