Financial markets in continuous time

1. The Discrete Case

1.1. A Model with Two Dates and Two States of the World

1.1.1. The Model

1.1.2. Hedging Portfolio, Value of the Option

1.1.3. The Risk-Neutral Measure, Put-Call Parity

1.1.4. No Arbitrage Opportunities

1.1.5. The Risk Attached to an Option

1.1.6. Incomplete Markets

1.2. A One-Period Model with (d + 1) Assets and k States of the World

1.2.1. No Arbitrage Opportunities

1.2.2. Complete Markets

1.2.3. Valuation by Arbitrage in the Case of a Complete Market

1.2.4. Incomplete Markets: the Arbitrage Interval

1.3. Optimal Consumption and Portfolio Choice in a One-Agent Model

1.3.1. The Maximization Problem

1.3.2. An Equilibrium Model with a Representative Agent

1.3.3. The Von Neumann-Morgenstern Model, Risk Aversion

1.3.4. Optimal Choice in the VNM Model

1.3.5. Equilibrium Models with Complete Financial Markets

2. Dynamic Models in Discrete Time

2.1. A Model with a Finite Horizon

2.2. Arbitrage with a Finite Horizon

2.2.1. Arbitrage Opportunities

2.2.2. Arbitrage and Martingales

2.3. Trees

2.4. Complete Markets with a Finite Horizon

2.4.1. Characterization

2.5. Valuation

2.5.1. The Complete Market Case

2.6. An Example

2.6.1. The Binomial Model

2.6.2. Option Valuation

2.6.3. Approaching the Black-Scholes Model

2.7. Maximization of the Final Wealth

2.8. Optimal Choice of Consumption and Portfolio

2.9. Infinite Horizon

3. The Black-Scholes Formula

3.1. Stochastic Calculus

3.1.1. Brownian Motion and the Stochastic Integral

3.1.2. Ito Processes. Girsanov's Theorem

3.1.3. Ito's Lemma

3.1.4. Multidimensional Processes

3.1.5. Multidimensional Ito's Lemma

3.1.6. Examples

3.2. Arbitrage and Valuation

3.2.1. Financing Strategies

3.2.2. Arbitrage and the Martingale Measure

3.2.3. Valuation

3.3. The Black-Scholes Formula: the One-Dimensional Case

3.3.1. The Model

3.3.2. The Black-Scholes Formula

3.3.3. The Risk-Neutral Measure

3.3.4. Explicit Calculations

3.3.5. Comments on the Black-Scholes Formula

3.4. Extension of the Black-Scholes Formula

3.4.1. Financing Strategies

3.4.2. The State Variable

3.4.3. The Black-Scholes Formula

3.4.4. Special Case

3.4.5. The Risk-Neutral Measure

3.4.6. Example

3.4.7. Applications of the Black-Scholes Formula

4. Portfolios Optimizing Wealth and Consumption

4.1. The Model

4.2. Optimization

4.3. Solution in the Case of Constant Coefficients

4.3.1. Dynamic Programming

4.3.2. The Hamilton-Jacobi-Bellman Equation

4.3.3. A Special Case

4.4. Admissible Strategies

4.5. Existence of an Optimal Pair

4.5.1. Construction of an Optimal Pair

4.5.2. The Value Function

4.5.3. A Special Case

4.6. Solution in the Case of Deterministic Coefficients

4.6.1. The Value Function and Partial Differential Equations

4.6.2. Optimal Wealth

4.6.3. Obtaining the Optimal Portfolio

4.7. Market Completeness and NAO

5. The Yield Curve

5.1. Discrete-Time Model

5.2. Continuous-Time Model

5.2.1. Definitions

5.2.2. Change of Numeraire

5.2.3. Valuation of an Option on a Coupon Bond

5.3. The Heath-Jarrow-Morton Model

5.3.1. The Model

5.3.2. The Linear Gaussian Case

5.4. When the Spot Rate is Given

5.5. The Vasicek Model

5.5.1. The Ornstein-Uhlenbeck Process

5.5.2. Determining P(t,T) when q is Constant

5.6. The Cox-Ingersoll-Ross Model

