The Valuation of Multivariate Options

Hassold, Christian

The Valuation of Multivariate Options

Zusammenfassung

Inhaltsangabe:Abstract:
During the last decades, capital markets have transformed rapidly. Derivative securities or more simply derivatives like swaps, futures, and options supplemented the trading of stocks and bonds. These financial products can frequently be seen in the media: Due to derivatives, Procter & Gamble lost $150 million in 1994, Barings Bank lost $1.3 billion in 1995 and Long-Term Capital Management (LTCM) lost $3.5 billion in 1998.
Though these figures seem daunting, derivatives can be useful financial instruments. Applications include risk management, speculation, reduced transaction costs, and regulatory arbitrage.
Theory and practice of option valuation were revolutionized in 1973, when Fischer Black and Myron Scholes published their celebrated Black Scholes formula in the landmark paper The pricing of options and corporate liabilities. Afterwards, a vast amount of papers on option valuation was published which employ all kinds of stochastic processes. Thereby, the special features of financial return data are tried to be taken into account.
Advancing option valuation theory to options with multiple underlyings, lead to the problem that the dependence structure of the underlying securities needs to be considered. Though linear correlation is a widely used dependence measure, it may be inappropriate for multivariate return data. Throughout the last years, dependence modelling through copulas has become common.
Copulas are multivariate distributions on the d-dimensional unit-hyper-square which couples d marginal distributions to a joint distribution. Copulas can be used to construct dependence measures like the rank correlation coefficients of Spearman or Kendall. They are also a useful tool in the context of option valuation.
The prices of multivariate options depend on the distributional assumptions of stock price changes and the dependence structure. This thesis exhibits the features of multivariate financial return data. Evidence of (multi-)non-normality is presented. A general overview on multivariate option valuation theory is given. A nonparametric model and two Esscher models are introduced in detail. Then, the multivariate normal and the multivariate normal inverse Gaussian distribution are assumed as return distributions for an empirical study. The study exhibits the influence of the dependence structure and the properties of the assumed return distribution on option prices.

Inhaltsverzeichnis:Table of […]

Leseprobe

Inhaltsverzeichnis

Christian Hassold

The Valuation of Multivariate Options

ISBN: 978-3-8324-8132-2

Druck Diplomica® GmbH, Hamburg, 2007

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Abstract

The prices of multivariate options depend on the distributional assumptions of stock

price changes and the dependence structure. This thesis exhibits the features of mul-

tivariate financial return data. Evidence of (multi-)non-normality is presented. A

general overview on multivariate option valuation theory is given. The nonparamet-

ric model of Rosenberg [65] and the Esscher models of Gerber and Shiu [36] and

Prause [58] are introduced in detail. Then, the multivariate normal and the multi-

variate normal inverse Gaussian distribution are assumed as return distributions for

an empirical study. The study exhibits the influence of the dependence structure

and the properties of the assumed return distribution on option prices.

Contents

List of Figures

List of Tables

VII

Frequently Used Notation

VIII

1 Introduction

2 Derivatives and Options in Particular

2.1 Standard Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2 Exotic Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.3 Multivariate Options . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 Characteristics of Financial Returns

3.1 Stylized Facts of Univariate Return Distributions . . . . . . . . . . .

3.2 Digression: Dependence & Copulas . . . . . . . . . . . . . . . . . . . 12

3.2.1

Shortcomings of Linear Correlation . . . . . . . . . . . . . . . 14

3.2.2

Copulas and Rank Based Dependence Concepts . . . . . . . . 16

3.2.3

Further Remarks on Copulas . . . . . . . . . . . . . . . . . . . 20

CONTENTS

III

3.3 Stylized Facts of Multivariate Return Distributions . . . . . . . . . . 24

4 Option Valuation Approaches

4.1 Binomial Option Pricing . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.2 The Black-Scholes Formula . . . . . . . . . . . . . . . . . . . . . . . . 34

