Stochastic Methods in Asset Pricing
by Lyasoff
ISBN: 9780262364034  Copyright 2017
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Contents (pg. vii)  
Preface (pg. xi)  
Notation (pg. xv)  
Preliminaries (pg. xvii)  
1 Probability Spaces and Related Structures (pg. 1)  
1.1 Randomness in the Financial Markets (pg. 1)  
1.2 A Bird’sEye View of the OnePeriod Binomial Model (pg. 4)  
1.3 Probability Spaces (pg. 8)  
1.4 Coin Toss Space and Random Walk (pg. 17)  
1.5 Borel (pg. 26)  
2 Integration (pg. 35)  
2.1 Measurable Functions and Random Variables (pg. 35)  
2.2 Distribution Laws (pg. 42)  
2.3 Lebesgue Integral (pg. 46)  
2.4 Convergence of Integrals (pg. 55)  
2.5 Integration Tools (pg. 57)  
2.6 The Inverse of an Increasing Function (pg. 66)  
3 Absolute Continuity, Conditioning, and Independence (pg. 69)  
3.1 Quasiinvariance of the Gaussian Distribution under Translation (pg. 69)  
3.2 MomentGenerating Functions, Laplace, and Fourier Transforms (pg. 72)  
3.3 Conditioning and Independence (pg. 76)  
3.4 Multivariate Gaussian Distribution (pg. 87)  
3.5 Hermite–Gauss Quadratures (pg. 92)  
4 Convergence of Random Variables (pg. 97)  
4.1 Types of Convergence for Sequences of Random Variables (pg. 97)  
4.2 Uniform Integrability (pg. 103)  
4.3 Sequences of Independent Random Variables and Events (pg. 105)  
4.4 Law of Large Numbers and the Central Limit Theorem (pg. 108)  
5 The Art of Random Sampling (pg. 113)  
5.1 Motivation (pg. 113)  
5.2 Layer Cake Formulas (pg. 114)  
5.3 The Antithetic Variates Method (pg. 117)  
5.4 The Importance Sampling Method (pg. 119)  
5.5 The Acceptance–Rejection Method (pg. 120)  
6 Equilibrium Asset Pricing in Finite Economies (pg. 123)  
6.1 Information Structure (pg. 124)  
6.2 Risk Preferences (pg. 126)  
6.3 The Multiperiod Endowment Economy (pg. 137)  
6.4 General Equilibrium (pg. 144)  
6.5 The Two Fundamental Theorems of Asset Pricing (pg. 150)  
6.6 From Stochastic Discount Factors to Equivalent Measures and Local Martingales (pg. 155)  
7 Crash Course on DiscreteTime Martingales (pg. 161)  
7.1 Basic Concepts and Definitions (pg. 161)  
7.2 Predictable Compensators (pg. 168)  
7.3 Fundamental Inequalities and Convergence (pg. 170)  
8 Stochastic Processes and Brownian Motion (pg. 175)  
8.1 General Properties and Definitions (pg. 176)  
8.2 Limit of the Binomial Asset Pricing Model (pg. 182)  
8.3 Construction of Brownian Motion and First Properties (pg. 186)  
8.4 The Wiener Measure (pg. 189)  
8.5 Filtrations, Stopping Times, and Such (pg. 191)  
8.6 Brownian Filtrations (pg. 206)  
8.7 Total Variation (pg. 208)  
8.8 Quadratic Variation (pg. 211)  
8.9 Brownian Sample Paths Are Nowhere Differentiable (pg. 213)  
8.10 Some Special Features of Brownian Sample Paths (pg. 215)  
9 Crash Course on ContinuousTime Martingales (pg. 221)  
9.1 Definitions and First Properties (pg. 221)  
9.2 Poisson Process and First Encounter with Lévy Processes (pg. 225)  
9.3 Regularity of Paths, Optional Stopping, and Convergence (pg. 227)  
9.4 Doob–Meyer Decomposition (pg. 237)  
9.5 Local Martingales and Semimartingales (pg. 240)  
9.6 The Space of (pg. 248)  
9.7 The Binomial Asset Pricing Model Revisited (pg. 252)  
10 Stochastic Integration (pg. 257)  
10.1 Basic Examples and Intuition (pg. 258)  