5.6.1. The Cox-Ingersoll-Ross Process

5.6.2. Valuation of a Zero Coupon Bond

6. Equilibrium of Financial Markets in Discrete Time

6.1. Equilibrium in a Static Exchange Economy

6.2. The Demand Approach

6.3. The Negishi Method

6.3.1. Pareto Optima

6.3.2. Two Characterizations of Pareto Optima

6.3.3. Existence of an Equilibrium

6.4. The Theory of Contingent Markets

6.5. The Arrow-Radner Equilibrium Exchange Economy with Financial Markets with Two Dates

6.6. The Complete Markets Case

6.7. The CAPM

7. Equilibrium of Financial Markets in Continuous Time. The Complete Markets Case

7.1. The Model

7.1.1. The Financial Market

7.1.2. The Economy

7.1.3. Admissible Pairs

7.1.4. Definition and Existence of a Radner Equilibrium

7.2. Existence of a Contingent Arrow-Debreu Equilibrium

7.2.1. Aggregate Utility

7.2.2. Definition and Characterization of Pareto Optima

7.2.3. Existence and Characterization of a Contingent Arrow-Debreu Equilibrium

7.2.4. Existence of a Radner Equilibrium

7.3. Applications

7.3.1. Arbitrage Price of Real Secondary Assets. Lucas' Formula

7.3.2. CCAPM (Consumption-based Capital Asset Pricing Model)

8. Incomplete Markets

8.1. Incomplete Markets

8.1.1. The Case of Constant Coefficients

8.1.2. No-Arbitrage Markets

8.1.3. The Price Range

8.1.4. Superhedging

8.1.5. The Minimal Probability Measure

8.1.6. Valuation Using Utility Functions

8.1.7. Transaction Costs

8.2. Stochastic Volatility

8.2.1. The Robustness of the Black-Scholes Formula

8.3. Wealth Optimization

9. Exotic Options

9.1. The Hitting Time and Supremum for Brownian Motion

9.1.1. Distribution of the Pair (B[subscript t], M[subscript t])

9.1.2. Distribution of Sup and of the Hitting Time

9.1.3. Distribution of Inf

9.1.4. Laplace Tranforms

9.1.5. Hitting Time for a Double Barrier

9.2. Drifted Brownian Motion

9.2.1. The Laplace Transform of a Hitting Time

9.2.2. Distribution of the Pair (Maximum, Minimum)

9.2.3. Evaluation of E(e [superscript -r T[subscript y]] 11 [subscript T[subscript y[less than a]]])

9.3. Barrier Options

9.3.1. Down-and-Out Options

9.3.2. Down-and-In Options

9.3.3. Up-and-Out and Up-and-In Options

9.3.4. Intermediate Calculations

9.3.5. The Value of the Compensation

9.3.6. Valuation of a DIC Option

9.3.7. Up-and-In Options

9.3.8. P. Carr 's Symmetry

9.4. Double Barriers

9.5. Lookback Options

9.6. Other Options

9.6.1. Options Linked to the Hitting Time of a Barrier

9.6.2. Options Linked to Occupation Times

9.7. Other Products

9.7.1. Asian Options or Average Rate Options

9.7.2. Products Depending on an Interim Date

9.7.3. Still More Products

A. Brownian Motion

A.1. Historical Background

A.2. Intuition

A.3. Random Walk

A.4. The Stochastic Integral

A.5. Ito's Formula

B. Numerical Methods

B.1. Finite Difference

B.1.1. Method

B.1.2. The Implicit Scheme Case

B.1.3. Solving the System

B.1.4. Other Schemes

B.2. Extrapolation Methods

B.2.1. The Heat Equation

B.2.2. Approximations

B.3. Simulation

B.3.1. Simulation of the Uniform Distribution on [0, 1]

B.3.2. Simulation of Discrete Variables

B.3.3. Simulation of a Random Variable

B.3.4. Simulation of an Expectation

B.3.5. Simulation of a Brownian Motion

B.3.6. Simulation of Solutions to Stochastic Differential Equations

B.3.7. Calculating E(f(X[subscript t]))

References

Index