4.3 Risk-Neutral Valuation . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5 Multivariate Option Pricing Models

5.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.1.1

Margrabe: The Value of an Option to Exchange one Asset for

Another . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.1.2

Johnson: Options on the Maximum or the Minimum of Several

Assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.1.3

Boyle: A Lattice Framework for Option Pricing With two

State Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.1.4

Carmona and Durrleman: Generalizing Black-Scholes to Mul-

tivariate Contingent Claims . . . . . . . . . . . . . . . . . . . 44

5.2 Nonparametric Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.2.1

Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.2.2

Estimation of the Risk-Neutral Margins . . . . . . . . . . . . . 48

5.2.3

Estimation of the Risk-Neutral Copula . . . . . . . . . . . . . 51

5.3 Esscher Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.3.1

The Esscher Transform . . . . . . . . . . . . . . . . . . . . . . 52

5.3.2

Implementation of Certain L´evy Processes . . . . . . . . . . . 55

5.3.2.1

Remarks on L´evy Processes . . . . . . . . . . . . . . 55

CONTENTS

5.3.2.2

Normally Distributed Returns . . . . . . . . . . . . . 56

5.3.2.3

NIG-Distributed Returns . . . . . . . . . . . . . . . 59

5.3.3

Empirical Application . . . . . . . . . . . . . . . . . . . . . . 61

6 Conclusion & Outlook

A Remarks on Options

A.1 Determinants of a Call Option . . . . . . . . . . . . . . . . . . . . . . 69

A.2 Boundary Conditions of Options Prices . . . . . . . . . . . . . . . . . 70

A.3 Proof of Put-Call-Parity . . . . . . . . . . . . . . . . . . . . . . . . . 73

B Data Description

C Derivation of the Black Scholes Formula

C.1 Differential Equation Approach . . . . . . . . . . . . . . . . . . . . . 77

C.2 Esscher Transform Approach . . . . . . . . . . . . . . . . . . . . . . . 78

D Derivation of the Nadaraya-Watson Regression Estimator

E Selected MATLAB Functions

E.1 MT

-Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

E.2 Multivariate Kernel Contourplot . . . . . . . . . . . . . . . . . . . . . 87

E.3 Manzotti's Ellipticity Test . . . . . . . . . . . . . . . . . . . . . . . . 89

E.4 Determination of the NIG Esscher Vector . . . . . . . . . . . . . . . . 91

Bibliography

List of Figures

3.1 DBK: Kernel vs. fitted normal density . . . . . . . . . . . . . . . . .

3.2 NQ-plots of DBK and SIE . . . . . . . . . . . . . . . . . . . . . . . . 10

3.3 T3-plots of DBK and SIE . . . . . . . . . . . . . . . . . . . . . . . . 11

3.4 DBK: Kernel vs. fitted normal and NIG densities . . . . . . . . . . . 13

3.5 3D- and contourplots of basic copulas . . . . . . . . . . . . . . . . . . 18

3.6 MT

-plots of DBK and SIE . . . . . . . . . . . . . . . . . . . . . . . 26

3.7 Kernel contourplot and scatterplot: DBK vs. SIE . . . . . . . . . . . 28

4.1 Stock prices after one period . . . . . . . . . . . . . . . . . . . . . . . 31

4.2 Call prices after one period . . . . . . . . . . . . . . . . . . . . . . . . 31

4.3 Stock prices after two periods . . . . . . . . . . . . . . . . . . . . . . 33

4.4 Call prices after two periods . . . . . . . . . . . . . . . . . . . . . . . 33

5.1 Difference of fitted NIG and normal density functions . . . . . . . . . 59

5.2 Difference of fitted NIG and kernel density . . . . . . . . . . . . . . . 65

A.1 Allowable range for call prices . . . . . . . . . . . . . . . . . . . . . . 72

B.1 Stock price and return plots of DBK and SIE . . . . . . . . . . . . . 76

List of Tables

3.1 P-values of

-, KS-, and J B-tests . . . . . . . . . . . . . . . . . . . 12

3.2 Properties of selected Archimedean copulas . . . . . . . . . . . . . . . 23

5.1 Portfolio comparison for different points in time . . . . . . . . . . . . 41

5.2 Five-point jump process summary . . . . . . . . . . . . . . . . . . . . 43

5.3 EURIBOR for different maturities . . . . . . . . . . . . . . . . . . . . 62

5.4 Esscher parameter vectors for different interest rates . . . . . . . . . . 63

5.5 Prices of exchange option . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.6 Prices of exchange option (inverted dependence structure) . . . . . . 64

