## Stochastic Methods in Asset Pricing

by Lyasoff

### Instructor Requests

This book presents a self-contained, comprehensive, and yet concise and condensed overview of the theory and methods of probability, integration, stochastic processes, optimal control, and their connections to the principles of asset pricing. The book is broader in scope than other introductory-level graduate texts on the subject, requires fewer prerequisites, and covers the relevant material at greater depth, mainly without rigorous technical proofs. The book brings to an introductory level certain concepts and topics that are usually found in advanced research monographs on stochastic processes and asset pricing, and it attempts to establish greater clarity on the connections between these two fields. The book begins with measure-theoretic probability and integration, and then develops the classical tools of stochastic calculus, including stochastic calculus with jumps and Lévy processes. For asset pricing, the book begins with a brief overview of risk preferences and general equilibrium in incomplete finite endowment economies, followed by the classical asset pricing setup in continuous time. The goal is to present a coherent single overview. For example, the text introduces discrete-time martingales as a consequence of market equilibrium considerations and connects them to the stochastic discount factors before offering a general definition. It covers concrete option pricing models (including stochastic volatility, exchange options, and the exercise of American options), Merton's investment--consumption problem, and several other applications. The book includes more than 450 exercises (with detailed hints). Appendixes cover analysis and topology and computer code related to the practical applications discussed in the text.
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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’s-Eye View of the One-Period 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 Quasi-invariance of the Gaussian Distribution under Translation (pg. 69)
3.2 Moment-Generating Functions, Laplace, and Fourier Transforms (pg. 72)
3.3 Conditioning and Independence (pg. 76)
3.4 Multivariate Gaussian Distribution (pg. 87)
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 Discrete-Time 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)
8.9 Brownian Sample Paths Are Nowhere Differentiable (pg. 213)
8.10 Some Special Features of Brownian Sample Paths (pg. 215)
9 Crash Course on Continuous-Time 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 European-Style Contingent Claims (pg. 360)
14.3 The Martingale Solution to Merton’s Problem (pg. 366)
14.4 American-Style 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 Moon-Landing 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 American-Style 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’s-Eye View of the One-Period 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 Quasi-invariance of the Gaussian Distribution under Translation (pg. 69)
3.2 Moment-Generating Functions, Laplace, and Fourier Transforms (pg. 72)
3.3 Conditioning and Independence (pg. 76)
3.4 Multivariate Gaussian Distribution (pg. 87)
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 Discrete-Time 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)
8.9 Brownian Sample Paths Are Nowhere Differentiable (pg. 213)
8.10 Some Special Features of Brownian Sample Paths (pg. 215)
9 Crash Course on Continuous-Time 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 European-Style Contingent Claims (pg. 360)
14.3 The Martingale Solution to Merton’s Problem (pg. 366)
14.4 American-Style 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 Moon-Landing 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 American-Style Call Options (pg. 546)
Select Bibliography (pg. 549)
Index (pg. 575)
1.1 Randomness in the Financial Markets (pg. 1)
1.2 A Bird’s-Eye View of the One-Period 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 Quasi-invariance of the Gaussian Distribution under Translation (pg. 69)
3.2 Moment-Generating Functions, Laplace, and Fourier Transforms (pg. 72)
3.3 Conditioning and Independence (pg. 76)
3.4 Multivariate Gaussian Distribution (pg. 87)
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)
8.9 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 European-Style Contingent Claims (pg. 360)
14.3 The Martingale Solution to Merton’s Problem (pg. 366)
14.4 American-Style 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 Moon-Landing 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 American-Style Call Options (pg. 546)
1.1 Randomness in the Financial Markets (pg. 1)
1.2 A Bird’s-Eye View of the One-Period 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 Quasi-invariance of the Gaussian Distribution under Translation (pg. 69)
3.2 Moment-Generating Functions, Laplace, and Fourier Transforms (pg. 72)
3.3 Conditioning and Independence (pg. 76)
3.4 Multivariate Gaussian Distribution (pg. 87)
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)
8.9 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 European-Style Contingent Claims (pg. 360)
14.3 The Martingale Solution to Merton’s Problem (pg. 366)
14.4 American-Style 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 Moon-Landing 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 American-Style Call Options (pg. 546)

#### Andrew Lyasoff

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