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TitleApplied Quantitative Finance
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Table of Contents
                            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
		Skewness Trades
			Performance
		Kurtosis Trades
			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
	Long Memory Effects Trading Strategy
		Introduction
		Hurst and Rescaled Range Analysis
		Stationary Long Memory Processes
			Fractional Brownian Motion and Noise
		Data Analysis
		Trading the Negative Persistence
	Locally time homogeneous time series modeling
		Intervals of homogeneity
			The adaptive estimator
			A small simulation study
		Estimating the coefficients of an exchange rate basket
			The Thai Baht basket
			Estimation results
		Estimating the volatility of financial time series
			The standard approach
			The locally time homogeneous approach
			Modeling volatility via power transformation
			Adaptive estimation under local time-homogeneity
		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
		Why Spreadsheets?
		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
spreadsheet, 385
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

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