INTRODUCTION TO THE ECONOMICS AND MATHEMATICS OF FINANCIAL MARKETS. Jakša Cvitanić and Fernando Zapatero

Transcription

1 INTRODUCTION TO THE ECONOMICS AND MATHEMATICS OF FINANCIAL MARKETS Jakša Cvitanić and Fernando Zapatero

2 INTRODUCTION TO THE ECONOMICS AND MATHEMATICS OF FINANCIAL MARKETS Table of Contents PREFACE...1 Part I. The Setting: Markets, Models, Interest Rates, Utility Maximization, Risk 1. FINANCIAL MARKETS Bonds Types of Bonds Reasons for Trading Bonds Risk of Trading Bonds 1.2. Stocks How are Stocks different from Bonds Going Long or Short 1.3. Derivatives Futures and Forwards Marking to Market Reasons for Trading Futures Options Calls and Puts Option Prices Reasons for Trading Options Swaps Mortgage Backed Securities; Callable Bonds 1.4. Organization of Financial Markets Exchanges Market Index 1.5. Margins Trades that involve Margin Requirements 1.6. Transaction Costs...32 SUMMARY...31 PROBLEMS...30 FURTHER READINGS...34

3 2. INTEREST RATES Computation of Interest Rates Simple versus Compound Interest; Annualized Rates Continuous Interest 2.2. Present Value Present/Future Values of Cash Flows Bond Yield Price-Yield Curves 2.3. Term Structure of Interest Rates and Forward Rates Yield Curve Calculating Spot Rates; Rates Arbitrage Forward Rates Term Structure Theories SUMMARY...52 PROBLEMS...53 FURTHER READINGS MODELS OF SECURITIES PRICES IN FINANCIAL MARKETS Single-Period Models Asset Dynamics Portfolio and Wealth Processes Arrow-Debreu Securities 3.2. Multi-Period Models General Model Specifications Cox-Ross-Rubinstein Binomial Model 3.3. Continuous-Time Models Simple Facts about the Merton-Black-Scholes Model Brownian Motion Process Diffusion Processes, Stochastic Integrals Technical Properties of Stochastic Integrals Itô s Rule Merton-Black-Scholes Model Wealth Process and Portfolio Process 3.4. Modeling Interest Rates Discrete-Time Models Continuous-Time Models 3.5. Nominal Rates and Real Rates Discrete-Time Models

4 Continuous-Time Models 3.6. Arbitrage and Market Completeness Notion of Arbitrage Arbitrage in Discrete-Time Models Arbitrage in Continuous-Time Models Notion of Complete Markets Complete Markets in Discrete-Time Models Complete Markets in Continuous-Time Models 3.7. Appendix More Details for the Proof of Itô s Formula Multi-Dimensional Itô s Rule SUMMARY...99 PROBLEMS...99 FURTHER READINGS OPTIMAL CONSUMPTION/PORTFOLIO STRATEGIES Preference Relations and Utility Functions Consumption Preferences Concept of Utility Functions Marginal Utility; Risk Aversion; Certainty Equivalent Utility Functions in Multi-Period Discrete-Time Models Utility Functions in Continuous-Time Models 4.2. Discrete-Time Utility Maximization Single Period Multi-Period Utility Maximization: Dynamic Programming Optimal Portfolios in Merton-Black-Scholes Model Utility from Consumption 4.3. Utility Maximization in Continuous Time Hamilton-Jacobi-Bellman PDE 4.4. Duality/Martingale Approach to Utility Maximization Martingale Approach in Single-Period Binomial Model Martingale Approach in Multi-Period Binomial Model Duality/Martingale Approach in Continuous Time 4.5. Transaction Costs Incomplete and Asymmetric Information Single Period Incomplete Information in Continuous Time

5 Power Utility and Normally Distributed Drift 4.7. Appendix: Proof of Dynamic Programming Principle SUMMARY PROBLEMS FURTHER READINGS RISK Risk vs. Return: Mean-Variance Analysis Mean and Variance of a Portfolio Mean-Variance Efficient Frontier Computing the Optimal Mean-Variance Portfolio Computing the Optimal Mutual Fund Mean-Variance Optimization in Continuous Time 5.2. VaR: Value at Risk Definition of VaR Computing VaR VaR of a Portfolio of Assets Alternatives to VaR The Story of Long-Term Capital Management SUMMARY PROBLEMS FURTHER READINGS Part II: Pricing and Hedging of Securities 6. ARBITRAGE AND RISK-NEUTRAL PRICING Arbitrage Relationships for Call and Put Options; Put-Call Parity Arbitrage Pricing of Forwards and Futures Forward Prices Futures Prices Futures on Commodities 6.3. Risk-Neutral Pricing Martingale Measures; Cox-Ross-Rubinstein (CRR) Model State Prices in Single-Period Models No Arbitrage and Risk-Neutral Probabilities Pricing by No Arbitrage Pricing by Risk-Neutral Expected Values

