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CALCUL STOCHASTIQUE ET FINANCE

PeterTankov

peter.tankov@ensae.frNizarTouzi nizar.touzi@polytechnique.edu

Ecole Polytechnique

D epartement de Mathematiques Appliquees

Last update: Septembre 2018

2

Contents

1 Introduction: discrete time derivatives pricing 9

1.1 Basic derivative products . . . . . . . . . . . . . . . . . . . . . .

11

1.1.1 European and American options . . . . . . . . . . . . . .

11

1.1.2 Bonds and term structure of interest rates . . . . . . . . .

12

1.1.3 Forward contrats . . . . . . . . . . . . . . . . . . . . . . .

13

1.2 No dominance principle and rst properties . . . . . . . . . . . .

14

1.2.1 Valuation of forward Contracts . . . . . . . . . . . . . . .

14

1.2.2 Some properties of options prices . . . . . . . . . . . . . .

14

1.3 Put-Call Parity . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

1.4 Bounds on call prices and early exercise of American calls . . . .

16

1.4.1 Risk eect on options prices . . . . . . . . . . . . . . . . .

17

1.5 Some popular examples of contingent claims . . . . . . . . . . . .

18

2 A rst approach to the Black-Scholes formula 21

2.1 The single period binomial model . . . . . . . . . . . . . . . . . .

21

2.2 The Cox-Ross-Rubinstein model . . . . . . . . . . . . . . . . . .

23

2.3 Valuation and hedging

in the Cox-Ross-Rubinstein model . . . . . . . . . . . . . . . . . 24

2.4 Continuous-time limit . . . . . . . . . . . . . . . . . . . . . . . .

25

3 Some preliminaries on continuous-time processes 29

3.1 Filtration and stopping times . . . . . . . . . . . . . . . . . . . .

29

3.1.1 Filtration . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

3.1.2 Stopping times . . . . . . . . . . . . . . . . . . . . . . . .

30

3.2 Martingales and optional sampling . . . . . . . . . . . . . . . . .

33

3.3 Maximal inequalities for submartingales . . . . . . . . . . . . . .

35

3.4 Submartingales with a.s. cad-lag versions . . . . . . . . . . . . .

36

3.5 Appendix: on discrete-time martingales . . . . . . . . . . . . . .

38

3.5.1 Doob's optional sampling for discrete martingales . . . . .

38

3.5.2 Upcrossings of discrete-time submartingales . . . . . . . .

39

4 The Brownian Motion 41

4.1 Denition of the Brownian motion . . . . . . . . . . . . . . . . .

42

4.2 The Brownian motion as a limit of a random walk . . . . . . . .

43
3 4

4.3 Distribution of the Brownian motion . . . . . . . . . . . . . . . .

46

4.4 Scaling, symmetry, and time reversal . . . . . . . . . . . . . . . .

49

4.5 Brownian ltration and the Zero-One law . . . . . . . . . . . . .

52

4.6 Small/large time behavior of the Brownian sample paths . . . . .

53

4.7 Quadratic variation . . . . . . . . . . . . . . . . . . . . . . . . . .

57

5 Stochastic integration with respect to the Brownian motion 61

5.1 Stochastic integrals of simple processes . . . . . . . . . . . . . . .

61

5.2 Stochastic integrals of processes inH2. . . . . . . . . . . . . . .62

5.2.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . .

