Financial Derivatives in Theory and PracticeRevised Edition
P. J. HUNT
WestLB AG, London, UK
J. E. KENNEDY
University of Warwick, UK

Contents
Preface to revised edition xv
Preface xvii
Acknowledgements xxi
Part I: Theory 1
1 Single-Period Option Pricing 3
1.1 Option pricing in a nutshell 3
1.2 The simplest setting 4
1.3 General one-period economy 5
1.3.1 Pricing 6
1.3.2 Conditions for no arbitrage: existence of Z 7
1.3.3 Completeness: uniqueness of Z 9
1.3.4 Probabilistic formulation 12
1.3.5 Units and numeraires 15
1.4 A two-period example 15
2 Brownian Motion 19
2.1 Introduction 19
2.2 Definition and existence 20
2.3 Basic properties of Brownian motion 21
2.3.1 Limit of a random walk 21
2.3.2 Deterministic transformations of Brownian motion 23
2.3.3 Some basic sample path properties 24
2.4 Strong Markov property 26
2.4.1 Reflection principle 28
3 Martingales 31
3.1 Definition and basic properties 32
3.2 Classes of martingales 35
3.2.1 Martingales bounded in L1 35
3.2.2 Uniformly integrable martingales 36
3.2.3 Square-integrable martingales 39
3.3 Stopping times and the optional sampling theorem 41
3.3.1 Stopping times 41
3.3.2 Optional sampling theorem 45
3.4 Variation, quadratic variation and integration 49
3.4.1 Total variation and Stieltjes integration 49
3.4.2 Quadratic variation 51
3.4.3 Quadratic covariation 55
viii Contents
3.5 Local martingales and semimartingales 56
3.5.1 The space cMloc 56
3.5.2 Semimartingales 59
3.6 Supermartingales and the Doob—Meyer decomposition 61
4 Stochastic Integration 63
4.1 Outline 63
4.2 Predictable processes 65
4.3 Stochastic integrals: the L2 theory 67
4.3.1 The simplest integral 68
4.3.2 The Hilbert space L2(M) 69
4.3.3 The L2 integral 70
4.3.4 Modes of convergence to H •M 72
4.4 Properties of the stochastic integral 74
4.5 Extensions via localization 77
4.5.1 Continuous local martingales as integrators 77
4.5.2 Semimartingales as integrators 78
4.5.3 The end of the road! 80
4.6 Stochastic calculus: Itˆo’s formula 81
4.6.1 Integration by parts and Itˆo’s formula 81
4.6.2 Differential notation 83
4.6.3 Multidimensional version of Itˆo’s formula 85
4.6.4 L´evy’s theorem 88
5 Girsanov and Martingale Representation 91
5.1 Equivalent probability measures and the Radon—Nikod´ym derivative 91
5.1.1 Basic results and properties 91
5.1.2 Equivalent and locally equivalent measures on a filtered space 95
5.1.3 Novikov’s condition 97
5.2 Girsanov’s theorem 99
5.2.1 Girsanov’s theorem for continuous semimartingales 99
5.2.2 Girsanov’s theorem for Brownian motion 101
5.3 Martingale representation theorem 105
5.3.1 The space I2(M) and its orthogonal complement 106
5.3.2 Martingale measures and the martingale representation
theorem 110
5.3.3 Extensions and the Brownian case 111
6 Stochastic Differential Equations 115
6.1 Introduction 115
6.2 Formal definition of an SDE 116
6.3 An aside on the canonical set-up 117
6.4 Weak and strong solutions 119
6.4.1 Weak solutions 119
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