A Factor Model Approach to Derivative Pricing

Cover -- Half Title -- Title Page -- Copyright Page -- Dedication -- Table of Contents -- List of Figures -- Preface -- Notation -- CHAPTER 1: Building Blocks and Stochastic Differential Equation Models -- 1.1 BROWNIAN MOTION AND POISSON PROCESSES -- 1.1.1 Gaussian Random Variables -- 1.1.2 Brownian Motion -- 1.1.3 Poisson Random Variables -- 1.1.4 Poisson Processes -- 1.1.5 Increments of Brownian Motion and Poisson Processes -- 1.1.5.1 Brownian Motion Increment -- 1.1.5.2 Poisson Process Increment -- 1.2 STOCHASTIC DIFFERENTIAL EQUATIONS -- 1.2.1 The Differential -- 1.2.1.1 The Problem with Jumps -- 1.2.1.2 Altering the Differential for Jumps -- 1.2.2 Compound Poisson Process -- 1.2.3 Ito Stochastic Differential Equations -- 1.2.3.1 Instantaneous Statistics -- 1.2.4 Poisson Driven Differential Equations -- 1.3 SUMMARY -- CHAPTER 2: Ito's Lemma -- 2.1 CHAIN RULE OF ORDINARY CALCULUS -- 2.1.1 Multivariable Taylor Series Expansions -- 2.2 ITO'S LEMMA FOR BROWNIAN MOTION -- 2.2.1 Replacing dz2 by dt -- 2.2.1.1 Computing the Mean -- 2.2.1.2 Computing the Variance -- 2.2.2 Discussion of Ito's Lemma -- 2.3 ITO'S LEMMA FOR POISSON PROCESSES -- 2.3.1 Interpretation of Ito's Lemma for Poisson -- 2.4 MORE VERSIONS OF ITO'S LEMMA -- 2.4.1 Ito's Lemma for Compound Poisson Processes -- 2.4.2 Ito's Lemma for Brownian and Compound Poisson -- 2.4.3 Ito's Lemma for Vector Processes -- 2.5 ITO'S LEMMA, THE PRODUCT RULE, AND A RECTANGLE -- 2.6 SUMMARY -- CHAPTER 3: Stochastic Differential Equations -- 3.1 GEOMETRIC BROWNIAN MOTION -- 3.1.1 Stock Price Interpretation -- 3.2 GEOMETRIC POISSON MOTION -- 3.2.1 Conditional Lognormal Version -- 3.3 JUMP DIFFUSION MODEL -- 3.3.1 Jump Diffusion as a Stock Price Model -- 3.4 A MORE GENERAL STOCHASTIC DIFFERENTIAL EQUATION -- 3.4.1 Ornstein-Uhlenbeck Process and Mean Reversion -- 3.5 COX-INGERSOLL-ROSS PROCESS -- 3.6 SUMMARY

CHAPTER 4: The Factor Model Approach to Arbitrage Pricing -- 4.1 RETURNS AND FACTOR MODELS -- 4.1.1 Returns from Price Changes -- 4.1.2 Stochastic Differential Equations and Factor Models -- 4.2 FACTOR APPROACH TO ARBITRAGE USING RETURNS -- 4.2.1 Arbitrage -- 4.2.2 Null and Range Space Relationship -- 4.2.3 A Useful Absence of Arbitrage Condition -- 4.2.4 Interpretations -- 4.2.4.1 Market Price of Risk -- 4.2.4.2 Market Price of Time -- 4.2.5 A Problem with Returns -- 4.3 FACTOR APPROACH USING PRICES AND VALUE CHANGES -- 4.3.1 Prices, Value Changes, and Arbitrage -- 4.3.2 Profit/Loss and Arbitrage -- 4.4 TWO STANDARD EXAMPLES -- 4.4.1 A Stock under Geometric Brownian Motion -- 4.4.1.1 Using the Return APT -- 4.4.1.2 Using the Price APT -- 4.4.2 Futures Contracts -- 4.5 SUMMARY -- CHAPTER 5: Constructing a Factor Model Pricing Framework -- 5.1 CLASSIFICATION OF QUANTITIES -- 5.1.1 Factors -- 5.1.2 Underlying Variables -- 5.1.3 Tradables -- 5.1.4 Importance of Tradables -- 5.2 DERIVATIVES -- 5.2.1 Examples of Derivatives -- 5.3 FACTOR MODELS FOR UNDERLYING VARIABLES AND TRADABLES -- 5.3.1 Direct Factor Models -- 5.3.2 Factor Models via Ito's Lemma -- 5.4 TRADABLE TABLES -- 5.4.1 Structure of a Tradable Table -- 5.5 APPLYING THE PRICE APT -- 5.5.1 Relative Pricing and Marketed Tradables -- 5.5.2 Pricing the Derivative -- 5.6 UNDERDETERMINED AND OVERDETERMINED SYSTEMS -- 5.6.1 Underdetermined and Incompleteness -- 5.6.2 Overdetermined and Calibration -- 5.7 THREE STEP PROCEDURE -- 5.8 SOLVING THE PRICE APT EQUATION FOR A SPECIFIC DERIVATIVE -- 5.9 SUMMARY -- CHAPTER 6: Equity Derivatives -- 6.1 BLACK-SCHOLES MODEL -- 6.1.1 Solution for European Call and Put Options -- 6.1.2 Implied Volatility -- 6.2 CONTINUOUS DIVIDENDS -- 6.2.1 European Calls and Puts under Continuous Dividends -- 6.2.2 What Pays a Continuous Dividend? -- 6.3 CASH DIVIDENDS

