Financial statistics and mathematical finance: methods, models and applications

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Title:

Personal Author:

Steland, Ansgar

Publication Information:

Chichester, West Sussex: Wiley, 2012

Physical Description:

xiv,415p.; 26cm.

ISBN:

9780470710586

Subject Term:

Business mathematics

Calculus

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Library	Item Barcode	Call Number	Material Type	Item Category 1	Status
Searching... PSZ JB	30000010307169	HF5691 S6585 2012	Open Access Book	Book	Searching... Unknown
Searching... Razak School	30000010306815	HF5691 S6585 2012	Open Access Book	Book	Searching... Unknown

Mathematical finance has grown into a huge area of research which requires a lot of care and a large number of sophisticated mathematical tools. Mathematically rigorous and yet accessible to advanced level practitioners and mathematicians alike, it considers various aspects of the application of statistical methods in finance and illustrates some of the many ways that statistical tools are used in financial applications.

Financial Statistics and Mathematical Finance:

Provides an introduction to the basics of financial statistics and mathematical finance. Explains the use and importance of statistical methods in econometrics and financial engineering. Illustrates the importance of derivatives and calculus to aid understanding in methods and results. Looks at advanced topics such as martingale theory, stochastic processes and stochastic integration. Features examples throughout to illustrate applications in mathematical and statistical finance. Is supported by an accompanying website featuring R code and data sets.

Financial Statistics and Mathematical Finance introduces the financial methodology and the relevant mathematical tools in a style that is both mathematically rigorous and yet accessible to advanced level practitioners and mathematicians alike, both graduate students and researchers in statistics, finance, econometrics and business administration will benefit from this book.

Author Notes

Ansgar Steland, Institute for Statistics and Economics, RWTH Aachen University, Germany.

