Title Applied Quantitative Finance 3.9 MB 423
```                            Preface
Contributors
Frequently Used Notation
Value at Risk
Approximating Value at Risk in Conditional Gaussian Models
Introduction
The Practical Need
Statistical Modeling for VaR
VaR Approximations
Pros and Cons of Delta-Gamma Approximations
General Properties of Delta-Gamma-Normal Models
Cornish-Fisher Approximations
Derivation
Properties
Fourier Inversion
Error Analysis
Tail Behavior
Inversion of the cdf minus the Gaussian Approximation
Variance Reduction Techniques in Monte-Carlo Simulation
Monte-Carlo Sampling Method
Partial Monte-Carlo with Importance Sampling
XploRe Examples
Applications of Copulas for the Calculation of Value-at-Risk
Copulas
Definition
Sklar's Theorem
Examples of Copulas
Further Important Properties of Copulas
Computing Value-at-Risk with Copulas
Selecting the Marginal Distributions
Selecting a Copula
Estimating the Copula Parameters
Generating Scenarios - Monte Carlo Value-at-Risk
Examples
Results
Quantification of Spread Risk by Means of Historical Simulation
Introduction
Risk Categories -- a Definition of Terms
Descriptive Statistics of Yield Spread Time Series
Data Analysis with XploRe
Discussion of Results
Historical Simulation and Value at Risk
Risk Factor: Full Yield
Risk Factor: Benchmark
Risk Factor: Spread over Benchmark Yield
Conservative Approach
Simultaneous Simulation
Mark-to-Model Backtesting
VaR Estimation and Backtesting with XploRe
P-P Plots
Q-Q Plots
Discussion of Simulation Results
Risk Factor: Full Yield
Risk Factor: Benchmark
Risk Factor: Spread over Benchmark Yield
Conservative Approach
Simultaneous Simulation
XploRe for Internal Risk Models
Credit Risk
Rating Migrations
Rating Transition Probabilities
From Credit Events to Migration Counts
Estimating Rating Transition Probabilities
Dependent Migrations
Computation and Quantlets
Analyzing the Time-Stability of Transition Probabilities
Aggregation over Periods
Are the Transition Probabilities Stationary?
Computation and Quantlets
Examples with Graphical Presentation
Multi-Period Transitions
Time Homogeneous Markov Chain
Bootstrapping Markov Chains
Computation and Quantlets
Rating Transitions of German Bank Borrowers
Portfolio Migration
Sensitivity analysis of credit portfolio models
Introduction
Construction of portfolio credit risk models
Dependence modelling
Factor modelling
Copula modelling
Simulations
Random sample generation
Portfolio results
Implied Volatility
The Analysis of Implied Volatilities
Introduction
The Implied Volatility Surface
Calculating the Implied Volatility
Surface smoothing
Dynamic Analysis
Data description
PCA of ATM Implied Volatilities
Common PCA of the Implied Volatility Surface
How Precise Are Price Distributions Predicted by IBT?
Implied Binomial Trees
The Derman and Kani (D & K) algorithm
Compensation
Barle and Cakici (B & C) algorithm
A Simulation and a Comparison of the SPDs
Simulation using Derman and Kani algorithm
Simulation using Barle and Cakici algorithm
Comparison with Monte-Carlo Simulation
Example -- Analysis of DAX data
Estimating State-Price Densities with Nonparametric Regression
Introduction
Extracting the SPD using Call-Options
Black-Scholes SPD
Semiparametric estimation of the SPD
Estimating the call pricing function
Further dimension reduction
Local Polynomial Estimation
An Example: Application to DAX data
Data
SPD, delta and gamma
Bootstrap confidence bands
Comparison to Implied Binomial Trees
Trading on Deviations of Implied and Historical Densities
Introduction
Estimation of the Option Implied SPD
Application to DAX Data
Estimation of the Historical SPD
The Estimation Method
Application to DAX Data
Comparison of Implied and Historical SPD
Performance
Performance
A Word of Caution
Econometrics
Multivariate Volatility Models
Introduction
Model specifications
Estimation of the BEKK-model
An empirical illustration
Data description
Estimating bivariate GARCH
Estimating the (co)variance processes
Forecasting exchange rate densities
Statistical Process Control
Control Charts
Chart characteristics
Average Run Length and Critical Values
Average Delay
Probability Mass and Cumulative Distribution Function
Comparison with existing methods
Two-sided EWMA and Lucas/Saccucci
Two-sided CUSUM and Crosier
Real data example -- monitoring CAPM
An Empirical Likelihood Goodness-of-Fit Test for Diffusions
Introduction
Discrete Time Approximation of a Diffusion
Hypothesis Testing
Kernel Estimator
The Empirical Likelihood concept
Introduction into Empirical Likelihood
Empirical Likelihood for Time Series Data
Goodness-of-Fit Statistic
Goodness-of-Fit test
Application
Simulation Study and Illustration
Appendix
A simple state space model of house prices
Introduction
A Statistical Model of House Prices
The Price Function
State Space Form
Estimation with Kalman Filter Techniques
Kalman Filtering given all parameters
Filtering and state smoothing
Maximum likelihood estimation of the parameters
Diagnostic checking
The Data
Estimating and filtering in XploRe
Overview
Setting the system matrices
Kalman filter and maximized log likelihood
Diagnostic checking with standardized residuals
Calculating the Kalman smoother
Appendix
Procedure equivalence
Smoothed constant state variables
Introduction
Hurst and Rescaled Range Analysis
Stationary Long Memory Processes
Fractional Brownian Motion and Noise
Data Analysis
Locally time homogeneous time series modeling
Intervals of homogeneity
A small simulation study
Estimating the coefficients of an exchange rate basket
Estimation results
Estimating the volatility of financial time series
The standard approach
The locally time homogeneous approach
Modeling volatility via power transformation
Technical appendix
Simulation based Option Pricing
Simulation techniques for option pricing
Introduction to simulation techniques
Pricing path independent European options on one underlying
Pricing path dependent European options on one underlying
Pricing options on multiple underlyings
Quasi Monte Carlo (QMC) techniques for option pricing
Introduction to Quasi Monte Carlo techniques
Error bounds
Construction of the Halton sequence
Experimental results
Pricing options with simulation techniques - a guideline
Construction of the payoff function
Integration of the payoff function in the simulation framework
Restrictions for the payoff functions
Nonparametric Estimators of GARCH Processes
Deconvolution density and regression estimates
Nonparametric ARMA Estimates
Nonparametric GARCH Estimates
Net Based Spreadsheets in Quantitative Finance
Introduction
Client/Server based Statistical Computing
Using MD*ReX
Applications
Value at Risk Calculations with Copulas
Implied Volatility Measures
Index
```
##### Document Text Contents
Page 1

