The Mathematics of Financial Derivatives: A Student Introduction,9780521497893

The Mathematics of Financial Derivatives: A Student Introduction

by
ISBN13: 9780521497893
ISBN10: 0521497892
Format: Paperback
Pub. Date: 1995-09-29
Publisher(s): Cambridge University Press
List Price: $86.10

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Summary

Finance is one of the fastest growing areas in the modern banking and corporate world. This, together with the sophistication of modern financial products, provides a rapidly growing impetus for new mathematical models and modern mathematical methods; the area is an expanding source for novel and relevant 'real-world' mathematics. In this book the authors describe the modelling of financial derivative products from an applied mathematician's viewpoint, from modelling through analysis to elementary computation. A unified approach to modelling derivative products as partial differential equations is presented, using numerical solutions where appropriate. Some mathematics is assumed, but clear explanations are provided for material beyond elementary calculus, probability, and algebra. Over 140 exercises are included. This volume will become the standard introduction to this exciting new field for advanced undergraduate students.

Table of Contents

Preface ix
Part One: Basic Option Theory 1(132)
1 An Introduction to Options and Markets
3(15)
1.1 Introduction
3(1)
1.2 What is an Option?
4(3)
1.3 Reading the Financial Press
7(4)
1.4 What are Options For?
11(2)
1.5 Other Types of Option
13(1)
1.6 Forward and Futures Contracts
14(1)
1.7 Interest Rates and Present Value
15(3)
2 Asset Price Random Walks
18(15)
2.1 Introduction
18(1)
2.2 A Simple Model for Asset Prices
19(6)
2.3 Ito's Lemma
25(5)
2.4 The Elimination of Randomness
30(3)
3 The Black-Scholes Model
33(25)
3.1 Introduction
33(1)
3.2 Arbitrage
33(2)
3.3 Option Values, Payoffs and Strategies
35(5)
3.4 Put-call Parity
40(1)
3.5 The Black-Scholes Analysis
41(3)
3.6 The Black-Scholes Equation
44(2)
3.7 Boundary and Final Conditions
46(2)
3.8 The Black-Scholes Formulae
48(3)
3.9 Hedging in Practice
51(1)
3.10 Implied Volatility
52(6)
4 Partial Differential Equations
58(13)
4.1 Introduction
58(1)
4.2 The Diffusion Equation
59(7)
4.3 Initial and Boundary Conditions
66(2)
4.4 Forward versus Backward
68(3)
5 The Black-Scholes Formulae
71(19)
5.1 Introduction
71(1)
5.2 Similarity Solutions
71(4)
5.3 An Initial Value Problem
75(1)
5.4 The Formulae Derived
76(5)
5.5 Binary Options
81(2)
5.6 Risk Neutrality
83(7)
6 Variations on the Black-Scholes Model
90(16)
6.1 Introduction
90(1)
6.2 Options on Dividend-paying Assets
90(8)
6.3 Forward and Futures Contracts
98(2)
6.4 Options on Futures
100(1)
6.5 Time-dependent Parameters
101(5)
7 American Options
106(27)
7.1 Introduction
106(2)
7.2 The Obstacle Problem
108(1)
7.3 American Options as Free Boundary Problems
109(1)
7.4 The American Put
110(4)
7.5 Other American Options
114(1)
7.6 Linear Complementarity Problems
115(6)
7.7 The American Call with Dividends
121(12)
Part Two: Numerical Methods 133(62)
8 Finite-difference Methods
135(30)
8.1 Introduction
135(1)
8.2 Finite-difference Approximations
136(2)
8.3 The Finite-difference Mesh
138(1)
8.4 The Explicit Finite-difference Method
139(5)
8.5 Implicit Finite-difference Methods
144(1)
8.6 The Fully-implicit Method
144(11)
8.7 The Crank-Nicolson Method
155(10)
9 Methods for American Options
165(15)
9.1 Introduction
165(2)
9.2 Finite-difference Formulation
167(1)
9.3 The Constrained Matrix Problem
168(1)
9.4 Projected SOR
169(3)
9.5 The Time-stepping Algorithm
172(2)
9.6 Numerical Examples
174(2)
9.7 Convergence of the Method
176(4)
10 Binomial Methods
180(15)
10.1 Introduction
180(3)
10.2 The Discrete Random Walk
183(4)
10.3 Valuing the Option
187(1)
10.4 European Options
187(2)
10.5 American Options
189(2)
10.6 Dividend Yields
191(4)
Part Three: Further Option Theory 195(68)
11 Exotic and Path-dependent Options
197(9)
11.1 Introduction
197(2)
11.2 Compound Options: Options on Options
199(2)
11.3 Chooser Options
201(1)
11.4 Barrier Options
201(1)
11.5 Asian Options
202(1)
11.6 Lookback Options
203(3)
12 Barrier Options
206(7)
12.1 Introduction
206(1)
12.2 Knock-outs
207(2)
12.3 Knock-ins
209(4)
13 A Unifying Framework for Path-dependent Options
213(9)
13.1 Introduction
213(1)
13.2 Time Integrals of the Random Walk
214(3)
13.3 Discrete Sampling
217(5)
14 Asian Options
222(14)
14.1 Introduction
222(1)
14.2 Continuously Sampled Averages
223(2)
14.3 Similarity Reductions
225(1)
14.4 The Average Strike Option
226(4)
14.5 Average Rate Options
230(3)
14.6 Discretely Sampled Averages
233(3)
15 Lookback Options
236(16)
15.1 Introduction
236(1)
15.2 Continuous Sampling of the Maximum
237(6)
15.3 Discrete Sampling of the Maximum
243(1)
15.4 Similarity Reductions
244(2)
15.5 Some Numerical Examples
246(2)
15.6 Two `Perpetual Options'
248(4)
16 Options with Transaction Costs
252(11)
16.1 Introduction
252(1)
16.2 Discrete Hedging
252(5)
16.3 Portfolios of Options
257(6)
Part Four: Interest Rate Derivative Products 263(32)
17 Interest Rate Derivatives
265(21)
17.1 Introduction
265(1)
17.2 Basics of Bond Pricing
265(3)
17.3 The Yield Curve
268(2)
17.4 Stochastic Interest Rates
270(1)
17.5 The Bond Pricing Equation
270(3)
17.6 Solutions of the Bond Pricing Equation
273(7)
17.7 The Extended Vasicek Model of Hull & White
280(1)
17.8 Bond Options
281(1)
17.9 Other Interest Rate Products
282(4)
18 Convertible Bonds
286(9)
18.1 Introduction
286(1)
18.2 Convertible Bonds
286(4)
18.3 Convertible Bonds with Random Interest Rate
290(5)
Hints to Selected Exercises 295(13)
Bibliography 308(4)
Index 312

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