## Statistics of Financial Markets: An IntroductionStatistics of Financial Markets offers a vivid yet concise introduction to the growing field of statistical applications in finance. The reader will learn the basic methods to evaluate option contracts, to analyse financial time series, to select portfolios and manage risks making realistic assumptions of the market behaviour. The focus is both on fundamentals of mathematical finance and financial time series analysis and on applications to given problems of financial markets, making the book the ideal basis for lectures, seminars and crash courses on the topic. For the second edition the book has been updated and extensively revised. Several new aspects have been included, among others a chapter on credit risk management. From the reviews of the first edition: "The book starts ... with five eye-catching pages that reproduce a student’s handwritten notes for the examination that is based on this book. ... The material is well presented with a good balance between theoretical and applied aspects. ... The book is an excellent demonstration of the power of stochastics ... . The author’s goal is well achieved: this book can satisfy the needs of different groups of readers ... . " (Jordan Stoyanov, Journal of the Royal Statistical Society, Vol. 168 (4), 2005) |

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### Contents

1 Derivatives | 5 |

11 Recommended Literature | 12 |

2 Introduction to Option Management | 13 |

22 Portfolio Insurance | 25 |

23 Recommended Literature | 33 |

3 Basic Concepts of Probability Theory | 35 |

32 Expectation and Variance | 38 |

33 Skewness and Kurtosis | 39 |

1152 Estimation of the Covariance Function | 190 |

1153 Estimation of the ACF | 191 |

116 Portmanteau Statistics | 192 |

117 Estimation of ARp Models | 193 |

118 Estimation of MAq and ARMApq Models | 194 |

119 Recommended Literature | 199 |

12 Time Series with Stochastic Volatility | 201 |

121 ARCH and GARCH Models | 203 |

34 Random Vectors Dependence Correlation | 40 |

35 Conditional Probabilities and Expectations | 41 |

36 Recommended Literature | 44 |

4 Stochastic Processes in Discrete Time | 45 |

42 Trinomial Processes | 49 |

43 General Random Walks | 51 |

44 Geometric Random Walks | 52 |

45 Binomial Models with State Dependent Increments | 53 |

5 Stochastic Integrals and Differential Equations | 55 |

52 Stochastic Integration | 58 |

53 Stochastic Differential Equations | 61 |

54 The Stock Price as a Stochastic Process | 63 |

55 Itos Lemma | 64 |

56 Recommended Literature | 67 |

6 BlackScholes Option Pricing Model | 69 |

62 BlackScholes Formulae for European Options | 76 |

63 Risk Management and Hedging | 82 |

631 Delta Hedging | 84 |

632 Gamma and Theta | 87 |

633 Rho and Vega | 90 |

634 Historical and Implied Volatility | 91 |

64 Recommended Literature | 95 |

7 Binomial Model for European Options | 97 |

71 CoxRossRubinstein Approach to Option Pricing | 98 |

72 Discrete Dividends | 103 |

722 Dividends as a Fixed Money Amount | 104 |

73 Recommended Literature | 107 |

8 American Options | 109 |

82 The Trinomial Model for American Options | 117 |

83 Recommended Literature | 121 |

9 Exotic Options and Interest Rate Derivatives | 123 |

912 Chooser Options or As you wish Options | 124 |

913 Barrier Options | 125 |

914 Asian Options | 126 |

915 Lookback Options | 128 |

92 Models for the Interest Rate and Interest Rate Derivatives | 129 |

921 Bond Value with Known Time Dependent Interest Rate | 130 |

923 The Bonds Value Equation | 131 |

924 Solving the Zero Bonds Value Equation | 133 |

Statistical Model of Financial Time Series | 137 |

Definitions and Concepts | 139 |

101 Certain Definitions | 140 |

102 Statistical Analysis of German Stock Returns | 147 |

103 Expectations and Efficient Markets | 149 |

A Brief Summary | 155 |

Theory of the Interest Rate Parity | 156 |

The CoxlngersollRoss Model | 158 |

The BlackScholes Model | 160 |

1045 The Market Price of Risk | 162 |

105 The Random Walk Hypothesis | 165 |

106 Unit Root Tests | 168 |

1062 The KPSS Test of Stationarity | 170 |

1063 Variance Ratio Tests | 172 |

11 ARIMA Time Series Models | 177 |