Preface	p. xi
Acknowledgements	p. xv
1 Elementary financial calculus	p. 1
1.1 Motivating examples	p. 1
1.2 Cashflows, interest rates, prices and returns	p. 2
1.2.1 Bonds and the term structure of interest rates	p. 5
1.2.2 Asset returns	p. 6
1.2.3 Some basic models for asset prices	p. 8
1.3 Elementary statistical analysis of returns	p. 11
1.3.1 Measuring location	p. 13
1.3.2 Measuring dispersion and risk	p. 16
1.3.3 Measuring skewness and kurtosis	p. 20
1.3.4 Estimation of the distribution	p. 21
1.3.5 Testing for normality	p. 27
1.4 Financial instruments	p. 28
1.4.1 Contingent claims	p. 28
1.4.2 Spot contracts and forwards	p. 29
1.4.3 Futures contracts	p. 29
1.4.4 Options	p. 30
1.4.5 Barrier options	p. 31
1.4.6 Financial engineering	p. 32
1.5 A primer on option pricing	p. 32
1.5.1 The no-arbitrage principle	p. 32
1.5.2 Risk-neutral evaluation	p. 33
1.5.3 Hedging and replication	p. 36
1.5.4 Nonexistence of a risk-neutral measure	p. 37
1.5.5 The Black-Scholes pricing formula	p. 37
1.5.6 The Greeks	p. 39
1.5.7 Calibration, implied volatility and the smile	p. 41
1.5.8 Option prices and the risk-neutral density	p. 41
1.6 Notes and further reading	p. 43
References	p. 43
2 Arbitrage theory for the one-period model	p. 45
2.1 Definitions and preliminaries	p. 45
2.2 Linear pricing measures	p. 47
2.3 More on arbitrage	p. 50
2.4 Separation theorems in R n	p. 53
2.5 No-arbitrage and martingale measures	p. 56
2.6 Arbitrage-free pricing of contingent claims	p. 65
2.7 Construction of martingale measures: general case	p. 70
2.8 Complete financial markets	p. 73
2.9 Notes and further reading	p. 76
References	p. 76
3 Financial models in discrete time	p. 79
3.1 Adapted stochastic processes in discrete time	p. 81
3.2 Martingales and martingale differences	p. 85
3.2.1 The martingale transformation	p. 91
3.2.2 Stopping times, optional sampling and a maximal inequality	p. 93
3.2.3 Extensions to R d	p. 101
3.3 Stationarity	p. 102
3.3.1 Weak and strict stationarity	p. 102
3.4 Linear processes and ARMA models	p. 111
3.4.1 Linear processes and the lag operator	p. 111
3.4.2 Inversion	p. 116
3.4.3 ARQ(p) and AR(∞) processes	p. 119
3.4.4 ARMA processes	p. 122
3.5 The frequency domain	p. 124
3.5.1 The spectrum	p. 124
3.5.2 The periodogram	p. 126
3.6 Estimation of ARMA processes	p. 132
3.7 (G)ARCH models	p. 133
3.8 Long-memory series	p. 139
3.8.1 Fractional differences	p. 139
3.8.2 Fractionally integrated processes	p. 144
3.9 Notes and further reading	p. 144
References	p. 145
4 Arbitrage theory for the multiperiod model	p. 147
4.1 Definitions and preliminaries	p. 148
4.2 Self-financing trading strategies	p. 148
4.3 No-arbitrage and martingale measures	p. 152
4.4 European claims on arbitrage-free markets	p. 154
4.5 The martingale representation theorem in discrete time	p. 159
4.6 The Cox-Ross-Rubinstein binomial model	p. 160
4.7 The Black-Scholes formula	p. 165
4.8 American options and contingent claims	p. 171
4.8.1 Arbitrage-free pricing and the optimal exercise strategy	p. 171
4.8.2 Pricing american options using binomial trees	p. 174
4.9 Notes and further reading	p. 175
References	p. 175
5 Brownian motion and related processes in continuous time	p. 177
5.1 Preliminaries	p. 177
5.2 Brownian motion	p. 181
5.2.1 Definition and basic properties	p. 181
5.2.2 Brownian motion and the central limit theorem	p. 188
5.2.3 Path properties	p. 190
5.2.4 Brownian motion in higher dimensions	p. 191
5.3 Continuity and differentiability	p. 192
5.4 Self-similarity and fractional Brownian motion	p. 193
5.5 Counting processes	p. 195
5.5.1 The poisson process	p. 195
5.5.2 The compound poisson process	p. 196
5.6 Lévy processes	p. 199
5.7 Notes and further reading	p. 201
References	p. 201
6 Itô Calculus	p. 203
6.1 Total and quadratic variation	p. 204
6.2 Stochastic Stieltjes integration	p. 208
6.3 The Itô integral	p. 212
6.4 Quadratic covariation	p. 225
6.5 Itô's formula	p. 226
6.6 Itô processes	p. 229
6.7 Diffusion processes and ergodicity	p. 236
6.8 Numerical approximations and statistical estimation	p. 238
6.9 Notes and further reading	p. 239
References	p. 240
7 The Black-Scholes model	p. 241
7.1 The model and first properties	p. 241
7.2 Girsanov's theorem	p. 247
7.3 Equivalent martingale measure	p. 251
7.4 Arbitrage-free pricing and hedging claims	p. 252
7.5 The delta hedge	p. 256
7.6 Time-dependent volatility	p. 257
7.7 The generalized Black-Scholes model	p. 259
7.8 Notes and further reading	p. 261
References	p. 262
8 Limit theory for discrete-time processes	p. 263
8.1 Limit theorems for correlated time series	p. 264
8.2 A regression model for financial time series	p. 273
8.2.1 Least squares estimation	p. 276
8.3 Limit theorems for martingale difference	p. 278
8.4 Asymptotics	p. 283
8.5 Density estimation and nonparametric regression	p. 287
8.5.1 Multivariate density estimation	p. 288
8.5.2 Nonparametric regression	p. 295
8.6 The CLT for linear processes	p. 302
8.7 Mixing processes	p. 306
8.7.1 Mixing coefficients	p. 306
8.7.2 Inequalities	p. 308
8.8 Limit theorems for mixing processes	p. 313
8.9 Notes and further reading	p. 323
References	p. 323
9 Special topics	p. 325
9.1 Copulas - and the 2008 financial crisis	p. 325
9.1.1 Copulas	p. 326
9.1.2 The financial crisis	p. 332
9.1.3 Models for credit defaults and CDOs	p. 335
9.2 Local Linear nonparametric regression	p. 338
9.2.1 Applications in finance: estimation of martingale measures and Ito diffusions	p. 339
9.2.2 Method and asymptotics	p. 340
9.3 Change-point detection and monitoring	p. 350
9.3.1 Offline detection	p. 351
9.3.2 Online detection	p. 359
9.4 Unit roots and random walk	p. 363
9.4.1 The OLS estimator in the stationary AR(1) model	p. 364
9.4.2 Nonparametric definitions for the degree of integration	p. 368
9.4.3 The Dickey-Fuller test	p. 370
9.4.4 Detecting unit roots and stationarity	p. 373
9.5 Notes and further reading	p. 381
References	p. 382
Appendix A	p. 385
A.1 (Stochastic) Landau symbols	p. 385
A.2 Bochner's lemma	p. 387
A.3 Conditional expectation	p. 387
A.4 Inequalities	p. 388
A.5 Random series	p. 389
A.6 Local martingales in discrete time	p. 389
Appendix B Weak convergence and central limit theorems	p. 391
B.1 Convergence in distribution	p. 391
B.2 Weak convergence	p. 392
B.3 Prohorov's theorem	p. 398
B.4 Sufficient criteria	p. 399
B.5 More on Skorohod spaces	p. 401
B.6 Central limit theorems for martingale differences	p. 402
B.7 Functional central limit theorems	p. 403
B.8 Strong approximations	p. 405
References	p. 407
Index	p. 409

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