5.7 Prices of a call on the maximum (K = 40) . . . . . . . . . . . . . . . 66

5.8 Prices of a call on the maximum (K = 60) . . . . . . . . . . . . . . . 66

5.9 Prices of a call on the maximum (K = 80) . . . . . . . . . . . . . . . 66

A.1 Portfolio comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

A.2 Arbitrage portfolio (1) . . . . . . . . . . . . . . . . . . . . . . . . . . 72

A.3 Arbitrage portfolio (2) . . . . . . . . . . . . . . . . . . . . . . . . . . 73

A.4 Put-call-parity portfolio . . . . . . . . . . . . . . . . . . . . . . . . . 74

LIST OF TABLES

VII

B.1 Minimum, maximum, and moments of DBK and SIE . . . . . . . . . 75

B.2 Dependency measures . . . . . . . . . . . . . . . . . . . . . . . . . . 76

Frequently Used Notation

I. General Notation

defined by

distributed

asymptotically distributed

equality in distribution

convolution

II. Sets and Spaces

real numbers

positive real numbers

III. Functions

Cov(x)

covariance

E(x)

expectation

F (x)

cumulative distribution function

f (x)

probability density function

{x>}

indicator function

(x)

modified bessel function of the second kind

VIII

FREQUENTLY USED NOTATION

ln(x)

natural logarithm

(x)

standard normal probability density function

(x)

standard normal cumulative distribution function

(x, )

d-dimensional standard normal cumulative distribution

function with correlation matrix

P (x)

probability measure

(x)

risk-neutral probability measure

Ran(x)

range

sgn(x)

signum function

Var(x)

variance

(x; k)

chi-square cumulative distribution function with k degrees of

freedom

IV. Options

price of a European call

price of an American call

dividend yield

Esscher parameter

strike price

future price

price of a European put

continuous risk-free interest rate

discrete risk-free interest rate

stock price

time to expiration

V. Abbreviations

c.p.

ceteris paribus

cdf

cumulative distribution function

cf.

confer

DBK

Deutsche Bank AG

e.g.

exempli gratia

FREQUENTLY USED NOTATION

et al.

et altera

etc.

etcetera

EURIBOR

Euro interbank offered rate

generalized hyperbolic

i.e.

id est

iid

independent identically distributed

mgf

moment generating function

NDRE

Nadaraya-Watson regression estimator

NIG

normal inverse Gaussian

page

pp.

pages

pdf

probability density function

SIE

Siemens AG

Chapter 1

Introduction

During the last decades, capital markets have transformed rapidly. Derivative secu-

rities -- or more simply derivatives -- like swaps, futures, and options supplemented

the trading of stocks and bonds. These financial products can frequently be seen in

the media: Due to derivatives, Procter & Gamble lost $150 million in 1994, Barings

Bank lost $1.3 billion in 1995 and Long-Term Capital Management (LTCM) lost

$3.5 billion in 1998.

Though these figures seem daunting, derivatives can be useful financial instruments.

Applications include (cf. McDonald [52, pp.2-3]):

· Risk management: Reducing risks by using derivatives with contrary pay-off

(hedging).

· Speculation: High gains or losses relative to the initial investment are possible

(leverage effect).

· Reduced transaction costs: In some cases trading costs may be reduced by

trading derivatives instead of stocks and bonds while achieving the same eco-

nomic effect.

· Regulatory arbitrage: Derivatives are often used to achieve the economic sale of

stocks (i.e. realizing gains/losses without being exposed to the risk of holding

CHAPTER 1. INTRODUCTION

the stock) while allowing the owner to retain voting rights and to defer taxes

on the sale of the stock.

Theory and practice of option valuation were revolutionized in 1973, when Fischer

Black and Myron Scholes published their celebrated Black Scholes formula in the

landmark paper "The pricing of options and corporate liabilities". Afterwards, a

vast amount of papers on option valuation was published which employ all kinds of

stochastic processes. Thereby, the special features of financial return data are tried

to be taken into account.

