BINOMIAL MODELS IN FINANCE VAN DER HOEK ELLIOTT

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BINOMIAL MODELS IN FINANCE

JOHNVAN DER HOEK, ROBERT J. ELLIOTT

SPRINGER, 206 PAGES, HARDCOVER

This book deals with many topics in modern financial mathematics in a way that does not use advanced mathematical tools and shows how these models can be numerically implemented in a practical way. The book is aimed at undergraduate students, MBA students, and executives who wish to understand and apply financial models in the spreadsheet computing environment.

The basic building block is the one-step binomial model where a known price today can take one of two possible values at the next time. In this simple situation, risk neutral pricing can be defined and the model can be applied to price forward contracts, exchange rate contracts, and interest rate derivatives. The simple one-period framework can then be extended to multi-period models. The authors show how binomial tree models can be constructed for several applications to bring about valuations consistent with market prices. The book closes with a novel discussion of real options.

Contents:

1 Introduction

1.1 No Arbitrage and Its Consequences .

1.2 Exercises

2 The Binomial Model for Stock Options

2.1 The Basic Model .

2.2 Why Is ? Called a Risk Neutral Probability? .

2.3 More on Arbitrage

2.4 The Model of Cox-Ross-Rubinstein .

2.5 Call-Put Parity Formula

2.6 Non Arbitrage Inequalities

2.7 Exercises

3 The Binomial Model for Other Contracts

3.1 Forward Contracts

3.2 Contingent Premium Options

3.3 Exchange Rates

3.4 Interest Rate Derivatives .

3.5 Exercises .

4 Multiperiod Binomial Models .

4.1 The Labelling of the Nodes

4.2 The Labelling of the Processes

4.3 Generalized Quantities

X Contents

4.4 Generalized Backward Induction Pricing Formula .

4.5 Pricing European Style Contingent Claims .

4.6 The CRR Multiperiod Model

4.7 Jamshidian’s Forward Induction Formula .

4.8 Application to CRR Model

4.9 The CRR Option Pricing Formula

4.10 Discussion of the CRR Formula

4.11 Exercises .

5 Hedging

5.1 Hedging .

5.2 Exercises .

6 Forward and Futures Contracts .

6.1 The Forward Contract .

6.2 The Futures Contract .

6.3 Exercises .

7 American and Exotic Option Pricing

7.1 American Style Options .

7.2 Barrier Options

7.3 Examples of the Application of Barrier Options

7.4 Exercises

8 Path-Dependent Options

8.1 Notation for Non-Recombing Trees .

8.2 Asian Options .

8.3 Floating Strike Options

8.4 Lookback Options .

8.5 More on Average Rate Options .

8.6 Exercises

Contents XI

9 The Greeks

9.1 The Delta (?) of an Option

9.2 The Gamma (?) of an Option .

9.3 The Theta (?) of an Option .

9.4 The Vega (?) of an Option

9.5 The Rho (?) of an Option

9.6 Exercises .

10 Dividends .

10.1 Some Basic Results about Forwards

10.2 Dividends as Percentage of Spot Price

10.3 Binomial Trees with Known Dollar Dividends .

10.4 Exercises

11 Implied Volatility Trees

11.1 The Recursive Calculation

11.2 The Inputs V put and V call

11.3 A Simple Smile Example

11.4 In General .

11.5 The Barle and Cakici Approach .

11.6 Exercises

12 Implied Binomial Trees

12.1 The Inputs

12.2 Time T Risk-Neutral Probabilities .

12.3 Constructing the Binomial Tree

12.4 A Basic Theorem and Applications

12.5 Choosing Time T Data .

12.6 Some Proofs and Discussion

12.7 Jackwerth’s Extension .

12.8 Exercises

XII Contents

13 Interest Rate Models .

13.1 P(0, T) from Treasury Data .

13.2 P(0, T) from Bank Data .

13.3 The Ho and Lee Model

13.4 The Pedersen, Shiu and Thorlacius Model .

13.5 The Morgan and Neave Model

13.6 The Black, Derman and Toy Model

13.7 Defaultable Bonds

13.8 Exercises

14 Real Options .

14.1 Examples

14.2 Options on Non-Tradeable Assets .

14.3 Correlation with Tradeable Assets

14.4 Approximate Methods .

14.5 Exercises

A The Binomial Distribution

A.1 Bernoulli Random Variables .

A.2 Bernoulli Trials

A.3 Binomial Distribution

A.4 Central Limit Theorem (CLT) .

A.5 Berry-Ess´een Theorem

A.6 Complementary Binomials and Normals

A.7 CRR and the Black and Scholes Formula .

B An Application of Linear Programming .

B.1 Incomplete Markets .

B.2 Solutions to Incomplete Markets. .

B.3 The Duality Theorem of Linear Programming .

B.4 The First Fundamental Theorem of Finance .

B.5 The Duality Theorem .

B.6 The Second Fundamental Theorem of Finance .

B.7 Transaction Costs

Contents XIII

C Volatility Estimation

C.1 Historical Volatility Estimation .

C.2 Implied Volatility Estimation

C.3 Exercises .

D Existence of a Solution .

D.1 Farkas’ Lemma .

D.2 An Application to the Problem .

E Some Generalizations

E.1 Preliminary Observations .

E.2 Solution to System in van der Hoek’s Method

E.3 Exercises

F Yield Curves and Splines .

F.1 An Alternative representation of Function (F.1) .

F.2 Imposing Smoothness .

F.3 Unknown Coe.cients

F.4 Observations .

F.5 Determination of Unknown Coe.cients .

F.6 Forward Interest Rates

F.7 Yield Curve

F.8 Other Issues

References .

Index .