6 Martingale Measure for the Merton-Black-Scholes Model Computing Expectations by Feynman-Kac PDE Risk-Neutral Pricing in Continuous Time Futures and Forwards Revisited 6.4. Appendix No Arbitrage Implies Existence of a Risk-Neutral Probability Completeness and Unique EMM Another Proof of Theorem?? Proof of Bayes Rule SUMMARY PROBLEMS FURTHER READINGS OPTION PRICING Option Pricing in the Binomial Model Backward Induction and Expectation Formula Black-Scholes Formula as a Limit of the Binomial Model Formula 7.2. Option Pricing in the Merton-Black-Scholes Model Black-Scholes Formula as Expected Value Black-Scholes Equation Black-Scholes Formula for the Call Option Implied Volatility 7.3. Pricing American Options Stopping Times and American Options Binomial Trees and American Options PDE s and American Options 7.4. Options on Dividend-Paying Securities Binomial Model Merton-Black-Scholes Model 7.5. Other Types of Options Currency Options Futures Options Exotic Options 7.6. Pricing in the Presence of Several Random Variables Options on Two Risky Assets Quantos Stochastic Volatility with Complete Markets Stochastic Volatility with Incomplete Markets; Market Price of Risk

7 Utility Pricing in Incomplete Markets 7.7. Merton s Jump-Diffusion Model Estimation of Variance and ARCH/GARCH Models Appendix: Derivation of the Black-Scholes Formula SUMMARY PROBLEMS FURTHER READINGS FIXED INCOME MARKET MODELS AND DERIVATIVES Discrete-Time Interest Rate Modeling Binomial Tree for the Interest Rate Black-Derman-Toy Model Ho-Lee Model 8.2. Interest Rate Models in Continuous Time One-Factor Short Rate Models Bond Pricing in Affine Models HJM Forward Rate Models Change of Numeraire Option Pricing with Random Interest Rate BGM Market Model 8.3. Swaps, Caps and Floors Interest Rate Swaps and Swaptions Caplets, Caps and Floors 8.4. Credit/Default Risk SUMMARY PROBLEMS FURTHER READINGS HEDGING Hedging with Futures Perfect Hedge Crosshedging; Basis Risk Rolling the Hedge Forward Quantity Uncertainty 9.2. Portfolios of Options as Trading Strategies Covered Calls and Protective Puts Bull Spreads and Bear Spreads Butterfly Spreads

8 Straddles and Strangles 9.3. Hedging Options Positions; Delta Hedging Delta Hedging in Discrete-Time Models Delta-Neutral Strategies Deltas of Calls and Puts Example: Hedging a Call Option Other Greeks Stochastic Volatility and Interest Rate Formulas for Greeks Portfolio Insurance 9.4. Perfect Hedging in a Multi-Variable Continuous-Time Model Hedging in Incomplete Markets SUMMARY PROBLEMS FURTHER READINGS BOND HEDGING Duration Definition and Interpretation Duration and Change in Yield Duration of a Portfolio of Bonds Immunization Matching Durations Duration and Immunization in Continuous Time Convexity SUMMARY PROBLEMS FURTHER READINGS NUMERICAL METHODS Binomial Tree Methods Computations in the Cox-Ross-Rubinstein Model Computing Option Sensitivities Extensions of the Tree Method Monte Carlo Simulation Monte Carlo Basics Generating Random Numbers 8

9 Variance Reduction Techniques Simulation in a Continuous-Time Multi-Variable Model Computation of Hedging Portfolios by Finite Differences Retrieval of Volatility Method for Hedging and Utility Maximization Numerical Solutions of PDE s; Finite Difference Methods Implicit Finite Difference Method Explicit Finite Difference Method SUMMARY PROBLEMS FURTHER READINGS Part III: Equilibrium Models 12. EQUILIBRIUM FUNDAMENTALS Concept of Equilibrium Definition and Single-Period Case A Two-Period Example Continuous-Time Equilibrium Single-Agent and Multi-Agent Equilibrium Representative Agent Single-Period Aggregation Pure Exchange Equilibrium Basic Idea and Single-Period Case Multi-Period Discrete-Time Model Continuous-Time Pure Exchange Equilibrium Existence of Equilibrium Equilibrium Existence in Discrete Time Equilibrium Existence in Continuous Time Determining Market Parameters in Equilibrium SUMMARY PROBLEMS FURTHER READINGS CAPM Basic CAPM...391

10 CAPM Equilibrium Argument Capital Market Line CAPM Formula Economic Interpretations Securities Market Line Systematic and Non-Systematic Risk Asset Pricing Implications: Performance Evaluation Pricing Formulas Empirical Tests Alternative Derivation of CAPM Continuous-Time, Intertemporal CAPM Consumption CAPM SUMMARY PROBLEMS FURTHER READINGS MULTIFACTOR MODELS Discrete-Time Multifactor Models Arbitrage Pricing Theory (APT) Multifactor Models in Continuous Time Model Parameters and Variables Value Function and Optimal Portfolio Separation Theorem Intertemporal Multifactor CAPM SUMMARY PROBLEMS FURTHER READINGS OTHER PURE EXCHANGE EQUILIBRIA Term Structure Equilibria Equilibrium Term Structure in Discrete-Time Equilibrium Term Structure in Continuous-Time; CIR Model Informational Equilibria Discrete-Time Models with Incomplete Information Continuous-Time Models with Incomplete Information Equilibrium with Heterogeneous Agents Discrete-Time Equilibrium with Heterogeneous Agents

11 Continuous-Time Equilibrium with Heterogeneous Agents International Equilibrium/Equilibrium with Two Prices Discrete-Time International Equilibrium Continuous-Time International Equilibrium SUMMARY PROBLEMS FURTHER READINGS APPENDIX: PROBABILITY THEORY ESSENTIALS Discrete Random Variables Expectation and Variance Continuous Random Variables Expectation and Variance Several Random Variables Independence Correlation and Covariance Normal Random Variables Properties of Conditional Expectations Martingale Definition Random Walk and Brownian Motion References Index...465