NO-ARBITRAGE THEORY FOR DERIVATIVES PRICING
Nizar TOUZI
Ecole Polytechnique Paris
Département de Mathématiques Appliquées
nizar.touzi@polytechnique.edu
March 29, 2007

Contents
1 Introduction 6
1.1 Finite discrete-time frictionless financial markets . . . . . . . . 6
1.1.1 Probabilistic setting . . . . . . . . . . . . . . . . . . . 6
1.1.2 The financial market . . . . . . . . . . . . . . . . . . . 7
1.1.3 Self-financing portfolio strategies . . . . . . . . . . . . 8
1.1.4 Wealth process . . . . . . . . . . . . . . . . . . . . . . 9
1.1.5 Change of numéraire . . . . . . . . . . . . . . . . . . . 10
1.2 Some examples of frictions . . . . . . . . . . . . . . . . . . . . 11
1.2.1 Portfolio constraints . . . . . . . . . . . . . . . . . . . 11
1.2.2 Solvency constraint . . . . . . . . . . . . . . . . . . . . 12
1.2.3 Transaction costs . . . . . . . . . . . . . . . . . . . . . 12
1.2.4 Taxes on capital gains . . . . . . . . . . . . . . . . . . 13
1.2.5 Large investor problem . . . . . . . . . . . . . . . . . . 13
1.3 Some problems in frictionless financial markets . . . . . . . . . 14
2 Arbitrage-free frictionless financial markets in discrete-time 19
2.1 The finite probability one-period model with a single risky asset 20
2.2 The fundamental theorem of asset pricing in general finite
discrete-time models . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.2 Reduction to a local one-period problem . . . . . . . . 23
2.2.3 Characterization of the local no arbitrage condition . . 25
2.2.4 On discrete-time martingales . . . . . . . . . . . . . . . 30
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2.2.5 Characterization of the no arbitrage condition . . . . . 35
2.3 Complement : the closeness property . . . . . . . . . . . . . . 38
2.4 Model-free implications of no-arbitrage . . . . . . . . . . . . . 41
2.4.1 American versus European options . . . . . . . . . . . 42
2.4.2 Put-Call Parity . . . . . . . . . . . . . . . . . . . . . . 43
2.4.3 Exercise price effects . . . . . . . . . . . . . . . . . . . 43
2.4.4 Maturity Effects for American call and put options . . 44
2.4.5 Bounds on call prices and early exercise of American
calls . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.4.6 Risk effect on options prices . . . . . . . . . . . . . . . 46
3 Super-hedging contingent claims in discrete-time financial
markets 48
3.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . 48
3.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.1.2 First properties . . . . . . . . . . . . . . . . . . . . . . 49
3.1.3 No arbitrage bounds . . . . . . . . . . . . . . . . . . . 50
3.2 One-period examples . . . . . . . . . . . . . . . . . . . . . . . 50
3.2.1 The binomial model . . . . . . . . . . . . . . . . . . . 50
3.2.2 The trinomial model . . . . . . . . . . . . . . . . . . . 53
3.3 Dual formulation
of the super-hedging problem . . . . . . . . . . . . . . . . . . 56
3.4 Application to the one-period examples . . . . . . . . . . . . . 59
3.5 The Cox-Ross-Rubinstein model . . . . . . . . . . . . . . . . . 60
3.5.1 Description of themodel . . . . . . . . . . . . . . . . . 60
3.5.2 Valuation and hedging of European options . . . . . . 62
3.5.3 Continuous-time limit : A first approach to the Black-
Schole model . . . . . . . . . . . . . . . . . . . . . . . 64
3.6 Complement : closeness of the set of hedgeable claims . . . . . 68
4 Complete financial markets 71
4.1 Financial markets with unique risk neutral measure . . . . . . 71
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4.2 Complete / incomplete financial markets . . . . . . . . . . . . 74
4.3 Martingale characterization of completemarkets . . . . . . . . 77
4.4 Utility indifference valuation . . . . . . . . . . . . . . . . . . . 79
4.5 Application : optimal portfolio management in complete markets 80
4.5.1 Problem formulation . . . . . . . . . . . . . . . . . . . 80
4.5.2 Explicit solution in a complete financial market . . . . 81