10.2 Stochastic Integrals with Respect to Continuous Local Martingales (pg. 262)  
10.3 Stochastic Integrals with Respect to Continuous Semimartingales (pg. 271)  
10.4 Itô’s Formula (pg. 273)  
10.5 Stochastic Integrals with Respect to Brownian Motion (pg. 275)  
10.6 Girsanov’s Theorem (pg. 281)  
10.7 Local Times and Tanaka’s Formula (pg. 285)  
10.8 Reflected Brownian Motion (pg. 290)  
11 Stochastic Differential Equations (pg. 295)  
11.1 An Example (pg. 295)  
11.2 Strong and Weak Solutions (pg. 296)  
11.3 Existence of Solutions (pg. 307)  
11.4 Linear Stochastic Differential Equations (pg. 315)  
11.5 Some Common Diffusion Models Used in Asset Pricing (pg. 319)  
12 The Connection between SDEs and PDEs (pg. 325)  
12.1 Feynman–Kac Formula (pg. 325)  
12.2 Fokker–Planck Equation (pg. 331)  
13 Brief Introduction to Asset Pricing in Continuous Time (pg. 337)  
13.1 Basic Concepts and Definitions (pg. 337)  
13.2 Trading Strategy and Wealth Dynamics (pg. 345)  
13.3 Equivalent Local Martingale Measures (pg. 348)  
13.4 The Two Fundamental Theorems of Asset Pricing (pg. 352)  
14 Replication and Arbitrage (pg. 357)  
14.1 Résumé of Malliavin Calculus (pg. 357)  
14.2 EuropeanStyle Contingent Claims (pg. 360)  
14.3 The Martingale Solution to Merton’s Problem (pg. 366)  
14.4 AmericanStyle Contingent Claims (pg. 370)  
14.5 Put–Call Symmetry and Foreign Exchange Options (pg. 378)  
14.6 Exchange Options (pg. 380)  
14.7 Stochastic Volatility Models (pg. 382)  
14.8 Dupire’s Formula (pg. 394)  
15 Résumé of Stochastic Calculus with Discontinuous Processes (pg. 397)  
15.1 Martingales, Local Martingales, and Semimartingales with Jumps (pg. 397)  
15.2 Stochastic Integrals with Respect to Semimartingales with Jumps (pg. 409)  
15.3 Quadratic Variation and Itô’s Formula (pg. 417)  
16 Random Measures and Lévy Processes (pg. 425)  
16.1 Poisson Random Measures (pg. 425)  
16.2 Lévy Processes (pg. 433)  
16.3 Stochastic Integrals with Respect to Lévy Processes (pg. 439)  
16.4 Stochastic Exponents (pg. 444)  
16.5 Change of Measure and Removal of the Drift (pg. 447)  
16.6 Lévy–Itô Diffusions (pg. 452)  
16.7 An Asset Pricing Model with Jumps in the Returns (pg. 456)  
17 Résumé of the Theory and Methods of Stochastic Optimal Control (pg. 463)  
17.1 The MoonLanding Problem (pg. 464)  
17.2 Principle of Dynamic Programming and the HJB Equation (pg. 465)  
17.3 Some Variations of the PDP and the HJB Equation (pg. 474)  
18 Applications to Dynamic Asset Pricing (pg. 481)  
18.1 Merton’s Problem with Intertemporal Consumption and No Rebalancing Costs (pg. 481)  
18.2 Merton’s Problem with Intertemporal Consumption and Rebalancing Costs (pg. 490)  
18.3 Real Options (pg. 502)  
18.4 The Exercise Boundary for American Calls and Puts (pg. 510)  
18.5 Corporate Debt, Equity, Dividend Policy, and the Modigliani–Miller Proposition (pg. 517)  
Appendix A: Résumé of Analysis and Topology (pg. 523)  
Appendix B: Computer Code (pg. 541)  
B.1 Working with Market Data (pg. 541)  
B.2 Simulation of Multivariate Gaussian Laws (pg. 543)  
B.3 Numerical Program for AmericanStyle Call Options (pg. 546)  
Select Bibliography (pg. 549)  
Index (pg. 575)  
Contents (pg. vii)  
Preface (pg. xi)  
Notation (pg. xv)  
Preliminaries (pg. xvii)  
1 Probability Spaces and Related Structures (pg. 1)  