Applied Quantitative Finance

Wolfgang Härdle
Torsten Kleinow

Gerhard Stahl

In cooperation with

Gökhan Aydınlı, Oliver Jim Blaskowitz, Song Xi Chen,
Matthias Fengler, Jürgen Franke, Christoph Frisch,
Helmut Herwartz, Harriet Holzberger, Steffi Höse,

Stefan Huschens, Kim Huynh, Stefan R. Jaschke, Yuze Jiang
Pierre Kervella, Rüdiger Kiesel, Germar Knöchlein,

Sven Knoth, Jens Lüssem, Danilo Mercurio,
Marlene Müller, Jörn Rank, Peter Schmidt,

Rainer Schulz, Jürgen Schumacher, Thomas Siegl,
Robert Wania, Axel Werwatz, Jun Zheng

June 20, 2002

Page 211

8.4 An Example: Application to DAX data 189

This result can be obtained using some theorems related to local polynomial
estimation, for example in Fan and Gijbels (1996), if some boundary conditions
are satisfied.

An asymptotic approximation of f̂∗n is complicated by the fact that f̂

n is a

non linear function of V , V ′ and V ′′. Analytical confidence intervals can be
obtained using delta methods proposed by Aı̈t-Sahalia (1996). However, an
alternative method is to use the bootstrap to construct confidence bands. The
idea for estimating the bootstrap bands is to approximate the distribution of

sup
k
|f̂∗(k)− f∗(k)|.

The following procedure illustrates how to construct bootstrap confidence
bands for local polynomial SPD estimation.

1. Collect daily option prices from MD*BASE, only choose those options
with the same expiration date, for example, those with time to maturity
49 days on Jan 3, 1997.

2. Use the local polynomial estimation method to obtain the empirical SPD.
Notice that when τ is fixed the forward price F is also fixed. So that the
implied volatility function σ(K/F ) can be considered as a fixed design
situation, where K is the strike price.

3. Obtain the confidence band using the wild bootstrap method. The wild
bootstrap method entails:

• Suppose that the regression model for the implied volatility function
σ(K/F ) is:

Yi = σ

Ki
F

+ εi, i = 1, · · · , n.