Advancing option valuation theory to options with multiple underlyings, lead to the

problem that the dependence structure of the underlying securities needs to be con-

sidered. Though linear correlation is a widely used dependence measure, it may be

inappropriate for multivariate return data. Throughout the last years, dependence

modelling through copulas has become common. Copulas are multivariate distribu-

tions on the d-dimensional unit-hyper-square which couples d marginal distributions

to a joint distribution. Copulas can be used to construct dependence measures like

the rank correlation coefficients of Spearman or Kendall. They are also a useful tool

in the context of option valuation.

The remainder of this thesis is structured as follows: Chapter 2 gives a general

introduction to derivatives and options. Chapter 3 exhibits the features of finan-

cial return data. Then, chapter 4 discusses univariate option valuation models.

Chapter 5 gives an overview of multivariate option pricing models. Rosenberg's

non-parametric pricing-approach [65] and valuation using the Esscher transform are

discussed in detail. An empirical study applying the Esscher methodology is pro-

vided. Finally, chapter 6 contains some concluding comments and an outlook.

Chapter 2

Derivatives and Options in

Particular

2.1

Standard Options

Definition 2.1 (Derivative securities) Derivative securities are financial instru-

ments which have a value determined by the value of other securities (cf. e.g. Mc-

Donald [52, p.1].

Forwards and futures

, options and swaps are examples of derivatives, which are also

known as contingent claims. For instance, a stock of Siemens is not a derivative.

However, suppose an investor enters into a contract. His contracting party pays him

e.g. e 5, if the price of e.g. one stock of Siemens is greater than e 60 in one year and

vice versa. As the value of this contract depends on the price of something else, it is

a derivative. One might also view this as a bet. Certainly, derivatives can be used

for speculation. However, derivatives can also be used to hedge existing risks. It is

not the derivative itself, but how it is used, that determines whether the derivative

increases or decreases the risk of a portfolio.

Both futures and forwards specify the purchase or sale of some underlying asset at some future

date. In contrast to futures which are exchange traded and highly standardized, forwards are

contracts that are traded over-the-counter and negotiated individually.

CHAPTER 2. DERIVATIVES AND OPTIONS IN PARTICULAR

Definition 2.2 (Option) A call (put) option gives its holder the right to purchase

(sell) an asset for a specified price (called strike or exercise price), on (European

style) or until (American style) some specified expiration date (see e.g. Bodie et al.

[8, p.649]).

The holder of an option is not required to exercise it. The holder will exercise the

call (put) if the market value of the underlying security at the expiration date is

higher (lower) than the strike price. Otherwise the option will be left unexercised.

In contrast to options, forwards give their holder the obligation (not the right (!))

to buy or sell some underlying asset at expiration. Thus, the holder has to buy (sell)

the underlying security, even in the case the agreement turns out to be unprofitable.

In general, option valuation models determine the price of an option by using the

following exogenous variables (e.g. see Bank and Gerke [2]):

· price of the underlying asset S

· strike price K

· time to expiration T

· risk-free interest rate r

· volatility of the underlying security, usually measured in annual standard de-

viations

In addition, option valuation models may also capture dividend payments, a stochas-

tic interest rate or stochastic volatility. The following partial derivatives

show the

impact of the exogenous variables on a the price c of a call option:

> 0;

< 0;

> 0;

> 0.

(2.1)

The payoff-functions of a call (c) and put option (p) at expiration are:

c = max (S - K, 0)

(2.2)

See appendix A.1 for the economic reasons of these partial derivatives.

CHAPTER 2. DERIVATIVES AND OPTIONS IN PARTICULAR

p = max (K - S, 0)

(2.3)

There are a couple of requirements on option prices in order to disable arbitrage

opportunities. An overview on these conditions is given in Appendix A.2.

The previous statements and their inherent remarks in the appendices focus on calls

only. However, similar statements can also be deduced for put options. For a com-

prehensive outline see Bank and Gerke [2]. The statements about put options were

neglected, because put-call-parity (see equation (2.4)) relates the prices of European

puts and calls on a non-dividend-paying underlying

to each other.

+ p - c = Ke

-rT

(2.4)

For otherwise identical puts and calls, Put-Call-Parity

gives a powerful tool to

derive the price of a put option given the price of a call, and vice versa. If the

put-call-parity relationship is violated, arbitrage is possible.