1.1 Randomness in the Financial Markets (pg. 1)  
1.2 A Bird’sEye View of the OnePeriod Binomial Model (pg. 4)  
1.3 Probability Spaces (pg. 8)  
1.4 Coin Toss Space and Random Walk (pg. 17)  
1.5 Borel (pg. 26)  
2 Integration (pg. 35)  
2.1 Measurable Functions and Random Variables (pg. 35)  
2.2 Distribution Laws (pg. 42)  
2.3 Lebesgue Integral (pg. 46)  
2.4 Convergence of Integrals (pg. 55)  
2.5 Integration Tools (pg. 57)  
2.6 The Inverse of an Increasing Function (pg. 66)  
3 Absolute Continuity, Conditioning, and Independence (pg. 69)  
3.1 Quasiinvariance of the Gaussian Distribution under Translation (pg. 69)  
3.2 MomentGenerating Functions, Laplace, and Fourier Transforms (pg. 72)  
3.3 Conditioning and Independence (pg. 76)  
3.4 Multivariate Gaussian Distribution (pg. 87)  
3.5 Hermite–Gauss Quadratures (pg. 92)  
4 Convergence of Random Variables (pg. 97)  
4.1 Types of Convergence for Sequences of Random Variables (pg. 97)  
4.2 Uniform Integrability (pg. 103)  
4.3 Sequences of Independent Random Variables and Events (pg. 105)  
4.4 Law of Large Numbers and the Central Limit Theorem (pg. 108)  
5 The Art of Random Sampling (pg. 113)  
5.1 Motivation (pg. 113)  
5.2 Layer Cake Formulas (pg. 114)  
5.3 The Antithetic Variates Method (pg. 117)  
5.4 The Importance Sampling Method (pg. 119)  
5.5 The Acceptance–Rejection Method (pg. 120)  
6 Equilibrium Asset Pricing in Finite Economies (pg. 123)  
6.1 Information Structure (pg. 124)  
6.2 Risk Preferences (pg. 126)  
6.3 The Multiperiod Endowment Economy (pg. 137)  
6.4 General Equilibrium (pg. 144)  
6.5 The Two Fundamental Theorems of Asset Pricing (pg. 150)  
6.6 From Stochastic Discount Factors to Equivalent Measures and Local Martingales (pg. 155)  
7 Crash Course on DiscreteTime Martingales (pg. 161)  
7.1 Basic Concepts and Definitions (pg. 161)  
7.2 Predictable Compensators (pg. 168)  
7.3 Fundamental Inequalities and Convergence (pg. 170)  
8 Stochastic Processes and Brownian Motion (pg. 175)  
8.1 General Properties and Definitions (pg. 176)  
8.2 Limit of the Binomial Asset Pricing Model (pg. 182)  
8.3 Construction of Brownian Motion and First Properties (pg. 186)  
8.4 The Wiener Measure (pg. 189)  
8.5 Filtrations, Stopping Times, and Such (pg. 191)  
8.6 Brownian Filtrations (pg. 206)  
8.7 Total Variation (pg. 208)  
8.8 Quadratic Variation (pg. 211)  
8.9 Brownian Sample Paths Are Nowhere Differentiable (pg. 213)  
8.10 Some Special Features of Brownian Sample Paths (pg. 215)  
9 Crash Course on ContinuousTime Martingales (pg. 221)  
9.1 Definitions and First Properties (pg. 221)  
9.2 Poisson Process and First Encounter with Lévy Processes (pg. 225)  
9.3 Regularity of Paths, Optional Stopping, and Convergence (pg. 227)  
9.4 Doob–Meyer Decomposition (pg. 237)  
9.5 Local Martingales and Semimartingales (pg. 240)  
9.6 The Space of (pg. 248)  
9.7 The Binomial Asset Pricing Model Revisited (pg. 252)  
10 Stochastic Integration (pg. 257)  
10.1 Basic Examples and Intuition (pg. 258)  
10.2 Stochastic Integrals with Respect to Continuous Local Martingales (pg. 262)  
10.3 Stochastic Integrals with Respect to Continuous Semimartingales (pg. 271)  
10.4 Itô’s Formula (pg. 273)  
10.5 Stochastic Integrals with Respect to Brownian Motion (pg. 275)  
10.6 Girsanov’s Theorem (pg. 281)  
