The Valuation of Multivariate Options
©2004
Diplomarbeit
114 Seiten
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 […]
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
Zugl. Friedrich-Alexander-Universität Erlangen-Nürnberg, Erlangen, Deutschland,
Diplomarbeit, 2004
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© Diplomica GmbH
http://www.diplom.de, Hamburg 2007
Printed in Germany
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.
c 2004 Christian Hassold -- All rights reserved.
I
Contents
List of Figures
VI
List of Tables
VII
Frequently Used Notation
VIII
1 Introduction
1
2 Derivatives and Options in Particular
3
2.1 Standard Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2.2 Exotic Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.3 Multivariate Options . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
3 Characteristics of Financial Returns
8
3.1 Stylized Facts of Univariate Return Distributions . . . . . . . . . . .
8
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
II
CONTENTS
III
3.3 Stylized Facts of Multivariate Return Distributions . . . . . . . . . . 24
4 Option Valuation Approaches
30
4.1 Binomial Option Pricing . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.2 The Black-Scholes Formula . . . . . . . . . . . . . . . . . . . . . . . . 34
4.3 Risk-Neutral Valuation . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5 Multivariate Option Pricing Models
39
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
IV
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
67
A Remarks on Options
69
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
75
C Derivation of the Black Scholes Formula
77
C.1 Differential Equation Approach . . . . . . . . . . . . . . . . . . . . . 77
C.2 Esscher Transform Approach . . . . . . . . . . . . . . . . . . . . . . . 78
D Derivation of the Nadaraya-Watson Regression Estimator
83
E Selected MATLAB Functions
85
E.1 MT
3
-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
94
List of Figures
3.1 DBK: Kernel vs. fitted normal density . . . . . . . . . . . . . . . . .
9
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
3
-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
V
List of Tables
3.1 P-values of
2
-, 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
VI
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
a
asymptotically distributed
d
=
equality in distribution
convolution
II. Sets and Spaces
R
real numbers
R
+
positive real numbers
III. Functions
Cov(x)
covariance
E(x)
expectation
F (x)
cumulative distribution function
f (x)
probability density function
I
{x>}
indicator function
K
(x)
modified bessel function of the second kind
VIII
FREQUENTLY USED NOTATION
IX
ln(x)
natural logarithm
(x)
standard normal probability density function
(x)
standard normal cumulative distribution function
d
(x, )
d-dimensional standard normal cumulative distribution
function with correlation matrix
P (x)
probability measure
P
(x)
risk-neutral probability measure
Ran(x)
range
sgn(x)
signum function
Var(x)
variance
2
(x; k)
chi-square cumulative distribution function with k degrees of
freedom
IV. Options
c
price of a European call
c
A
price of an American call
dividend yield
h
Esscher parameter
K
strike price
future price
p
price of a European put
r
continuous risk-free interest rate
r
D
discrete risk-free interest rate
S
stock price
T
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
X
et al.
et altera
etc.
etcetera
EURIBOR
Euro interbank offered rate
GH
generalized hyperbolic
i.e.
id est
iid
independent identically distributed
mgf
moment generating function
NDRE
Nadaraya-Watson regression estimator
NIG
normal inverse Gaussian
p.
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
1
CHAPTER 1. INTRODUCTION
2
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
1
, 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.
1
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.
3
CHAPTER 2. DERIVATIVES AND OPTIONS IN PARTICULAR
4
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
2
show the
impact of the exogenous variables on a the price c of a call option:
c
S
> 0;
c
K
< 0;
c
> 0;
c
T
> 0;
c
r
> 0.
(2.1)
The payoff-functions of a call (c) and put option (p) at expiration are:
c = max (S - K, 0)
(2.2)
2
See appendix A.1 for the economic reasons of these partial derivatives.
CHAPTER 2. DERIVATIVES AND OPTIONS IN PARTICULAR
5
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
3
to each other.
S
0
+ p - c = Ke
-rT
(2.4)
For otherwise identical puts and calls, Put-Call-Parity
4
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:
3
An extension of put-call-parity to dividend-paying European stocks is straightforward. The
more general formula is S
0
+ p - c = Ke
-rT
+ P V (D) , where P V (D) denotes the present value of
all dividend payments which occur until expiration.
4
See Appendix A.3 for a proof of the Put-Call-Parity theorem.