6.3.1 European Calls and Puts under Cash Dividends -- 6.4 POISSON PROCESSES -- 6.4.1 Closed Form Solution for European Call Option -- 6.5 OPTIONS ON FUTURES -- 6.5.1 Solution for European Call and Put Options -- 6.6 JUMP DIFFUSION MODEL -- 6.6.1 Bankruptcy! -- 6.6.2 Lognormal Jumps -- 6.7 EXCHANGE ONE ASSET FOR ANOTHER -- 6.7.1 Closed Form Solution -- 6.7.2 Using a Different Currency (Change of Numeraire) -- 6.7.3 Recovering the Closed Form Solution -- 6.8 STOCHASTIC VOLATILITY -- 6.9 SUMMARY -- CHAPTER 7: Interest Rate and Credit Derivatives -- 7.1 INTEREST RATE MODELING AND DERIVATIVES -- 7.2 MODELING THE TERM STRUCTURE OF INTEREST RATES -- 7.2.1 Spot Rates -- 7.2.2 Zero-Coupon Bonds -- 7.2.3 Forward Rates -- 7.2.4 Discount Factors -- 7.2.5 Instantaneous Rates -- 7.2.6 Two Choices for Underlying Variables -- 7.3 MODELS BASED ON THE SHORT RATE -- 7.3.1 The Short Rate of Interest and a Money Market Account -- 7.3.2 Single Factor Short Rate Models -- 7.3.2.1 Vasicek Model -- 7.3.2.2 Cox-Ingersoll-Ross Model -- 7.3.2.3 Additional Single Factor Short Rate Models -- 7.3.3 Multifactor Short Rate Models -- 7.4 MODELS BASED ON FORWARD RATES -- 7.4.1 An Important Detour on Forwards -- 7.4.1.1 Market Price of Risk for Forwards -- 7.4.1.2 Pricing Options on Forwards -- 7.4.1.3 Simplified Pricing in Units of the Bond -- 7.5 LIBOR MARKET MODEL -- 7.5.1 Discount Factors are Forward Prices -- 7.5.2 LMM with Discount Factors as Underlying Variables -- 7.5.2.1 Calibration of the Market Prices of Risk -- 7.5.2.2 Using the Forward Analogy -- 7.5.2.3 Formal Three Step Approach -- 7.5.3 Forward Rates as Underlying Variables -- 7.5.4 Pricing a Caplet in the Right Units -- 7.5.4.1 LMM Caplet Formula -- 7.5.4.2 From Caplets to Floorlets and Beyond -- 7.6 HEATH-JARROW-MORTON MODEL -- 7.7 CREDIT DERIVATIVES -- 7.7.1 Defaultable Bonds -- 7.7.1.1 Different Types of Recovery