10.7 Local Times and Tanaka’s Formula (pg. 285)  
10.8 Reflected Brownian Motion (pg. 290)  
11 Stochastic Differential Equations (pg. 295)  
11.1 An Example (pg. 295)  
11.2 Strong and Weak Solutions (pg. 296)  
11.3 Existence of Solutions (pg. 307)  
11.4 Linear Stochastic Differential Equations (pg. 315)  
11.5 Some Common Diffusion Models Used in Asset Pricing (pg. 319)  
12 The Connection between SDEs and PDEs (pg. 325)  
12.1 Feynman–Kac Formula (pg. 325)  
12.2 Fokker–Planck Equation (pg. 331)  
13 Brief Introduction to Asset Pricing in Continuous Time (pg. 337)  
13.1 Basic Concepts and Definitions (pg. 337)  
13.2 Trading Strategy and Wealth Dynamics (pg. 345)  
13.3 Equivalent Local Martingale Measures (pg. 348)  
13.4 The Two Fundamental Theorems of Asset Pricing (pg. 352)  
14 Replication and Arbitrage (pg. 357)  
14.1 Résumé of Malliavin Calculus (pg. 357)  
14.2 EuropeanStyle Contingent Claims (pg. 360)  
14.3 The Martingale Solution to Merton’s Problem (pg. 366)  
14.4 AmericanStyle Contingent Claims (pg. 370)  
14.5 Put–Call Symmetry and Foreign Exchange Options (pg. 378)  
14.6 Exchange Options (pg. 380)  
14.7 Stochastic Volatility Models (pg. 382)  
14.8 Dupire’s Formula (pg. 394)  
15 Résumé of Stochastic Calculus with Discontinuous Processes (pg. 397)  
15.1 Martingales, Local Martingales, and Semimartingales with Jumps (pg. 397)  
15.2 Stochastic Integrals with Respect to Semimartingales with Jumps (pg. 409)  
15.3 Quadratic Variation and Itô’s Formula (pg. 417)  
16 Random Measures and Lévy Processes (pg. 425)  
16.1 Poisson Random Measures (pg. 425)  
16.2 Lévy Processes (pg. 433)  
16.3 Stochastic Integrals with Respect to Lévy Processes (pg. 439)  
16.4 Stochastic Exponents (pg. 444)  
16.5 Change of Measure and Removal of the Drift (pg. 447)  
16.6 Lévy–Itô Diffusions (pg. 452)  
16.7 An Asset Pricing Model with Jumps in the Returns (pg. 456)  
17 Résumé of the Theory and Methods of Stochastic Optimal Control (pg. 463)  
17.1 The MoonLanding Problem (pg. 464)  
17.2 Principle of Dynamic Programming and the HJB Equation (pg. 465)  
17.3 Some Variations of the PDP and the HJB Equation (pg. 474)  
18 Applications to Dynamic Asset Pricing (pg. 481)  
18.1 Merton’s Problem with Intertemporal Consumption and No Rebalancing Costs (pg. 481)  
18.2 Merton’s Problem with Intertemporal Consumption and Rebalancing Costs (pg. 490)  
18.3 Real Options (pg. 502)  
18.4 The Exercise Boundary for American Calls and Puts (pg. 510)  
18.5 Corporate Debt, Equity, Dividend Policy, and the Modigliani–Miller Proposition (pg. 517)  
Appendix A: Résumé of Analysis and Topology (pg. 523)  
Appendix B: Computer Code (pg. 541)  
B.1 Working with Market Data (pg. 541)  
B.2 Simulation of Multivariate Gaussian Laws (pg. 543)  
B.3 Numerical Program for AmericanStyle Call Options (pg. 546)  
Select Bibliography (pg. 549)  
Index (pg. 575)  
1.1 Randomness in the Financial Markets (pg. 1)  
1.2 A Bird’sEye View of the OnePeriod Binomial Model (pg. 4)  
1.3 Probability Spaces (pg. 8)  
1.4 Coin Toss Space and Random Walk (pg. 17)  
1.5 Borel (pg. 26)  
2.1 Measurable Functions and Random Variables (pg. 35)  
2.2 Distribution Laws (pg. 42)  
2.3 Lebesgue Integral (pg. 46)  
2.4 Convergence of Integrals (pg. 55)  
2.5 Integration Tools (pg. 57)  
