Stochastic processes with applications to finance

Content: 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.