2.2

Exotic Options

The previous section explained how standard options work. This section outlines

exotic options and is mainly based on McDonald [52]. Nonstandard or exotic options

are obtained by changing the contractual terms. Exotic options are tailored to hedge

specific risks. Of course, they can also be used to speculate.

The payoff of Asian options is based on the average price of the underlying asset

over some period of time. Therefore, Asian options exhibit a lower volatility than

the underlying asset itself. Due to (2.1), the Asian option needs to be worth less

than its standard equivalent.

Another class of exotic options are Barrier options. The payoff of a barrier op-

tion depends on whether the price of the underlying reaches a specific level until

expiration. The basic kinds are:

An extension of put-call-parity to dividend-paying European stocks is straightforward. The

more general formula is S

+ p - c = Ke

-rT

+ P V (D) , where P V (D) denotes the present value of

all dividend payments which occur until expiration.

See Appendix A.3 for a proof of the Put-Call-Parity theorem.

CHAPTER 2. DERIVATIVES AND OPTIONS IN PARTICULAR

· Knock-out options automatically become worthless, when the specified barrier

is hit.

· Knock-in options become effective, when the barrier is hit.

· Rebate options pay a specified amount of money either immediately or at

expiration, when the barrier is reached.

For instance, a down-and-in knock-in call with a barrier of e 50 and a strike price

of e 70 is only worthwhile if the stock price falls bellow e 50 and is above e 70 at

expiration.

Gap options separate the condition for the payoff and the size of the payoff. A

standard option pays S

- K, when S

> K. In contrast, a gap option pays

- K

, when S

> K

Furthermore, there are compound options, cash-or-nothing options, asset-or-nothing

options, and many more. For a more comprehensive introduction and valuation

approaches, see McDonald [52]. Another class of exotic options are multivariate

options, which are the focus of this thesis and are introduced in the next section.

2.3

Multivariate Options

Multivariate options depend on several risky assets, which can be represented in the

payoff function as underlying and/or strike asset. Clearly, the dependence structure

of risky securities used in the payoff function is a key determinant in the valuation

of multivariate options. For instance, an option which gives its holder the right

to buy the more valuable of two underlying securities for a specified strike price at

expiration, has a greater value if the underlying securities move in different directions

than if they move together.

Examples of multivariate options are calls/puts on the maximum/minimum of sev-

eral assets or multivariate digital calls/puts. Their payoffs at expiration are:

· Call on the maximum: c

max

= max max (S

, S

, . . . , S

) - K, 0

CHAPTER 2. DERIVATIVES AND OPTIONS IN PARTICULAR

· Put on the maximum: p

max

= max K - max (S

, S

, . . . , S

), 0

· Call on the minimum: c

min

= max min (S

, S

, . . . , S

) - K, 0

· Put on the minimum: p

min

= max K - min (S

, S

, . . . , S

), 0

· Digital call: c

Digital

= I

{(S

)...(S

)}

· Digital put: p

Digital

= I

{(S

)...(S

)}

Furthermore, a commonly used type of multivariate option is the exchange option.

An exchange option pays off if the underlying asset outperforms some other risky

security, which functions as the strike asset. I.e., an executive who receives such

options as (variable) part of his salary, is only compensated if the stock of his

company outperforms the stock of the competitor's company (the benchmark). The

payoff function is:

= max (S

- S

, 0),

(2.5)

where S

denotes the stock of the own company and S

the benchmark. The ex-

change option may also be interpreted as a call on S

with (stochastic) strike price

or as put on S

with (stochastic) exercise price S

Chapter 3

Characteristics of Financial

Returns

3.1

Stylized Facts of Univariate Return Distribu-

tions

Definition 3.1 (Discrete and continuous return) Let S

denote the price of an

asset at time t. Then

- S

t-1

- 1

(3.1)

is the asset's discrete return in the time interval [t - 1, t]. For n - equidis-

tant subperiods and uniform changes in value in each subperiod, the discrete return

becomes a continuous return, which is defined as the difference of the prices' loga-

rithms:

= ln(S

) - ln(S

t-1

(3.2)

Econometric researchers commonly use continuously compounded returns for their

calculations, as the difference between discrete and continuous returns is negligible

small and because of their choice computational properties.