62

5.2.2 The stochastic integral as a continuous process . . . . . .

63

5.2.3 Martingale property and the It^o isometry . . . . . . . . .

65

5.2.4 Deterministic integrands . . . . . . . . . . . . . . . . . . .

65

5.3 Stochastic integration beyondH2and It^o processes . . . . . . . .66

5.4 Complement: density of simple processes inH2. . . . . . . . . .68

6 It^o Dierential Calculus 71

6.1 It^o's formula for the Brownian motion . . . . . . . . . . . . . . .

72

6.2 Extension to It^o processes . . . . . . . . . . . . . . . . . . . . . .

75

6.3 Levy's characterization of Brownian motion . . . . . . . . . . . .

78

6.4 A verication approach to the Black-Scholes model . . . . . . . .

79

6.5 The Ornstein-Uhlenbeck process . . . . . . . . . . . . . . . . . .

82

6.5.1 Distribution . . . . . . . . . . . . . . . . . . . . . . . . . .

82

6.5.2 Dierential representation . . . . . . . . . . . . . . . . . .

84

6.6 The Merton optimal portfolio allocation . . . . . . . . . . . . . .

85

6.6.1 Problem formulation . . . . . . . . . . . . . . . . . . . . .

85

6.6.2 The dynamic programming equation . . . . . . . . . . . .

85

6.6.3 Solving the Merton problem . . . . . . . . . . . . . . . . .

87

7 Martingale representation and change of measure 89

7.1 Martingale representation . . . . . . . . . . . . . . . . . . . . . .

89

7.2 The Cameron-Martin change of measure . . . . . . . . . . . . . .

93

7.3 The Girsanov's theorem . . . . . . . . . . . . . . . . . . . . . . .

94

7.3.1 The Novikov's criterion . . . . . . . . . . . . . . . . . . .

97

7.4 Application: the martingale approach to the Black-Scholes model

98

7.4.1 The continuous-time nancial market . . . . . . . . . . .

98

7.4.2 Portfolio and wealth process . . . . . . . . . . . . . . . . .

99

7.4.3 Admissible portfolios and no-arbitrage . . . . . . . . . . .

101

7.4.4 Super-hedging and no-arbitrage bounds . . . . . . . . . .

101

7.4.5 Heuristics from linear programming . . . . . . . . . . . .

102

7.4.6 The no-arbitrage valuation formula . . . . . . . . . . . . .

104

7.5 The continuous time Kalman-Bucy lter . . . . . . . . . . . . . .

104

7.5.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . .

105

7.5.2 Main result . . . . . . . . . . . . . . . . . . . . . . . . . .

105

7.5.3 The innovation process . . . . . . . . . . . . . . . . . . . .

106

7.5.4 Dynamics of the best estimate . . . . . . . . . . . . . . .

107
5

7.5.5 ODE characterization of the variance . . . . . . . . . . . .

110

8 Stochastic dierential equations 111

8.1 First examples . . . . . . . . . . . . . . . . . . . . . . . . . . . .

111

8.2 Strong solution of a stochastic dierential equation . . . . . . . .

113

8.2.1 Existence and uniqueness . . . . . . . . . . . . . . . . . .

113

8.2.2 The Markov property . . . . . . . . . . . . . . . . . . . .

116

8.3 More results for scalar stochastic dierential equations . . . . . .

116

8.4 Linear stochastic dierential equations . . . . . . . . . . . . . . .

120

8.4.1 An explicit representation . . . . . . . . . . . . . . . . . .

120

8.4.2 The Brownian bridge . . . . . . . . . . . . . . . . . . . . .

121

8.5 Connection with linear partial dierential equations . . . . . . .

122

8.5.1 Generator . . . . . . . . . . . . . . . . . . . . . . . . . . .

122

8.5.2 Cauchy problem and the Feynman-Kac representation . .

123

8.5.3 Representation of the Dirichlet problem . . . . . . . . . .

125

8.6 The hedging portfolio in a Markov nancial market . . . . . . . .

126

8.7 Application to importance sampling . . . . . . . . . . . . . . . .

127

8.7.1 Importance sampling for random variables . . . . . . . . .

127

8.7.2 Importance sampling for stochastic dierential equations .

129

9 The Black-Scholes model and its extensions 131

9.1 The Black-Scholes approach for the Black-Scholes formula . . . .

131

9.2 The Black and Scholes model for European call options . . . . .

132

9.2.1 The Black-Scholes formula . . . . . . . . . . . . . . . . . .

132

9.2.2 The Black's formula . . . . . . . . . . . . . . . . . . . . .

135

9.2.3 Option on a dividend paying stock . . . . . . . . . . . . .

135

9.2.4 The Garman-Kohlhagen model for exchange rate options

137

9.2.5 The practice of the Black-Scholes model . . . . . . . . . .

139

9.2.6 Hedging with constant volatility: robustness of the Black-

Scholes model . . . . . . . . . . . . . . . . . . . . . . . . . 144

9.3 Complement: barrier options in the Black-Scholes model . . . . .

146

9.3.1 Barrier options prices . . . . . . . . . . . . . . . . . . . .

147

9.3.2 Dynamic hedging of barrier options . . . . . . . . . . . .

150

9.3.3 Static hedging of barrier options . . . . . . . . . . . . . .

150

10 Local volatility models and Dupire's formula 153

10.1 Implied volatility . . . . . . . . . . . . . . . . . . . . . . . . . . .

153

10.2 Local volatility models . . . . . . . . . . . . . . . . . . . . . . . .

155

10.2.1 CEV model . . . . . . . . . . . . . . . . . . . . . . . . . .

156

10.3 Dupire's formula . . . . . . . . . . . . . . . . . . . . . . . . . . .

158

10.3.1 Dupire's formula in practice . . . . . . . . . . . . . . . . .

162

10.3.2 Link between local and implied volatility . . . . . . . . .

162
6

11 Backward SDEs and funding problems 165

11.1 Preliminaries: the BDG inequality . . . . . . . . . . . . . . . . .

165

11.1.1 The smooth power case . . . . . . . . . . . . . . . . . . .

166

11.1.2 The case of an arbitrary power . . . . . . . . . . . . . . .

167

11.2 Backward SDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . .

168

11.2.1 Martingale representation for zero generator . . . . . . . .

168

11.2.2 BSDEs with ane generator . . . . . . . . . . . . . . . .

169

11.2.3 The main existence and uniqueness result . . . . . . . . .

169

11.2.4 Complementary properties . . . . . . . . . . . . . . . . . .

172

11.3 Application: Funding Value Adjustment of the Black-Scholes the-

ory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

11.3.1 The BSDE point of view for the Black-Scholes model . . .

176

11.3.2 Funding Value Adjustment (FVA) . . . . . . . . . . . . .

177

12 Doob-Meyer decomposition, optimal stopping and American

options 181

12.1 The Doob-Meyer decomposition . . . . . . . . . . . . . . . . . . .

181

12.1.1 The discrete-time Doob decomposition . . . . . . . . . . .

182

12.1.2 Convergence of uniformly integrable sequences . . . . . .

183

12.1.3 The continuous-time Doob-Meyer decomposition . . . . .

184

12.2 Optimal stopping . . . . . . . . . . . . . . . . . . . . . . . . . . .

185

12.2.1 The dynamic programming principle . . . . . . . . . . . .

186

12.2.2 Optimal stopping rule . . . . . . . . . . . . . . . . . . . .

188

12.3 Pricing and hedging American derivatives . . . . . . . . . . . . .

189

12.3.1 Denition and rst properties . . . . . . . . . . . . . . . .

189

12.3.2 No-arbitrage valuation and hedging . . . . . . . . . . . .

189

12.3.3 The valuation equation . . . . . . . . . . . . . . . . . . .

190