2.6 The Inverse of an Increasing Function (pg. 66)  
3.1 Quasiinvariance of the Gaussian Distribution under Translation (pg. 69)  
3.2 MomentGenerating Functions, Laplace, and Fourier Transforms (pg. 72)  
3.3 Conditioning and Independence (pg. 76)  
3.4 Multivariate Gaussian Distribution (pg. 87)  
3.5 Hermite–Gauss Quadratures (pg. 92)  
4.1 Types of Convergence for Sequences of Random Variables (pg. 97)  
4.2 Uniform Integrability (pg. 103)  
4.3 Sequences of Independent Random Variables and Events (pg. 105)  
4.4 Law of Large Numbers and the Central Limit Theorem (pg. 108)  
5.1 Motivation (pg. 113)  
5.2 Layer Cake Formulas (pg. 114)  
5.3 The Antithetic Variates Method (pg. 117)  
5.4 The Importance Sampling Method (pg. 119)  
5.5 The Acceptance–Rejection Method (pg. 120)  
6.1 Information Structure (pg. 124)  
6.2 Risk Preferences (pg. 126)  
6.3 The Multiperiod Endowment Economy (pg. 137)  
6.4 General Equilibrium (pg. 144)  
6.5 The Two Fundamental Theorems of Asset Pricing (pg. 150)  
6.6 From Stochastic Discount Factors to Equivalent Measures and Local Martingales (pg. 155)  
7.1 Basic Concepts and Definitions (pg. 161)  
7.2 Predictable Compensators (pg. 168)  
7.3 Fundamental Inequalities and Convergence (pg. 170)  
8.1 General Properties and Definitions (pg. 176)  
8.2 Limit of the Binomial Asset Pricing Model (pg. 182)  
8.3 Construction of Brownian Motion and First Properties (pg. 186)  
8.4 The Wiener Measure (pg. 189)  
8.5 Filtrations, Stopping Times, and Such (pg. 191)  
8.6 Brownian Filtrations (pg. 206)  
8.7 Total Variation (pg. 208)  
8.8 Quadratic Variation (pg. 211)  
8.9 Brownian Sample Paths Are Nowhere Differentiable (pg. 213)  
8.10 Some Special Features of Brownian Sample Paths (pg. 215)  
9.1 Definitions and First Properties (pg. 221)  
9.2 Poisson Process and First Encounter with Lévy Processes (pg. 225)  
9.3 Regularity of Paths, Optional Stopping, and Convergence (pg. 227)  
9.4 Doob–Meyer Decomposition (pg. 237)  
9.5 Local Martingales and Semimartingales (pg. 240)  
9.6 The Space of (pg. 248)  
9.7 The Binomial Asset Pricing Model Revisited (pg. 252)  
10.1 Basic Examples and Intuition (pg. 258)  
10.2 Stochastic Integrals with Respect to Continuous Local Martingales (pg. 262)  
10.3 Stochastic Integrals with Respect to Continuous Semimartingales (pg. 271)  
10.4 Itô’s Formula (pg. 273)  
10.5 Stochastic Integrals with Respect to Brownian Motion (pg. 275)  
10.6 Girsanov’s Theorem (pg. 281)  
10.7 Local Times and Tanaka’s Formula (pg. 285)  
10.8 Reflected Brownian Motion (pg. 290)  
11.1 An Example (pg. 295)  
11.2 Strong and Weak Solutions (pg. 296)  
11.3 Existence of Solutions (pg. 307)  
11.4 Linear Stochastic Differential Equations (pg. 315)  
11.5 Some Common Diffusion Models Used in Asset Pricing (pg. 319)  
12.1 Feynman–Kac Formula (pg. 325)  
12.2 Fokker–Planck Equation (pg. 331)  
13.1 Basic Concepts and Definitions (pg. 337)  
13.2 Trading Strategy and Wealth Dynamics (pg. 345)  
13.3 Equivalent Local Martingale Measures (pg. 348)  
13.4 The Two Fundamental Theorems of Asset Pricing (pg. 352)  
14.1 Résumé of Malliavin Calculus (pg. 357)  
14.2 EuropeanStyle Contingent Claims (pg. 360)  
14.3 The Martingale Solution to Merton’s Problem (pg. 366)  
14.4 AmericanStyle Contingent Claims (pg. 370)  