CHAPTER 3. CHARACTERISTICS OF FINANCIAL RETURNS

Financial return data exhibit the so-called stylized facts. In comparison to a fitted

normal density

, the plot of a financial return kernel density estimation exhibits

high peakedness, heavy tails, and -- depending on the asset -- skewness. Figure

(3.1) gives a graphical impression of these characteristics for the daily returns

Deutsche Bank (DBK), by comparing the kernel density with the densities of a fitted

normal distribution.

-0.1

-0.08

-0.06

-0.04

-0.02

0.02

0.04

0.06

0.08

0.1

DBK: Kernel vs. Fitted Normal Density

Return

Density

Kernel Density

Fitted Normal Density

-0.15

-0.14

-0.13

-0.12

-0.11

-0.1

-0.09

-0.08

-0.07

-0.06

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Return

Density

DBK: Left Tails of Kernel vs. Fitted Normal Density

Kernel Density

Fitted Normal Density

Figure 3.1: DBK: Kernel vs. fitted normal density; the right figure highlights the

(left-hand-side) heavy tails (a similar picture can be obtained for the right-hand-

side).

Graphical frameworks to reveal deviations of financial returns from normality are the

so-called normal plots and T

-plots. The normal plots the empirical quantiles (the

quantiles of the returns) against the theoretical quantiles of a normal distribution.

Therefore, it is also called NQ-plot. Deviations from the line through origin with

gradient one are a sign for non-normality. Figure (3.2) shows NQ-plots for DBK

and SIE.

The normal distribution is used as a reference, since it is common to assume that financial

return data are normally distributed.

See Appendix B for a summary of the used data. All plots and calculations were performed

using MATLAB Version 6.5 [69].

CHAPTER 3. CHARACTERISTICS OF FINANCIAL RETURNS

-0.1

-0.05

0.05

0.1

0.001

0.003

0.01

0.02

0.05

0.10

0.25

0.50

0.75

0.90

0.95

0.98

0.99

0.997

0.999

DBK

Normal

DBK: NQ-Plot

-0.1

-0.05

0.05

0.1

0.15

0.001

0.003

0.01

0.02

0.05

0.10

0.25

0.50

0.75

0.90

0.95

0.98

0.99

0.997

0.999

SIE

Normal

SIE: NQ-Plot

Figure 3.2: NQ-plots of DBK and SIE. Departures from normality are obvious.

Particularly, the apparent 's'-shape signals excess-kurtosis.

The moment generating function (mgf) of a normal distribution is given by

(t) = exp

µt +

Hence, the logarithm of the mgf is ln M

(t) = µt +

. Apparently all derivatives

of order three and higher are zero:

ln M

(t)

= 0 for k 3.

This property is utilized by the T

-plot. The idea is to compute the third derivative

of the logarithm of the empirical mgf and compare to the case of normality, in which

it should be zero. Significant deviations from zero can be identified by the confidence

ties derived by Gosh [39], which are given by

= ±

1.38 -

5.92

12.8

(t, t),

(3.3)

where denotes the significance level,

is the 1 - quantile of the standard

normal distribution, N the sample size, and K

(t, s) is the covariance function of

a zero-mean Gaussian process. Figure (3.3) shows T

-plots for the returns of DBK

and SIE.

Moreover, several tests are available to test the normality hypothesis. The most

common are the

-test of goodness-of-fit, the Kolmogorov-Smirnov test, and the

Jarque-Bera test. The test statistics are given by

CHAPTER 3. CHARACTERISTICS OF FINANCIAL RETURNS

Figure 3.3: T3-plots of DBK and SIE. The dotted and dash-dotted lines mark the

5% and 10% confidence ties, respectively. The charts in the lower right corners

are the same curves, but with different scales, in order to exhibit the tremendous

deviations from normality.

-test of goodness-of-fit:

i=1

- E

)

(k - 1),

(3.4)

where k is the number of classes into which the ordered observations are di-

vided, O

is the number of observations in this class and E

is the number of

expected (under normality) observations.

· Kolmogorov-Smirnov test:

K = sup

(x) - F (x)|,

(3.5)

where F

(x) denotes the empirical distribution function and F (x) the theoret-

ical (normal) distribution function. K does not follow a standard distribution,

but critical values are tabulated (see for instance Fischer, Grottke, and Klein

[33, p.21]).