CHAPTER 2. DERIVATIVES AND OPTIONS IN PARTICULAR
6
· 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
T
- K, when S
T
> K. In contrast, a gap option pays
S
T
- K
1
, when S
T
> K
2
.
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
1
, S
2
, . . . , S
n
) - K, 0
CHAPTER 2. DERIVATIVES AND OPTIONS IN PARTICULAR
7
· Put on the maximum: p
max
= max K - max (S
1
, S
2
, . . . , S
n
), 0
· Call on the minimum: c
min
= max min (S
1
, S
2
, . . . , S
n
) - K, 0
· Put on the minimum: p
min
= max K - min (S
1
, S
2
, . . . , S
n
), 0
· Digital call: c
Digital
= I
{(S
1
K
1
)...(S
n
K
n
)}
· Digital put: p
Digital
= I
{(S
1
K
1
)...(S
n
K
n
)}
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:
D
E
= max (S
1
- S
2
, 0),
(2.5)
where S
1
denotes the stock of the own company and S
2
the benchmark. The ex-
change option may also be interpreted as a call on S
1
with (stochastic) strike price
S
2
or as put on S
2
with (stochastic) exercise price S
1
.
Chapter 3
Characteristics of Financial
Returns
3.1
Stylized Facts of Univariate Return Distribu-
tions
Definition 3.1 (Discrete and continuous return) Let S
t
denote the price of an
asset at time t. Then
R
d
t
=
S
t
- S
t-1
S
t-1
=
S
t
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:
R
c
t
= ln(S
t
) - 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.
8
CHAPTER 3. CHARACTERISTICS OF FINANCIAL RETURNS
9
Financial return data exhibit the so-called stylized facts. In comparison to a fitted
normal density
1
, 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
2
of
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
0.02
0.04
0.06
0.08
0.1
0
5
10
15
20
25
30
35
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
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
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
3
-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.
1
The normal distribution is used as a reference, since it is common to assume that financial
return data are normally distributed.
2
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
10
-0.1
-0.05
0
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
q
DBK
q
Normal
DBK: NQ-Plot
-0.1
-0.05
0
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
q
SIE
q
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
M
X
(t) = exp
µt +
1
2
2
t
2
.
Hence, the logarithm of the mgf is ln M
X
(t) = µt +
1
2
2
t
2
. Apparently all derivatives
of order three and higher are zero:
k
ln M
X
(t)
t
k
= 0 for k 3.
This property is utilized by the T
3
-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
q
,N
= ±
1.38 -
5.92
N
+
12.8
N
+
1-
K
3
(t, t),
(3.3)
where denotes the significance level,
1-
is the 1 - quantile of the standard
normal distribution, N the sample size, and K
3
(t, s) is the covariance function of
a zero-mean Gaussian process. Figure (3.3) shows T
3
-plots for the returns of DBK
and SIE.
Moreover, several tests are available to test the normality hypothesis. The most
common are the
2
-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
11
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.
·
2
-test of goodness-of-fit:
2
=
k
i=1
(O
i
- E
i
)
2
E
i
a
2
(k - 1),
(3.4)
where k is the number of classes into which the ordered observations are di-
vided, O
i
is the number of observations in this class and E
i
is the number of
expected (under normality) observations.
· Kolmogorov-Smirnov test:
K = sup
x
|F
n
(x) - F (x)|,
(3.5)
where F
n
(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 =
n
6
1
n
n
i=1
(x
i
- x)
3
1
n
n
i=1
(x
i
- x)
2
3
+
n
24
1
n
n
i=1
(x
i
- x)
4
1
n
n
i=1
(x
i
- x)
2
4
a
2
(2), (3.6)
CHAPTER 3. CHARACTERISTICS OF FINANCIAL RETURNS
12
where x =
1
n
n
i=1
x
i
.
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
3
. Apparently,
the null hypothesis (normally distributed returns), can be clearly rejected on all
common significance levels.
2
KS
J B
DBK
0
1 · 10
-23
0
SIE
0
5.5 · 10
-21
0
Table 3.1: P-values of
2
-, 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
4
, 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-
3
The p-value of the
2
-test of goodness-of-fit is stable for all common class segmentations.
4
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
13
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
0
5
10
15
20
25
30
35
40
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
0.2
0.4
0.6
0.8
1
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
- Erscheinungsjahr
- 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
- Produktsicherheit
- Diplom.de