12.3.4 The exercise boundary . . . . . . . . . . . . . . . . . . . .

193

12.3.5 Perpetual American derivatives . . . . . . . . . . . . . . .

194

12.4 Appendix: essential supremum . . . . . . . . . . . . . . . . . . .

197

12.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

198

13 Gaussian interest rates models 199

13.1 Fixed income terminology . . . . . . . . . . . . . . . . . . . . . .

200

13.1.1 Zero-coupon bonds . . . . . . . . . . . . . . . . . . . . . .

200

13.1.2 Interest rates swaps . . . . . . . . . . . . . . . . . . . . .

201

13.1.3 Yields from zero-coupon bonds . . . . . . . . . . . . . . .

202

13.1.4 Forward Interest Rates . . . . . . . . . . . . . . . . . . . .

202

13.1.5 Instantaneous interest rates . . . . . . . . . . . . . . . . .

203

13.2 The Vasicek model . . . . . . . . . . . . . . . . . . . . . . . . . .

204

13.3 Zero-coupon bonds prices . . . . . . . . . . . . . . . . . . . . . .

205

13.4 Calibration to the spot yield curve and the generalized Vasicek

model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

13.5 Multiple Gaussian factors models . . . . . . . . . . . . . . . . . .

209

13.6 Introduction to the Heath-Jarrow-Morton model . . . . . . . . .

211

13.6.1 Dynamics of the forward rates curve . . . . . . . . . . . .

211
7

13.6.2 The Heath-Jarrow-Morton drift condition . . . . . . . . .

212

13.6.3 The Ho-Lee model . . . . . . . . . . . . . . . . . . . . . .

214

13.6.4 The Hull-White model . . . . . . . . . . . . . . . . . . . .

214

13.7 The forward neutral measure . . . . . . . . . . . . . . . . . . . .

215

13.8 Derivatives pricing under stochastic interest rates and volatility

calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

13.8.1 European options on zero-coupon bonds . . . . . . . . . .

216

13.8.2 The Black-Scholes formula under stochastic interest rates

217

14 Introduction to nancial risk management 219

14.1 Classication of risk exposures . . . . . . . . . . . . . . . . . . .

220

14.1.1 Market risk . . . . . . . . . . . . . . . . . . . . . . . . . .

220

14.1.2 Credit risk . . . . . . . . . . . . . . . . . . . . . . . . . .

222

14.1.3 Liquidity risk . . . . . . . . . . . . . . . . . . . . . . . . .

223

14.1.4 Operational risk . . . . . . . . . . . . . . . . . . . . . . .

224

14.1.5 Model risk . . . . . . . . . . . . . . . . . . . . . . . . . . .

224

14.2 Risk exposures and risk limits: sensitivity approach to risk man-

agement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

14.3 Value at Risk and the global approach . . . . . . . . . . . . . . .

227

14.4 Convex and coherent risk measures . . . . . . . . . . . . . . . . .

231

14.5 Regulatory capital and the Basel framework . . . . . . . . . . . .

233

A Preliminaires de la theorie des mesures 237

A.1 Espaces mesurables et mesures . . . . . . . . . . . . . . . . . . . 237
A.1.1 Algebres,algebres . . . . . . . . . . . . . . . . . . . . .237 A.1.2 Mesures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
A.1.3 Proprietes elementaires des mesures . . . . . . . . . . . . 239
A.2 L'integrale de Lebesgue . . . . . . . . . . . . . . . . . . . . . . . 241
A.2.1 Fonction mesurable . . . . . . . . . . . . . . . . . . . . . . 241
A.2.2 Integration des fonctions positives . . . . . . . . . . . . . 242
A.2.3 Integration des fonctions reelles . . . . . . . . . . . . . . . 245
A.2.4 De la convergence p.p. a la convergenceL1. . . . . . . .245 A.2.5 Integrale de Lebesgue et integrale de Riemann . . . . . . 247
A.3 Transformees de mesures . . . . . . . . . . . . . . . . . . . . . . . 248
A.3.1 Mesure image . . . . . . . . . . . . . . . . . . . . . . . . . 248
A.3.2 Mesures denies par des densites . . . . . . . . . . . . . . 248
A.4 Inegalites remarquables . . . . . . . . . . . . . . . . . . . . . . . 249
A.5 Espaces produits . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
A.5.1 Construction et integration . . . . . . . . . . . . . . . . . 250
A.5.2 Mesure image et changement de variable . . . . . . . . . . 252
A.6 Annexe du chapitre A . . . . . . . . . . . . . . . . . . . . . . . . 253
A.6.1systeme,dsysteme et unicite des mesures . . . . . . .253 A.6.2 Mesure exterieure et extension des mesures . . . . . . . . 254
A.6.3 Demonstration du theoreme des classes monotones . . . . 25 6
8 B Preliminaires de la theorie des probabilites 259 B.1 Variables aleatoires . . . . . . . . . . . . . . . . . . . . . . . . . . 259
B.1.1algebre engendree par une v.a. . . . . . . . . . . . . . .259 B.1.2 Distribution d'une v.a. . . . . . . . . . . . . . . . . . . . . 260
B.2 Esperance de variables aleatoires . . . . . . . . . . . . . . . . . . 261
B.2.1 Variables aleatoires a densite . . . . . . . . . . . . . . . . 261
B.2.2 Inegalites de Jensen . . . . . . . . . . . . . . . . . . . . . 262
B.2.3 Fonction caracteristique . . . . . . . . . . . . . . . . . . . 263

