Table of contents for Financial statistics and mathematical finance : methods, models and applications / Ansgar Steland.


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Preface 7
1 Elementary Financial Calculus 1
1.1 Motivating Examples 1
1.2 Cashflows, interest rates, prices and returns 2
1.2.1 Bonds and the term structure of interest rates 5
1.2.2 Asset returns 6
1.2.3 Some basic models for asset prices 7
1.3 Elementary statistical analysis of returns 11
1.3.1 Measuring location 13
1.3.2 Measuring dispersion and risk 15
1.3.3 Measuring skewness and kurtosis 19
1.3.4 Estimation of the distribution 20
1.3.5 Testing for normality 26
1.4 Financial instruments 28
1.4.1 Contingent claims 28
1.4.2 Spot contracts and forwards 28
1.4.3 Futures contracts 29
1.4.4 Options 29
1.4.5 Barrier Options 30
1.4.6 Financial engineering 31
1.5 A Primer on Option Pricing 32
1.5.1 The no-arbitrage principle 32
1.5.2 Risk-neutral evaluation 33
1.5.3 Hedging and replication 35
1.5.4 Non-existence of a risk-neutral measure 36
1.5.5 The Black-Scholes pricing formula 36
1.5.6 The Greeks 38
1.5.7 Calibration, implied volatility and the smile 40
1.5.8 Option prices and the risk-neutral density 41
1.6 Notes and further reading 42
References 43
2 Arbitrage Theory for the One-Period Model 45
2.1 Definitions and preliminaries 45
2.2 Linear pricing measures 47
2.3 More on arbitrage 50
2.4 Separation theorems in Rn 52
2.5 No-arbitrage and martingale measures 55
2.6 Arbitrage-free pricing of contingent claims 63
2.7 Construction of MartingaleMeasures: General Case 68
2.8 Complete financial markets 71
2.9 Notes and further reading 74
References 74
3 Financial Models in Discrete Time 75
3.1 Adapted stochastic processes in discrete time 77
3.2 Martingales and martingale differences 81
3.2.1 The martingale transformation 87
3.2.2 Stopping times, optional sampling and a maximal inequality 88
3.2.3 Extensions to Rd 97
3.3 Stationarity 97
3.3.1 Weak and strict stationarity 98
3.4 Linear Processes and ARMA Models 106
3.4.1 Linear processes and the lag operator 106
3.4.2 Inversion 111
3.4.3 AR(p) and AR(∞) processes 113
3.4.4 ARMA processes 117
3.5 The frequency domain 118
3.5.1 The spectrum 118
3.5.2 The periodogram 121
3.6 Estimation of ARMA processes 126
3.7 (G)ARCH models 127
3.8 Long memory series 133
3.8.1 Fractional differences 133
3.8.2 Fractionally integrated processes 137
3.9 Notes and further reading 137
References 138
4 Arbitrage Theory for the Multi-Period Model 139
4.1 Definitions and preliminaries 139
4.2 Self-financing trading strategies 140
4.3 No-arbitrage and martingale measures 143
4.4 European claims on arbitrage-free markets 146
4.5 The martingale representation theorem in discrete time 150
4.6 The Cox-Ross-Rubinstein binomial model 151
4.7 The Black-Scholes formula 156
4.8 American options and contingent claims 161
4.8.1 Arbitrage-free pricing and the optimal exercise strategy 161
4.8.2 Pricing American options using binomial trees 164
4.9 Notes and further reading 165
References 166
5 Brownian Motion and Related Processes in Continuous Time 167
5.1 Preliminaries 167
5.2 Brownian Motion 170
5.2.1 Definition and basic properties 170
5.2.2 Brownian motion and the central limit theorem 177
5.2.3 Path properties 179
5.2.4 Brownian motion in higher dimensions 180
5.3 Continuity and differentiability 181
5.4 Self-similarity and fractional Brownian motion 183
5.5 Counting processes 184
5.5.1 The Poisson process 184
5.5.2 The compound Poisson process 186
5.6 L´evy processes 188
5.7 Notes and further reading 190
References 190
6 Itˆo Calculus 191
6.1 Total and quadratic variation 191
6.2 Stochastic Stieltjes integration 196
6.3 The Itˆo integral 199
6.4 Quadratic covariation 211
6.5 Itˆo’s formula 212
6.6 Itˆo processes 215
6.7 Diffusion processes and ergodicity 222
6.8 Numerical approximations and statistical estimation 223
6.9 Notes and further reading 225
References 225
7 The Black-Scholes-Model 227
7.1 The model and first properties 227
7.2 Girsanov’s theorem 233
7.3 Equivalent martingale measure 237
7.4 Arbitrage-free pricing and hedging claims 238
7.5 The delta hedge 241
7.6 Time-dependent volatility 242
7.7 The generalized Black-Scholes model 244
7.8 Notes and further reading 246
References 246
8 Limit Theory for Discrete-Time Processes 249
8.1 Limit theorems for correlated time series 250
8.2 A regression model for financial time series 259
8.2.1 Least squares estimation 261
8.3 Limit theorems for martingale difference 263
8.4 Asymptotics 268
8.5 Density estimation and nonparametric regression 272
8.5.1 Multivariate density estimation 272
8.5.2 Nonparametric regression 280
8.6 The CLT for linear processes 287
8.7 Mixing Processes 290
8.7.1 Mixing coefficients 290
8.7.2 Inequalities 292
8.8 Limit Theorems for Mixing Processes 297
8.9 Notes and further reading 306
References 306
9 Special Topics 309
9.1 Copulas - and the 2008 financial crisis 309
9.1.1 Copulas 310
9.1.2 The financial crisis 316
9.1.3 Models for credit defaults and CDOs 319
9.2 Local linear nonparametric regression 322
9.2.1 Applications in finance: Estimation of martingale measures and Itˆo
diffusions 322
9.2.2 Method and asymptotics 324
9.3 Change-point detection and monitoring 333
9.3.1 Offline detection 334
9.3.2 Online detection 342
9.4 Unit roots and random walk 345
9.4.1 The OLS estimator in the stationary AR(1) model 347
9.4.2 Nonparametric definitions for the degree of integration 351
9.4.3 The Dickey-Fuller test 352
9.4.4 Detecting unit roots and stationarity 355
9.5 Notes and further reading 363
References 363
A Appendix A 365
A.1 (Stochastic) Landau Symbols 365
A.2 Bochner’s Lemma 366
A.3 Conditional Expectation 367
A.4 Inequalities 368
A.5 Random Series 369
A.6 Local martingales in discrete time 369
Appendix B Weak Convergence and Central Limit Theorems 371
B.1 Convergence in distribution 371
B.2 Weak convergence 372
B.3 Prohorov’s theorem 377
B.4 Sufficient criteria 379
B.5 More on Skorohod spaces 381
B.6 Central Limit Theorems for Martingale Differences 381
B.7 Functional central limit theorems 382
B.8 Strong Approximations 384
References 386
Index


Library of Congress subject headings for this publication:
Business mathematics.
Calculus.
BUSINESS & ECONOMICS / Econometrics. -- bisacsh