· Jarque-Bera test:

J B =

i=1

- x)

i=1

- x)

i=1

- x)

i=1

- x)

(2), (3.6)

CHAPTER 3. CHARACTERISTICS OF FINANCIAL RETURNS

where x =

i=1

Further remarks on the introduced and other tests can be found in Fischer [32,

pp.11-17]. Table (3.1) exhibits the p-values of the introduced tests

. Apparently,

the null hypothesis (normally distributed returns), can be clearly rejected on all

common significance levels.

J B

DBK

1 · 10

-23

SIE

5.5 · 10

-21

Table 3.1: P-values of

-, KS-, and J B-tests applied on DBK and SIE returns.

Both the graphical frameworks and the tests showed that the normal distribution

is not able to capture the characteristics of financial return data. Thus, the often

made assumption of normally distributed returns is questionable. Fischer [32] and

Grottke [40] show that for instance the generalized hyperbolic distribution (GH), the

exponential beta distribution of the second kind (EGB2) and the skewed generalized

t-distribution of the second kind (SGT2) provide a significantly better fit than the

normal distribution. Figure (3.4) highlights the better fit of the normal inverse

Gaussian (NIG) distribution

, which is contained in the class of GH distribution as

a special case.

3.2

Digression: Dependence & Copulas

One of the basic problems in statistics is understanding the relationship between

multivariate outcomes. Linear correlation is the central concept to model depen-

The p-value of the

-test of goodness-of-fit is stable for all common class segmentations.

The class of GH distributions includes the NIG distribution as a special case. The NIG

distribution was selected for comparison, because it has several useful properties which ease the

application in option valuation models. See chapter 5 for details.

CHAPTER 3. CHARACTERISTICS OF FINANCIAL RETURNS

-0.1

-0.08

-0.06

-0.04

-0.02

0.02

0.04

0.06

0.08

0.1

DBK: Kernel vs. Fitted Normal and NIG Densities

Return

Density

Kernel Density

Fitted Normal Density

Fitted NIG Density

-0.15

-0.14

-0.13

-0.12

-0.11

-0.1

-0.09

-0.08

-0.07

-0.06

0.2

0.4

0.6

0.8

1.2

1.4

DBK: Left Tail of Kernel vs. Fitted Normal and NIG Densities

Return

Density

Kernel Density

Fitted Normal Density

Fitted NIG Density

Figure 3.4: DBK: Kernel vs. fitted normal and NIG density; the right figure high-

lights the (left-hand-side) heavy tails (a similar picture can be obtained for the

right-hand-side tails). Apparently, the NIG distribution provides a significant bet-

ter fit.

dence in financial theory. For instance, the dependence structure between financial

assets in the widely used Capital Asset Pricing Model (CAPM) and the Arbitrage

Pricing Theory (APT) is modelled by correlation.

However, correlation has several shortcomings. Linear correlation is only an ade-

quate dependence measure if the multivariate distribution is Gaussian or -- more

general -- elliptical. The falsity of several common views on linear correlation is

demonstrated in detail in Embrechts et al. [26].

This section is mainly based on Bouye et al. [9], Embrechts et al. [26], and [25],

Matteis [51], and McNeil [53]. Subsection 3.2.1 deals with the shortcomings of

linear correlation. In subsection 3.2.2, other dependence measures are presented.

Subsection 3.2.3 introduces the concept of copulas.

Details

Seiten
Erscheinungsform: Originalausgabe
Jahr: 2004
ISBN (eBook): 9783832481322
ISBN (Paperback): 9783838681320
DOI: 10.3239/9783832481322
Dateigröße: 2.3 MB
Sprache: Englisch
Institution / Hochschule: Friedrich-Alexander-Universität Erlangen-Nürnberg – Wirtschafts- und Sozialwissenschaftliche Fakultät
Erscheinungsdatum: 2004 (Juli)
Note: 1,0
Schlagworte: derivative derivate esscher transform

Autor

Christian Hassold (Autor:in)

The Valuation of Multivariate Options

Zusammenfassung

Leseprobe

Inhaltsverzeichnis

Details

Autor

Christian Hassold (Autor:in)