7.7.2 Defaultable Bonds with Random Intensity of Default -- 7.8 SUMMARY -- 7.9 APPENDIX: HEATH-JARROW-MORTON DERIVATION -- CHAPTER 8: Hedging -- 8.1 HEDGING FROM A FACTOR PERSPECTIVE -- 8.1.1 Description Using a Tradable Table -- 8.1.2 Relationship between Hedging and Arbitrage -- 8.1.2.1 Creating a No-Cost Hedge -- 8.1.2.2 Simple Explanation -- 8.1.3 Hedging Examples -- 8.1.3.1 Hedging in Black-Scholes -- 8.1.3.2 Pricing from the Black-Scholes Hedge -- 8.1.3.3 Hedging in Bonds -- 8.1.3.4 Pricing from the Bond Hedge -- 8.1.4 Hedging under Incompleteness -- 8.1.5 A Question of Consistency -- 8.2 HEDGING FROM AN UNDERLYING VARIABLE PERSPECTIVE -- 8.2.1 Black-Scholes Hedging -- 8.2.2 Hedging Bonds -- 8.2.3 Derivatives Imply Small Changes -- 8.3 HIGHER ORDER APPROXIMATIONS -- 8.3.1 The Greeks -- 8.3.2 Delta-Gamma Hedge -- 8.3.3 Determining What the Error Looks Like -- 8.4 SUMMARY -- CHAPTER 9: Computation of Solutions -- 9.1 DISCRETIZING THE MODELING PARADIGM -- 9.1.1 Discretizing Factor Models -- 9.1.2 Discretizing Geometric Brownian Motion -- 9.1.2.1 Direct Discretization of the Factor Model -- 9.1.2.2 Discretizing the Exact Solution -- 9.1.3 More General Modeling Induced by Binary Factor -- 9.1.4 From the Binary Model to a Factor Model -- 9.1.4.1 Note on Time Scaling of the Factor -- 9.2 THREE STEP PROCEDURE UNDER BINARY MODELING -- 9.2.1 Single Step Calibration Equation -- 9.2.2 Single Step Pricing Equation -- 9.2.2.1 Interpreting the Pricing Equation -- 9.2.3 Risk Neutral Parameterization -- 9.3 FINAL STEP: PRICING ON MULTISTEP BINOMIAL TREES -- 9.3.1 Collapsing Trees to Recombining Lattices -- 9.4 PRICING OPTIONS ON A STOCK LATTICE -- 9.4.1 Modeling the Stock -- 9.4.2 Calibration Using Marketed Tradables -- 9.4.3 Pricing a European Call Using the Calibrated Lattice -- 9.4.4 Pricing American Options -- 9.5 PRICING ON A CIR SHORT RATE TREE

9.5.1 Modeling and Classification of Discretized Variables -- 9.5.2 Representation of Marketed Tradables -- 9.5.3 Calibration Using Marketed Tradables -- 9.5.4 Market Price of Risk Calibration Algorithm -- 9.5.5 Pricing Derivatives on CIR Calibrated Trees -- 9.6 SINGLE FACTOR LIBOR MARKET MODEL -- 9.6.1 LMM Forward Rates and Calibration -- 9.6.2 Modeling and Classification of Discretized Variables -- 9.6.3 Specification of Bond Prices and Forward Rates -- 9.6.4 Calibration Using Marketed Tradables -- 9.6.5 Calibration Equation for the Forward Rate Parameters -- 9.6.5.1 Solving for an(i) -- 9.6.6 Calibration Algorithm for LMM -- 9.6.7 Pricing Derivatives with LMM Calibrated Trees -- 9.7 ADDITIONAL COMPUTATIONAL METHODS -- 9.8 SUMMARY -- CHAPTER 10: The Road to Risk Neutrality -- 10.1 DO THE FACTORS MATTER? -- 10.1.1 Brownian Factors -- 10.1.1.1 Adding a Drift Works -- 10.1.1.2 Changing the Variance Does Not Work -- 10.1.2 Poisson Factors -- 10.1.2.1 Changing the Intensity Works -- 10.1.2.2 Adding a Drift Does Not Work -- 10.2 RISK NEUTRAL REPRESENTATIONS -- 10.2.1 Risk Neutrality under Brownian Factors -- 10.2.2 Market Price of Risk Interpretation -- 10.2.3 Risk Neutrality under Poisson Factors -- 10.2.4 Market Price of Risk under Poisson Factors -- 10.3 PRICING AS AN EXPECTATION -- 10.4 APPLICATIONS OF RISK NEUTRAL PRICING -- 10.4.1 How to Apply Risk Neutral Pricing -- 10.4.2 Black-Scholes -- 10.4.3 Poisson Model -- 10.4.4 Vasicek Single Factor Short Rate Model -- 10.4.5 Heath-Jarrow-Morton Calibration -- 10.5 SUMMARY -- Bibliography -- Index