The Mathematics of Financial Derivatives: A Student Introduction

by Paul Wilmott , Sam Howison , Jeff Dewynne

• ISBN13: 9780521497893
• ISBN10: 0521497892
• Format: Paperback
• Copyright: 1995-09-29
• Publisher: Cambridge University Press

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