111 Moving Average Processes | 178 |

112 Autoregressive Process | 179 |

113 ARMA Models | 183 |

114 Partial Autocorrelation | 185 |

115 Estimation Moments | 188 |

1151 Estimation of the Mean Function | 189 |

Definition and Properties | 206 |

1212 Estimation of ARCH1 Models | 213 |

Definition and Properties | 217 |

1214 Estimation of an ARCHq Model | 219 |

1215 Generalized ARCH GARCH | 220 |

1216 Estimation of GARCHpq Models | 223 |

122 Extensions of the GARCH Model | 226 |

1221 Exponential GARCH | 227 |

1222 Threshold ARCH Models | 229 |

1223 Risk and Returns | 230 |

1224 Estimation Results for the DAX Returns | 231 |

123 Multivariate GARCH models | 232 |

1231 The Vec Specification | 233 |

1232 Die BEKK Spezifikation | 236 |

1233 The CCC model | 237 |

13 Nonparametric Concepts for Financial Time Series | 245 |

131 Nonparametric Regression | 246 |

132 Construction of the Estimator | 249 |

133 Asymptotic Normality | 252 |

134 Recommended Literature | 267 |

Selected Financial Applications | 269 |

14 Valuing Options with Flexible Volatility Estimators | 271 |

141 Valuing Options with ARCHModels | 272 |

142 A Monte Carlo Study | 278 |

143 Application to the Valuation of DAX Calls | 283 |

15 Value at Risk and Backtesting | 289 |

151 Forecast and VaR Models | 290 |

152 Backtesting with Expected Shortfall | 293 |

153 Backtesting in Action | 295 |

16 Copulas and ValueatRisk | 303 |

161 Copulas | 304 |

162 The Calculation of VaR and Copulas | 307 |

163 Recommended Literature | 311 |

17 Statistics of Extreme Risks | 313 |

172 Statistics of Extreme Events | 323 |

1721 The POT peaksoverthreshold Method | 324 |

1722 The Hill Estimator | 330 |

173 Estimators for Risk Measurements | 332 |

174 Extreme Value Theory for Time Series | 335 |

18 Neural Networks | 341 |

181 From Perceptron to Nonlinear Neuron | 343 |

182 Back Propagation | 351 |

183 Neural Networks in Nonparametric Regression Analysis | 354 |

184 Forecasts of Financial Time Series with Neural Networks | 360 |

185 Quantifying Risk with Neural Networks | 364 |

186 Recommended Literature | 369 |

19 Volatility Risk of Option Portfolios | 371 |

191 Description of the Data | 372 |

192 Principal Component Analysis of the VDAXs Dynamics | 376 |

193 Stability Analysis of the VDAXs Dynamics | 379 |

194 Measure of the Implied Volatilitys Risk | 380 |

195 Recommended Literature | 383 |

20 Nonparametric Estimators for the Probability of Default | 385 |

202 Semiparametric Model for Credit Rating | 387 |

203 Credit Ratings with Neural Networks | 391 |

A Technical Appendix | 395 |

A2 Portfolio Strategies | 400 |

407 | |

423 | |

### Common terms and phrases

American options applied approximation arbitrage ARCH models assume assumption asymptotic autocorrelation binomial model Black-Scholes formula calculated call option conditional variance consider converges copula corresponding covariance defined Definition delivery price delta density dependent derivative differential equation distribution function dividend empirical estimator European options example exchange rate expectation Figure financial time series forecast forward contract future contracts GARCH geometric Brownian motion given hedging hidden layer holds implied volatility increments independent interest rate investment investor Lemma likelihood function linear martingale matrix method neural network neurons nonparametric normal distribution observations obtain option price parameter perceptron position price St principal components probability process Xt put option quantile random variables random walk Recommended Literature regression respect returns risk sample Section stationary statistics stochastic process stock price strategy Table Theorem threshold trinomial underlying vector white noise Wiener process zero bond

### Popular passages

Page xvii - A T transpose of matrix A X ~ D the random variable X has distribution D E[X] expected value of random variable X Var(X) variance of random variable X Cov(X, Y) covariance

### References to this book

Multivariate Statistics:: Exercises and Solutions Wolfgang Karl Härdle,Zdenek Hlávka Limited preview - 2007 |