4.6 Complement: American contingent claims in complete financial
markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.6.1 Problem formulation . . . . . . . . . . . . . . . . . . . 84
4.6.2 Decomposition of supermartingales . . . . . . . . . . . 85
4.6.3 The Snell envelope . . . . . . . . . . . . . . . . . . . . 87
4.6.4 Valuation of American contingent claims in a complete
market . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5 Introduction to stochastic calculus 93
5.1 Filtration and stopping times . . . . . . . . . . . . . . . . . . 93
5.1.1 Filtration . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.1.2 Stopping times . . . . . . . . . . . . . . . . . . . . . . 95
5.1.3 Martingales and optional sampling . . . . . . . . . . . 96
5.2 The Brownian motion . . . . . . . . . . . . . . . . . . . . . . 97
5.2.1 Definition and discrete-time approximation . . . . . . . 98
5.2.2 Distribution of the Brownian motion . . . . . . . . . . 103
5.2.3 Scaling, symmetry, and time reversal . . . . . . . . . . 105
5.2.4 Small/large time behavior of the Brownian sample paths110
5.2.5 Quadratic variation . . . . . . . . . . . . . . . . . . . . 112
5.2.6 On the filtration of the Brownian motion . . . . . . . . 115
5.3 Itô’s lemma and stochastic integration . . . . . . . . . . . . . 116
5.4 Local martingale property of stochastic integrals . . . . . . . . 121
5.5 The Feynman-Kac representation formula . . . . . . . . . . . 122
5.6 The Cameron-Martin change ofmeasure . . . . . . . . . . . . 125
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6 The Black-Scholes valuation theory 127
6.1 The continuous-time financial market . . . . . . . . . . . . . . 127
6.2 Portfolio and wealth process . . . . . . . . . . . . . . . . . . . 128
6.3 Admissible portfolios and no-arbitrage . . . . . . . . . . . . . 130
6.4 Hedging and pricing Vanilla options . . . . . . . . . . . . . . . 134
6.5 The PDE approach for the valuation problem . . . . . . . . . 136
6.6 The Black and Scholes model for European call options . . . . 137
6.6.1 The Balck-Scholes formula . . . . . . . . . . . . . . . . 137
6.6.2 The Black’s formula . . . . . . . . . . . . . . . . . . . 140
6.6.3 Option on a dividend paying stock . . . . . . . . . . . 141
6.6.4 The Garman-Kolhagen model for exchange rate options 143
6.6.5 The practice of the Black-Scholes model . . . . . . . . 146
6.7 Complement: barrier options in the Black-Scholes model . . . 152
7 Gaussian interest rates models 154
7.1 Fixed income vocabulary . . . . . . . . . . . . . . . . . . . . . 155
7.1.1 Zero-coupon bonds . . . . . . . . . . . . . . . . . . . . 155
7.1.2 Interest rates swaps . . . . . . . . . . . . . . . . . . . . 157
7.1.3 Yields fromzero-coupon bonds . . . . . . . . . . . . . 158
7.1.4 Forward Interest Rates . . . . . . . . . . . . . . . . . . 159
7.1.5 Instantaneous interest rates . . . . . . . . . . . . . . . 161
7.2 The Vasicek model . . . . . . . . . . . . . . . . . . . . . . . . 162
7.3 Zero-coupon bonds prices . . . . . . . . . . . . . . . . . . . . . 163
7.4 Calibration to the spot yield curve and the generalized Vasicek
model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
7.5 Multiple Gaussian factorsmodels . . . . . . . . . . . . . . . . 167
7.6 Introduction to the Heath-Jarrow-Mortonmodel . . . . . . . . 169
7.6.1 Dynamics of the forward rates curve . . . . . . . . . . 169
7.6.2 The Heath-Jarrow-Morton drift condition . . . . . . . 170
7.6.3 The Ho-Leemodel . . . . . . . . . . . . . . . . . . . . 173
7.6.4 The Hull-Whitemodel . . . . . . . . . . . . . . . . . . 173
7.7 The forward neutral measure . . . . . . . . . . . . . . . . . . . 174
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7.8 Derivatives pricing under stochastic interest rates and volatility
calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
7.8.1 European options on zero-coupon bonds . . . . . . . . 176
7.8.2 The Black-Scholes formula under stochastic interest rates178