The Mathematics of Derivatives: Tools for Designing Numerical Algorithms

by Robert L. Navin

• ISBN13: 9780470099773
• ISBN10: 0470099771
• Additional ISBN(s): 9780470047255
• Format: eBook
• Copyright: 2007-03-01
• Publisher: Wiley
• More Book Details
• Purchase Benefits

Summary

Praise for The Mathematics of Derivatives "The Mathematics of Derivatives provides a concise pedagogical discussion of both fundamental and very recent developments in mathematical finance, and is particularly well suited for readers with a science or engineering background. It is written from the point of view of a physicist focused on providing an understanding of the methodology and the assumptions behind derivative pricing. Navin has a unique and elegant viewpoint, and will help mathematically sophisticated readers rapidly get up to speed in the latest Wall Street financial innovations." -David Montano, Managing Director JPMorgan Securities A stylish and practical introduction to the key concepts in financial mathematics, this book tackles key fundamentals in the subject in an intuitive and refreshing manner whilst also providing detailed analytical and numerical schema for solving interesting derivatives pricing problems. If Richard Feynman wrote an introduction to financial mathematics, it might look similar. The problem and solution sets are first rate." -Barry Ryan, Partner Bhramavira Capital Partners, London "This is a great book for anyone beginning (or contemplating), a career in financial research or analytic programming. Navin dissects a huge, complex topic into a series of discrete, concise, accessible lectures that combine the required mathematical theory with relevant applications to real-world markets. I wish this book was around when I started in finance. It would have saved me a lot of time and aggravation." -Larry Magargal

Table of Contents

Preface

Acknowledgments

The Models

Introduction to the Techniques of Derivative Modeling

Introduction

Models

What Is a Derivative?

What Is a Model?

Two Initial Methods for Modeling Derivatives

Price Processes

The Archetypal Security Process: Normal Returns

Book Outline

Preliminary Mathematical Tools

Probability Distributions

n-Dimensional Jacobians and n-Form Algebra

Functional Analysis and Fourier Transforms

Normal (Central) Limit Theorem

Random Walks

Correlation

Functions of Two/More Variables: Path Integrals

Differential Forms

Stochastic Calculus

Wiener Process

Ito's Lemma

Variable Changes to Get the Martingale

Other Processes: Multivariable Correlations

Applications of Stochastic Calculus to Finance

Risk Premium Derivation

Analytic Formula for the Expected Payoff of a European Option

From Stochastic Processes Formalism to Differential Equation Formalism

Backward and Forward Kolmogorov Equations

Derivation of Black-Scholes Equation, Risk-Neutral Pricing

Risks and Trading Strategies

Understanding the Black-Scholes Equation

Black-Scholes Equation: A Type of Backward Kolmogorov Equation

Forward Price

Black-Scholes Equation: Risk-Neutral Pricing

Black-Scholes Equation: Relation to Risk Premium Definition

Black-Scholes Equation Applies to Currency Options: Hidden Symmetry 1

Black-Scholes Equation in Martingale Variables: Hidden Symmetry 2

Black-Scholes Equation with Stock as a 'Derivative' of Option Price: Hidden Symmetry 3

Interest Rate Hedging

Euler's Relation

Interest Rate Dependence

Term-Structured Rates Hedging: Duration Bucketing

Algorithm for Deciding Which Hedging Instruments to Use

Interest Rate Derivatives: HJM Models

Hull-White Model Derivation

Process and Pricing Equation

Analytic Zero-Coupon Bond Valuation

Analytic Bond Call Option

Calibration

Arbitrage-Free Pricing for Interest Rate Derivatives: HJM

Differential Equations, Boundary Conditions, and Solutions

Boundary Conditions and Unique Solutions to Differential Equations

Solving the Black-Scholes or Heat Equation Analytically

Green's Functions

Separation of Variables

Solving the Black-Scholes Equation Numerically

Finite Difference Methods: Explicit/Implicit Methods, Variable Choice

Gaussian Kurtosis (and Skew = 0), Faster Convergence

Call/Put Options:Grid Point Shift Factor for Higher Accuracy

Dividends on the Underlying Equity

American Exercise

2-D Models, Correlation and Variable Changes

Credit Spreads

Credit Default Swaps (CDS) and the Continuous CDS Curve

Valuing Bonds Using the Continuous CDS Curve

Equations of Motion for Bonds and Credit Default Swaps

Specific Models

Stochastic Rates and Default

Convertible Bonds

Index Options versus Single Name Options: Trading Equity Correlation

Max of n Stocks: Trading Equity Correlation

Collateralized Debt Obligations (CDOs): Trading Credit Correlation

CDO Backed by Three Bonds

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