# Modelling Extreme Events in Finance

### Description

The Rmetrics "fExtemes" package is a collection of functions to analyze and model extreme events in Finance and Insurance.

### Details

1 2 3 4 5 6 7 8 |

### 1 Introduction

The `fExtremes`

package provides functions for analyzing
and modeling extreme events in financial time Series. The
topics include: (i) data proeprocessing, (ii) explorative
data analysis, (iii) peak over threshold modeling, (iv) block
maxima modeling, (v) estimation of VaR and CVaR, and (vi) the
computation of the extreme index.

### 2 Data and their Preprocessing

*Data Sets:*

Data sets used in the examples of the timeSeries packages.

*Data Preprocessing:*

These are tools for data preprocessing, including functions to separate data beyond a threshold value, to compute blockwise data like block maxima, and to decluster point process data.

1 2 3 4 5 | ```
blockMaxima extracts block maxima from a vector or a time series
findThreshold finds upper threshold for a given number of extremes
pointProcess extracts peaks over Threshold from a vector or a time series
deCluster de-clusters clustered point process data
``` |

### 2 Explorative Data Analysis of Extremes

This section contains a collection of functions for explorative data analysis of extreme values in financial time series. The tools include plot functions for emprical distributions, quantile plots, graphs exploring the properties of exceedences over a threshold, plots for mean/sum ratio and for the development of records. The functions are:

1 2 3 4 5 6 7 |

1 2 3 4 5 | ```
recordsPlot Record development compared with iid data
ssrecordsPlot another variant, investigates subsamples
sllnPlot verifies Kolmogorov's strong law of large numbers
lilPlot verifies Hartman-Wintner's law of the iterated logarithm
``` |

1 2 |

*Parameter Fitting of Mean Excesses:*

1 2 3 4 5 6 | ```
normMeanExcessFit fits mean excesses with a normal density
ghMeanExcessFit fits mean excesses with a GH density
hypMeanExcessFit fits mean excesses with a HYP density
nigMeanExcessFit fits mean excesses with a NIG density
ghtMeanExcessFit fits mean excesses with a GHT density
``` |

### 3 GPD Peak over Threshold Modeling

*GPD Distribution:*

A collection of functions to compute the generalized Pareto distribution. The functions compute density, distribution function, quantile function and generate random deviates for the GPD. In addition functions to compute the true moments and to display the distribution and random variates changing parameters interactively are available.

1 2 3 4 5 6 | ```
dgpd returns the density of the GPD distribution
pgpd returns the probability function of the GPD
qgpd returns quantile function of the GPD distribution
rgpd generates random variates from the GPD distribution
gpdSlider displays density or rvs from a GPD
``` |

*GPD Moments:*

1 2 | ```
gpdMoments computes true mean and variance of GDP
``` |

*GPD Parameter Estimation:*

This section contains functions to fit and to simulate processes that are generated from the generalized Pareto distribution. Two approaches for parameter estimation are provided: Maximum likelihood estimation and the probability weighted moment method.

1 2 3 | ```
gpdSim generates data from the GPD distribution
gpdFit fits data to the GPD istribution
``` |

*GPD print, plot and summary methods:*

1 2 3 4 |

*GDP Tail Risk:*

The following functions compute tail risk under the GPD approach.

1 2 3 4 5 6 7 | ```
gpdQPlot estimation of high quantiles
gpdQuantPlot variation of high quantiles with threshold
gpdRiskMeasures prescribed quantiles and expected shortfalls
gpdSfallPlot expected shortfall with confidence intervals
gpdShapePlot variation of GPD shape with threshold
gpdTailPlot plot of the GPD tail
``` |

### 4 GEV Block Maxima Modeling

*GEV Distribution:*

This section contains functions to fit and to simulate processes that are generated from the generalized extreme value distribution including the Frechet, Gumbel, and Weibull distributions.

1 2 3 4 5 6 | ```
dgev returns density of the GEV distribution
pgev returns probability function of the GEV
qgev returns quantile function of the GEV distribution
rgev generates random variates from the GEV distribution
gevSlider displays density or rvs from a GEV
``` |

*GEV Moments:*

1 2 | ```
gevMoments computes true mean and variance
``` |

*GEV Parameter Estimation:*

A collection to simulate and to estimate the parameters of processes generated from GEV distribution.

1 2 3 4 5 | ```
gevSim generates data from the GEV distribution
gumbelSim generates data from the Gumbel distribution
gevFit fits data to the GEV distribution
gumbelFit fits data to the Gumbel distribution
``` |

1 2 3 4 |

*GEV MDA Estimation:*

Here we provide Maximum Domain of Attraction estimators and visualize the results by a Hill plot and a common shape parameter plot from the Pickands, Einmal-Decker-deHaan, and Hill estimators.

1 2 3 |

*GEV Risk Estimation:*

1 2 | ```
gevrlevelPlot k-block return level with confidence intervals
``` |

### 4 Value at Risk

Two functions to compute Value-at-Risk and conditional Value-at-Risk.

1 2 3 |

### 5 Extreme Index

A collection of functions to simulate time series with a known extremal index, and to estimate the extremal index by four different kind of methods, the blocks method, the reciprocal mean cluster size method, the runs method, and the method of Ferro and Segers.

1 2 3 4 5 6 | ```
thetaSim simulates a time Series with known theta
blockTheta computes theta from Block Method
clusterTheta computes theta from Reciprocal Cluster Method
runTheta computes theta from Run Method
ferrosegersTheta computes theta according to Ferro and Seegers
``` |

1 2 3 | ```
exindexPlot calculatess and plots Theta(1,2,3)
exindexesPlot calculates Theta(1,2) and plots Theta(1)
``` |

### About Rmetrics

The `fExtremes`

Rmetrics package is written for educational
support in teaching "Computational Finance and Financial Engineering"
and licensed under the GPL.