B.3 EspacesLpet convergences

fonctionnelles des variables aleatoires . . . . . . . . . . . . . . . . 265
B.3.1 Geometrie de l'espaceL2. . . . . . . . . . . . . . . . . .265 B.3.2 EspacesLpetLp. . . . . . . . . . . . . . . . . . . . . . .266 B.3.3 EspacesL0etL0. . . . . . . . . . . . . . . . . . . . . . .267 B.3.4 Lien entre les convergencesLp, en proba et p.s. . . . . . .268 B.4 Convergence en loi . . . . . . . . . . . . . . . . . . . . . . . . . . 270
B.4.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . 271
B.4.2 Caracterisation de la convergence en loi par les fonctions de repartition . . . . . . . . . . . . . . . . . . . . . . . . . 271
B.4.3 Convergence des fonctions de repartition . . . . . . . . . . 272
B.4.4 Convergence en loi et fonctions caracteristiques . . . . . . 273
B.5 Independance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
B.5.1algebres independantes . . . . . . . . . . . . . . . . . .275 B.5.2 variables aleatoires independantes . . . . . . . . . . . . . 276
B.5.3 Asymptotique des suites d'evenements independants . . . 276
B.5.4 Asymptotique des moyennes de v.a. independantes . . . . 278

C Conditional expectation 279

C.1 Premieres intuitions . . . . . . . . . . . . . . . . . . . . . . . . . 279
C.1.1 Esperance conditionnelle en espace d'etats ni . . . . . . 279
C.1.2 Cas des variables a densites . . . . . . . . . . . . . . . . . 280
C.2 Denition et premieres proprietes . . . . . . . . . . . . . . . . . . 281
C.3 Proprietes de l'esperance conditionnelle . . . . . . . . . . . . . . 283
C.4 Application au ltre de Kalman-Bucy . . . . . . . . . . . . . . . 285
C.4.1 Lois conditionnelles pour les vecteurs gaussiens . . . . . . 286
C.4.2 Filtre de Kalman-Bucy . . . . . . . . . . . . . . . . . . . 287

Chapter 1

Introduction: discrete time

derivatives pricing Financial mathematics is a young eld of applications of mathematics which experienced a huge growth during the last thirty years. It is by now considered as one of the most challenging elds of applied mathematics by the diversity of the questions which are raised, and the high technical skills that it requires. These lecture notes provide an introduction to stochastic nance for the students of third year of Ecole Polytechnique. Our objective is to cover the basic Black-Scholes theory from the modern martingale approach. This requires the development of the necessary tools from stochastic calculus and their connection with partial dierential equations. Modeling nancial markets by continuous-time stochastic processes was ini- tiated by Louis Bachelier (1900) in his thesis dissertation under the supervision of Henri Poincare. Bachelier's work was not recognized until the recent his- tory. Sixty years later, Samuelson (Nobel Prize in economics 1970) came back to this idea, suggesting a Brownian motion with constant drift as a model for stock prices. However, the real success of Brownian motion in the nancial applications was realized by Fisher Black, Myron Scholes, et Robert Merton (Nobel Prize in economics 1997) who founded between 1969 and 1973 the mod- ern theory of nancial mathematics by introducing the portfolio theory and the no-arbitrage pricing arguments. Since then, this theory gained an impor- tant amount of rigor and precision, essentially thanks to the martingale theory developed in the eighties. Although continuous-time models are more demanding from the technical viewpoint, they are widely used in the nancial industry because of the sim- plicity of the resulting formulae for pricing and hedging. This is related to the powerful tools of dierential calculus which are available only in continuous- time. We shall rst provide a self-contained introduction of the main concept from stochastic analysis: Brownian motion, stochastic integration with respect to the Brownian motion, It^o's formula, Girsanov change of measure Theorem, 9

10CHAPTER 1. INTRODUCTION

connection with the heat equation, and stochastic dierential equations. We then consider the Black-Scholes continuous-time nancial market where the no- arbitrage concept is sucient for the determination of market prices of derivative securities. Prices are expressed in terms of the unique risk-neutral measure, and can be expressed in closed form for a large set of relevant derivative securities. The nal chapter provides the main concepts in interest rates models in the gaussian case. In order to motivate the remaining content of theses lecture notes, we wouldquotesdbs_dbs42.pdfusesText_42

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