14.5 Put–Call Symmetry and Foreign Exchange Options (pg. 378)  
14.6 Exchange Options (pg. 380)  
14.7 Stochastic Volatility Models (pg. 382)  
14.8 Dupire’s Formula (pg. 394)  
15.1 Martingales, Local Martingales, and Semimartingales with Jumps (pg. 397)  
15.2 Stochastic Integrals with Respect to Semimartingales with Jumps (pg. 409)  
15.3 Quadratic Variation and Itô’s Formula (pg. 417)  
16.1 Poisson Random Measures (pg. 425)  
16.2 Lévy Processes (pg. 433)  
16.3 Stochastic Integrals with Respect to Lévy Processes (pg. 439)  
16.4 Stochastic Exponents (pg. 444)  
16.5 Change of Measure and Removal of the Drift (pg. 447)  
16.6 Lévy–Itô Diffusions (pg. 452)  
16.7 An Asset Pricing Model with Jumps in the Returns (pg. 456)  
17.1 The MoonLanding Problem (pg. 464)  
17.2 Principle of Dynamic Programming and the HJB Equation (pg. 465)  
17.3 Some Variations of the PDP and the HJB Equation (pg. 474)  
18.1 Merton’s Problem with Intertemporal Consumption and No Rebalancing Costs (pg. 481)  
18.2 Merton’s Problem with Intertemporal Consumption and Rebalancing Costs (pg. 490)  
18.3 Real Options (pg. 502)  
18.4 The Exercise Boundary for American Calls and Puts (pg. 510)  
18.5 Corporate Debt, Equity, Dividend Policy, and the Modigliani–Miller Proposition (pg. 517)  
B.1 Working with Market Data (pg. 541)  
B.2 Simulation of Multivariate Gaussian Laws (pg. 543)  
B.3 Numerical Program for AmericanStyle Call Options (pg. 546)  
1.1 Randomness in the Financial Markets (pg. 1)  
1.2 A Bird’sEye View of the OnePeriod Binomial Model (pg. 4)  
1.3 Probability Spaces (pg. 8)  
1.4 Coin Toss Space and Random Walk (pg. 17)  
1.5 Borel (pg. 26)  
2.1 Measurable Functions and Random Variables (pg. 35)  
2.2 Distribution Laws (pg. 42)  
2.3 Lebesgue Integral (pg. 46)  
2.4 Convergence of Integrals (pg. 55)  
2.5 Integration Tools (pg. 57)  
2.6 The Inverse of an Increasing Function (pg. 66)  
3.1 Quasiinvariance of the Gaussian Distribution under Translation (pg. 69)  
3.2 MomentGenerating Functions, Laplace, and Fourier Transforms (pg. 72)  
3.3 Conditioning and Independence (pg. 76)  
3.4 Multivariate Gaussian Distribution (pg. 87)  
3.5 Hermite–Gauss Quadratures (pg. 92)  
4.1 Types of Convergence for Sequences of Random Variables (pg. 97)  
4.2 Uniform Integrability (pg. 103)  
4.3 Sequences of Independent Random Variables and Events (pg. 105)  
4.4 Law of Large Numbers and the Central Limit Theorem (pg. 108)  
5.1 Motivation (pg. 113)  
5.2 Layer Cake Formulas (pg. 114)  
5.3 The Antithetic Variates Method (pg. 117)  
5.4 The Importance Sampling Method (pg. 119)  
5.5 The Acceptance–Rejection Method (pg. 120)  
6.1 Information Structure (pg. 124)  
6.2 Risk Preferences (pg. 126)  
6.3 The Multiperiod Endowment Economy (pg. 137)  
6.4 General Equilibrium (pg. 144)  
6.5 The Two Fundamental Theorems of Asset Pricing (pg. 150)  
6.6 From Stochastic Discount Factors to Equivalent Measures and Local Martingales (pg. 155)  
7.1 Basic Concepts and Definitions (pg. 161)  
7.2 Predictable Compensators (pg. 168)  
7.3 Fundamental Inequalities and Convergence (pg. 170)  
8.1 General Properties and Definitions (pg. 176)  
8.2 Limit of the Binomial Asset Pricing Model (pg. 182)  
8.3 Construction of Brownian Motion and First Properties (pg. 186)  
8.4 The Wiener Measure (pg. 189)  