• Choose a bandwidth g which is larger than the optimal h in or-
der to have oversmoothing. Estimate the implied volatility function
σ(K/F ) nonparametrically and then calculate the residual errors:

ε̃i = Yi − σ̂h

Ki
F

.

• Replicate B times the series of the {ε̃i} with wild bootstrap ob-
taining {ε∗,ji } for j = 1, · · · , B, Härdle (1990), and build B new

Page 212

190 8 Estimating State-Price Densities with Nonparametric Regression

bootstrapped samples:

Y
∗,j
i = σ̂g

(
Ki
F

)
+ ε∗,ji .

• Estimate the SPD f∗,j using bootstrap samples, Rookley’s method
and the bandwidth h, and build the statistics

T ∗f = sup
z
|f∗,j(z)− f̂∗(z)|.

• Form the (1− α) bands [f̂∗(z)− tf∗,1−α, f̂∗(z) + tf∗,1−α],
where tf∗,1−α denotes the empirical (1− α)-quantile of T ∗f .

Two SPDs (Jan 3 and Jan 31, 1997) whose times to maturity are 49 days
were estimated and are plotted in Figure (8.5). The bootstrap confidence
band corresponding to the first SPD (Jan 3) is also visible on the chart. In
Figure (8.6), the SPDs are displayed on a moneyness metric. It seems that the
differences between the SPDs can be eliminated by switching to the moneyness
metric. Indeed, as can be extracted from Figure 8.6, both SPDs lie within
the 95 percent confidence bands. The number of bootstrap samples is set to
B = 100. The local polynomial estimation was done on standardized data, h
is then set to 0.75 for both plots and g is equal to 1.1 times h. Notice that
greater values of g are tried and the conclusion is that the confidence bands
are stable to an increase of g.

8.4.4 Comparison to Implied Binomial Trees

In Chapter 7, the Implied Binomial Trees (IBT) are discussed. This method is a
close approach to estimate the SPD. It also recovers the SPD nonparametrically
from market option prices and uses the Black Scholes formula to establish the
relationship between the option prices and implied volatilities as in Rookely’s
method. In Chapter 7, the Black Scholes formula is only used for Barle and
Cakici IBT procedure, but the CRR binomial tree method used by Derman
and Kani (1994) has no large difference with it in nature. However, IBT and
nonparametric regression methods have some differences caused by different
modelling strategies.

The IBT method might be less data-intensive than the nonparametric regres-
sion method. By construction, it only requires one cross section of prices. In the

Page 422

400 Index

VaR, 41, 42
finance, 185, 362
nummath, 298
spc, 239, 247

Markov chain, 87, 101
bootstrapping, 102

mcopt, 349
migration

correlation, 90, 92
counts, 89
events, 88
probability,
→ transition probability

rates, 90
MMPL data, 71
model

Implied Binomial Trees,
→ IBT

multivariate volatility,
→ BiGARCH

State-Price Densities,
→ XFGSPD

VaR,
→ XFGVAR

Monte Carlo option pricing,
→ option pricing

multi-period transitions, 101
Multivariate Volatility Models,

→ BiGARCH

nonparametric estimates of
GARCH processes,
→ NPGARCH

NPGARCH, 367
nummath library, 298

option pricing
Monte Carlo option pricing,
→ mcopt

PL data, 70
portfolio

composition, 89
migration, 106
weights, 107

Quasi Monte Carlo simulation, 356

randomized algorithm, 349
rating, 87

migrations, 87
dependence, 90
independence, 90

transition probability,
→ transition probability

risk horizon, 88
risk neutral model, 351
RiskMetrics, 367

sigmaprocess data, 229
SPC, 237
spc library, 239, 247
star-discrepancy, 356
State-Price Densities,

→ XFGSPD
statistical process control,

→ SPC

TGARCH, 368
threshold normal model, 91
time homogeneity, 323
transition matrix, 87
transition probability, 87, 89

chi-square test, 95
estimator, 90
simultaneous, 92
standard deviation, 90, 92
test of homogeneity, 95
time-stability, 94

Page 423

Index 401

USTF data, 55

Value at Risk,
→ XFGVAR

Value-at-Risk, 35
value-at-risk, 367
VaR, 367
VaR library, 41, 42
volatility, 323
volsurf01 data, 141
volsurf02 data, 141
volsurf03 data, 141
volsurfdata2 data, 130, 393

XFGData9701 data, 184
XFGhouseprice data, 290
XFGhousequality data, 290, 294
XFGSPD, 171
XFGVAR, 3