Financial Statistics and Mathematical Finance Methods, Models and Applications

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ISBN: 9780470710586 | 0470710586
Cover: Hardcover
Copyright: 8/13/2012

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Financial Statistics and Mathematical Finance: Methods, Models and Applications introduces the financial methodology and the relevant mathematical tools in a style that is mathematically rigorous and yet accessible to advanced level practitioners and mathematicians alike. The book will focus on elementary financial calculus, statistical models for financial data, option pricing. Elements of the statistical analysis of financial time series, central limit theorems for time series as well as Ito calculus and stochastic integration are discussed.

Preface	p. xi
Acknowledgements	p. xv
Elementary financial calculus	p. 1
Motivating examples	p. 1
Cashflows, interest rates, prices and returns	p. 2
Bonds and the term structure of interest rates	p. 5
Asset returns	p. 6
Some basic models for asset prices	p. 8
Elementary statistical analysis of returns	p. 11
Measuring location	p. 13
Measuring dispersion and risk	p. 16
Measuring skewness and kurtosis	p. 20
Estimation of the distribution	p. 21
Testing for normality	p. 27
Financial instruments	p. 28
Contingent claims	p. 28
Spot contracts and forwards	p. 29
Futures contracts	p. 29
Options	p. 30
Barrier options	p. 31
Financial engineering	p. 32
A primer on option pricing	p. 32
The no-arbitrage principle	p. 32
Risk-neutral evaluation	p. 33
Hedging and replication	p. 36
Nonexistence of a risk-neutral measure	p. 37
The Black-Scholes pricing formula	p. 37
The Greeks	p. 39
Calibration, implied volatility and the smile	p. 41
Option prices and the risk-neutral density	p. 41
Notes and further reading	p. 43
References	p. 43
Arbitrage theory for the one-period model	p. 45
Definitions and preliminaries	p. 45
Linear pricing measures	p. 47
More on arbitrage	p. 50
Separation theorems in Rⁿ	p. 53
No-arbitrage and martingale measures	p. 56
Arbitrage-free pricing of contingent claims	p. 65
Construction of martingale measures: general case	p. 70
Complete financial markets	p. 73
Notes and further reading	p. 76
References	p. 76
Financial models in discrete time	p. 79
Adapted stochastic processes in discrete time	p. 81
Martingales and martingale differences	p. 85
The martingale transformation	p. 91
Stopping times, optional sampling and a maximal inequality	p. 93
Extensions to R^d	p. 101
Stationarity	p. 102
Weak and strict stationarity	p. 102
Linear processes and ARMA models	p. 111
Linear processes and the lag operator	p. 111
Inversion	p. 116
ARQ(p) and AR(∞) processes	p. 119
ARMA processes	p. 122
The frequency domain	p. 124
The spectrum	p. 124
The periodogram	p. 126
Estimation of ARMA processes	p. 132
(G)ARCH models	p. 133
Long-memory series	p. 139
Fractional differences	p. 139
Fractionally integrated processes	p. 144
Notes and further reading	p. 144
References	p. 145
Arbitrage theory for the multiperiod model	p. 147
Definitions and preliminaries	p. 148
Self-financing trading strategies	p. 148
No-arbitrage and martingale measures	p. 152
European claims on arbitrage-free markets	p. 154
The martingale representation theorem in discrete time	p. 159
The Cox-Ross-Rubinstein binomial model	p. 160
The Black-Scholes formula	p. 165
American options and contingent claims	p. 171
Arbitrage-free pricing and the optimal exercise strategy	p. 171
Pricing american options using binomial trees	p. 174
Notes and further reading	p. 175
References	p. 175
Brownian motion and related processes in continuous time	p. 177
Preliminaries	p. 177
Brownian motion	p. 181
Definition and basic properties	p. 181
Brownian motion and the central limit theorem	p. 188
Path properties	p. 190
Brownian motion in higher dimensions	p. 191
Continuity and differentiability	p. 192
Self-similarity and fractional Brownian motion	p. 193
Counting processes	p. 195
The poisson process	p. 195
The compound poisson process	p. 196
Lévy processes	p. 199
Notes and further reading	p. 201
References	p. 201
Itô Calculus	p. 203
Total and quadratic variation	p. 204
Stochastic Stieltjes integration	p. 208
The Itô integral	p. 212
Quadratic covariation	p. 225
Itô's formula	p. 226
Itô processes	p. 229
Diffusion processes and ergodicity	p. 236
Numerical approximations and statistical estimation	p. 238
Notes and further reading	p. 239
References	p. 240
The Black-Scholes model	p. 241
The model and first properties	p. 241
Girsanov's theorem	p. 247
Equivalent martingale measure	p. 251
Arbitrage-free pricing and hedging claims	p. 252
The delta hedge	p. 256
Time-dependent volatility	p. 257
The generalized Black-Scholes model	p. 259
Notes and further reading	p. 261
References	p. 262
Limit theory for discrete-time processes	p. 263
Limit theorems for correlated time series	p. 264
A regression model for financial time series	p. 273
Least squares estimation	p. 276
Limit theorems for martingale difference	p. 278
Asymptotics	p. 283
Density estimation and nonparametric regression	p. 287
Multivariate density estimation	p. 288
Nonparametric regression	p. 295
The CLT for linear processes	p. 302
Mixing processes	p. 306
Mixing coefficients	p. 306
Inequalities	p. 308
Limit theorems for mixing processes	p. 313
Notes and further reading	p. 323
References	p. 323
Special topics	p. 325
Copulas - and the 2008 financial crisis	p. 325
Copulas	p. 326
The financial crisis	p. 332
Models for credit defaults and CDOs	p. 335
Local Linear nonparametric regression	p. 338
Applications in finance: estimation of martingale measures and Ito diffusions	p. 339
Method and asymptotics	p. 340
Change-point detection and monitoring	p. 350
Offline detection	p. 351
Online detection	p. 359
Unit roots and random walk	p. 363
The OLS estimator in the stationary AR(1) model	p. 364
Nonparametric definitions for the degree of integration	p. 368
The Dickey-Fuller test	p. 370
Detecting unit roots and stationarity	p. 373
Notes and further reading	p. 381
References	p. 382
Appendix A	p. 385
(Stochastic) Landau symbols	p. 385
Bochner's lemma	p. 387
Conditional expectation	p. 387
Inequalities	p. 388
Random series	p. 389
Local martingales in discrete time	p. 389
Weak convergence and central limit theorems	p. 391
Convergence in distribution	p. 391
Weak convergence	p. 392
Prohorov's theorem	p. 398
Sufficient criteria	p. 399
More on Skorohod spaces	p. 401
Central limit theorems for martingale differences	p. 402
Functional central limit theorems	p. 403
Strong approximations	p. 405
References	p. 407
Index	p. 409
Table of Contents provided by Ingram. All Rights Reserved.

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Financial Statistics and Mathematical Finance Methods, Models and Applications

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