8.5 Filtrations, Stopping Times, and Such (pg. 191)  
8.6 Brownian Filtrations (pg. 206)  
8.7 Total Variation (pg. 208)  
8.8 Quadratic Variation (pg. 211)  
8.9 Brownian Sample Paths Are Nowhere Differentiable (pg. 213)  
8.10 Some Special Features of Brownian Sample Paths (pg. 215)  
9.1 Definitions and First Properties (pg. 221)  
9.2 Poisson Process and First Encounter with Lévy Processes (pg. 225)  
9.3 Regularity of Paths, Optional Stopping, and Convergence (pg. 227)  
9.4 Doob–Meyer Decomposition (pg. 237)  
9.5 Local Martingales and Semimartingales (pg. 240)  
9.6 The Space of (pg. 248)  
9.7 The Binomial Asset Pricing Model Revisited (pg. 252)  
10.1 Basic Examples and Intuition (pg. 258)  
10.2 Stochastic Integrals with Respect to Continuous Local Martingales (pg. 262)  
10.3 Stochastic Integrals with Respect to Continuous Semimartingales (pg. 271)  
10.4 Itô’s Formula (pg. 273)  
10.5 Stochastic Integrals with Respect to Brownian Motion (pg. 275)  
10.6 Girsanov’s Theorem (pg. 281)  
10.7 Local Times and Tanaka’s Formula (pg. 285)  
10.8 Reflected Brownian Motion (pg. 290)  
11.1 An Example (pg. 295)  
11.2 Strong and Weak Solutions (pg. 296)  
11.3 Existence of Solutions (pg. 307)  
11.4 Linear Stochastic Differential Equations (pg. 315)  
11.5 Some Common Diffusion Models Used in Asset Pricing (pg. 319)  
12.1 Feynman–Kac Formula (pg. 325)  
12.2 Fokker–Planck Equation (pg. 331)  
13.1 Basic Concepts and Definitions (pg. 337)  
13.2 Trading Strategy and Wealth Dynamics (pg. 345)  
13.3 Equivalent Local Martingale Measures (pg. 348)  
13.4 The Two Fundamental Theorems of Asset Pricing (pg. 352)  
14.1 Résumé of Malliavin Calculus (pg. 357)  
14.2 EuropeanStyle Contingent Claims (pg. 360)  
14.3 The Martingale Solution to Merton’s Problem (pg. 366)  
14.4 AmericanStyle Contingent Claims (pg. 370)  
14.5 Put–Call Symmetry and Foreign Exchange Options (pg. 378)  
14.6 Exchange Options (pg. 380)  
14.7 Stochastic Volatility Models (pg. 382)  
14.8 Dupire’s Formula (pg. 394)  
15.1 Martingales, Local Martingales, and Semimartingales with Jumps (pg. 397)  
15.2 Stochastic Integrals with Respect to Semimartingales with Jumps (pg. 409)  
15.3 Quadratic Variation and Itô’s Formula (pg. 417)  
16.1 Poisson Random Measures (pg. 425)  
16.2 Lévy Processes (pg. 433)  
16.3 Stochastic Integrals with Respect to Lévy Processes (pg. 439)  
16.4 Stochastic Exponents (pg. 444)  
16.5 Change of Measure and Removal of the Drift (pg. 447)  
16.6 Lévy–Itô Diffusions (pg. 452)  
16.7 An Asset Pricing Model with Jumps in the Returns (pg. 456)  
17.1 The MoonLanding Problem (pg. 464)  
17.2 Principle of Dynamic Programming and the HJB Equation (pg. 465)  
17.3 Some Variations of the PDP and the HJB Equation (pg. 474)  
18.1 Merton’s Problem with Intertemporal Consumption and No Rebalancing Costs (pg. 481)  
18.2 Merton’s Problem with Intertemporal Consumption and Rebalancing Costs (pg. 490)  
18.3 Real Options (pg. 502)  
18.4 The Exercise Boundary for American Calls and Puts (pg. 510)  
18.5 Corporate Debt, Equity, Dividend Policy, and the Modigliani–Miller Proposition (pg. 517)  
B.1 Working with Market Data (pg. 541)  
B.2 Simulation of Multivariate Gaussian Laws (pg. 543)  
B.3 Numerical Program for AmericanStyle Call Options (pg. 546) 
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