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Preface to the Second Edition | |

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Introduction | |

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Financial Derivatives: A Brief Introduction | |

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Introduction | |

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Definitions | |

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Types of Derivatives | |

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Forwards and Futures | |

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Options | |

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Swaps | |

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Conclusions | |

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References | |

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Exercises | |

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A Primer on the Arbitrage Theorem | |

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Introduction | |

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Notation | |

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A Basic Example of Asset Pricing | |

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A Numerical Example | |

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An Application: Lattice Models | |

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Payouts and Foreign Currencies | |

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Some Generalizations | |

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Conclusions: A Methodology for Pricing Assets | |

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References | |

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Appendix: Generalization of the Arbitrage Theorem | |

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Exercises | |

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Calculus in Deterministic and Stochastic Environments | |

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Introduction | |

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Some Tools of Standard Calculus | |

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Functions | |

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Convergence and Limit | |

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Partial Derivatives | |

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Conclusions | |

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References | |

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Exercises | |

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Pricing Derivatives: Models and Notation | |

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Introduction | |

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Pricing Functions | |

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Application: Another Pricing Method | |

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The Problem | |

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References | |

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Exercises | |

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Tools in Probability Theory | |

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Introduction | |

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Probability | |

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Moments | |

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Conditional Expectations | |

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Some Important Models | |

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Markov Processes and Their Relevance | |

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Convergence of Random Variables | |

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Conclusions | |

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References | |

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Exercises | |

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Martingales and Martingale Representations | |

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Introduction | |

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Definitions | |

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The Use of Martingales in Asset Pricing | |

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Relevance of Martingales in Stochastic Modeling | |

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Properties of Martingale Trajectories | |

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Examples of Martingales | |

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The Simplest Martingale | |

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Martingale Representations | |

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The First Stochastic Integral | |

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Martingale Methods and Pricing | |

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A Pricing Methodology | |

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Conclusions | |

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References | |

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Exercises | |

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Differentiation in Stochastic Environments | |

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Introduction | |

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Motivation | |

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A Framework for Discussing Differentiation | |

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The "Size" of Incremental Errors | |

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One Implication | |

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Putting the Results Together | |

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Conclusions | |

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References | |

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Exercises | |

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The Wiener Process and Rare Events in Financial Markets | |

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Introduction | |

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Two Generic Models | |

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SDE in Discrete Intervals, Again | |

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Characterizing Rare and Normal Events | |

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A Model for Rare Events | |

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Moments That Matter | |

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Conclusions | |

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Rare and Normal Events in Practice | |

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References | |

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Exercises | |

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Integration in Stochastic Environments: The Ito Integral | |

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Introduction | |

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The Ito Integral | |

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Properties of the Ito Integral | |

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Other Properties of the Ito Integral | |

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Integrals with Respect to Jump Processes | |

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Conclusions | |

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References | |

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Exercises | |

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Ito's Lemma | |

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Introduction | |

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Types of Derivatives | |

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Ito's Lemma | |

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The Ito Formula | |

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Uses of Ito's Lemma | |

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Integral Form of Ito's Lemma | |

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Ito's Formula in More Complex Settings | |

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Conclusions | |

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References | |

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Exercises | |

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The Dynamics of Derivative Prices: Stochastic Differential Equations | |

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Introduction | |

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A Geometric Description of Paths Implied by SDEs | |

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Solution of SDEs | |

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Major Models of SDEs | |

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Stochastic Volatility | |

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Conclusions | |

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References | |

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Exercises | |

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Pricing Derivative Products: Partial Differential Equations | |

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Introduction | |

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Forming Risk-Free Portfolios | |

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Accuracy of the Method | |

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Partial Differential Equations | |

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Classification of PDEs | |

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A Reminder: Bivariate, Second-Degree Equations | |

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Types of PDEs | |

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Conclusions | |

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References | |

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Exercises | |

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The Black--Scholes PDE: An Application | |

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Introduction | |

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The Black--Scholes PDE | |

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PDEs in Asset Pricing | |

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Exotic Options | |

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Solving PDEs in Practice | |

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Conclusions | |

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References | |

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Exercises | |

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Pricing Derivative Products: Equivalent Martingale Measures | |

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Translations of Probabilities | |

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Changing Means | |

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The Girsanov Theorem | |

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Statement of the Girsanov Theorem | |

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A Discussion of the Girsanov Theorem | |

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Which Probabilities? | |

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A Method for Generating Equivalent Probabilities | |

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Conclusions | |

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References | |

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Exercises | |

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Equivalent Martingale Measures: Applications | |

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Introduction | |

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A Martingale Measure | |

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Converting Asset Prices into Martingales | |

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Application: The Black--Scholes Formula | |

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Comparing Martingale and PDE Approaches | |

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Conclusions | |

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References | |

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Exercises | |

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New Results and Tools for Interest-Sensitive Securities | |

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Introduction | |

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A Summary | |

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Interest Rate Derivatives | |

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Complications | |

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Conclusions | |

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References | |

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Exercises | |

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Arbitrage Theorem in a New Setting: Normalization and Random Interest Rates | |

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Introduction | |

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A Model for New Instruments | |

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Conclusions | |

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References | |

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Exercises | |

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Modeling Term Structure and Related Concepts | |

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Introduction | |

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Main Concepts | |

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A Bond Pricing Equation | |

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Forward Rates and Bond Prices | |

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Conclusions: Relevance of the Relationships | |

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References | |

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Exercises | |

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Classical and HJM Approaches to Fixed Income | |

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Introduction | |

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The Classical Approach | |

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The HJM Approach to Term Structure | |

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How to Fit r[subscript t] to Initial Term Structure | |

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Conclusions | |

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References | |

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Exercises | |

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Classical PDE Analysis for Interest Rate Derivatives | |

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Introduction | |

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The Framework | |

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Market Price of Interest Rate Risk | |

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Derivation of the PDE | |

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Closed-Form Solutions of the PDE | |

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Conclusions | |

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References | |

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Exercises | |

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Relating Conditional Expectations to PDEs | |

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Introduction | |

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From Conditional Expectations to PDEs | |