Weather risk management [Elektronische Ressource] : CAT bonds and weather derivatives / von Brenda López Cabrera

Frau M.Sc. Brenda López Cabrera

Informations

• Publié par: humboldt-universitat_zu_berlin
• Publié le: 01 janvier 2010
• Nombre de lectures: 35
• Langue: English
• Poids de l'ouvrage: 5 Mo

Extrait

Weather Risk Management: CAT bonds and Weather
Derivatives
DISSERTATION
zur Erlangung des akademischen Grades
doctor rerum politicarum
(Doktor der Wirtschaftswissenschaft)
eingereicht an der
Wirtschaftwissenschaftlichen Fakultaet
Humboldt-Universitaet zu Berlin
von
Frau M.Sc. Brenda López Cabrera
13.03.1980 in Puebla, Mexiko
Präsident der Humboldt-Universitaet zu Berlin:
Prof. Dr. Christoph Markschies
Dekan der Wirtschaftwissenschaftlichen Fakultaet:
Prof. Oliver Guenther, Ph.D.
Gutachter:
1. Prof. Dr. Wolfgang Haerdle
2. Prof. Dr. Vladimir Spokoiny
eingereicht am: 17 März 2010
Tag des Kolloquiums: 27 April 2010Abstract
CAT bonds and weather derivatives are end-products of a process known as se-
curitization that transform non-tradable (natural catastrophes or weather related)
risk factors into tradable financial assets. As a result the markets for such prod-
ucts are typically incomplete. Since appropiate measures of the risk associated to
a particular price become necessary for pricing, one essentially needs to incorpo-
rate the market price of risk (MPR), which is an important parameter of the as-
sociated equivalent martingale measure. The majority of papers so far has priced
non-tradable assets assuming zero MPR, but this assumption yields biased prices
and has never been quantified earlier. This thesis deals with the differences be-
tween historical and risk neutral behaviors of the non-tradable underlyings and
gives insights into the behaviour of the market price of weather risk and weather
risk premium. The thesis starts by introducing the risk transfering instruments,
the financial - statistical techniques and ends up by examining the real data appli-
cations with particular focus on the implied trigger intensity rates of a parametric
CAT bond for earthquakes and the MPR of temperature derivatives.
iiZusammenfassung
CAT-Bonds und Wetterderivate sind die Endprodukte eines Verbriefungprozes-
ses, der nicht handelbare Risikofaktoren (Wetterschäden oder Naturkatastrophen-
schäden) ine Finanzanlagen verwandelt. Als Ergebnis sind die Märkte
für diese Produkte in der Regel unvollständig. Da geeignete Risikomaße in Bezug
auf einen bestimmten Preis Voraussetzung sind zur Preisbestimmung, ist es not-
wendig den Marktpreis des Risikos (MPR), welcher ein wichtiger Parameter des
zugehörigen äquivalenten Martingalmaß ist, zu berücksichtigen. Die Mehrheit der
bisherigen Veröffentlichungen haben die Preise der nicht handelbaren Vermögens-
werte mittels der Annahme geschätzt, dass der MPR gleich null ist. Diese Annahme
verzerrt allerdings die Preise und wurde bisher noch nicht quantifiziert. Diese Dok-
torarbeit beschäftigt sich mit den Unterschieden zwischen dem historischen und
dem risikoneutralen Verhalten der nicht handelbaren Basiswerte und gibt Einblicke
in den Marktpreis für Wetterrisiko und die Wetterrisikoprämie. Diese Arbeit be-
ginnt mit einer Darstellung der Instrumente zur Übertragung der Risiken, gefolgt
von den finanziellen - statistischen Verfahren und endet mit einer Untersuchung
reeller Daten, wobei der Schwerpunkt auf die implizierten Trigger-Intensitätsraten
eines parametrischen CAT-Bond für Erdbeben und auf den MPR der Temperatur
Derivate gelegt wird.Acknowledgement
I would like to thank to Professor Dr. Wolfgang Haerdle for supervising and support-
ing me through the whole time of my Ph.D. studies. He introduced me to the world of
financial statistics and encouraged me to work on the analysis of weather risk manage-
ment.
I am thankful to Professor Dr. Spokoiny for willing accepting to evaluate my thesis and
sit in the examination commitee.
I would like to thank all those people with whom I collaborated during the preparation
of the thesis. The theoretical part of the thesis is based on the results of close coopera-
tion with Professor Fred Espen Benth, whose extraordinary deep knowledge and expe-
rience in financial mathematics and energy markets helped me a lot in understanding
of these new methods. I also appreciate him the dicussions and comments to improve
the estimation algorithms and hospitality during my visits at the University of Oslo.
I am grateful to Professor Jianqing Fan for inviting me to come to Princeton University
and giving me valuable suggestions.
I owe much to many colleagues and researchers for sharing their time with me by
numberless discussions and consultations during my work, among other these were:
Szymon Borak, Enzo Giacomini, Jelena Bradic, and of course my thanks goes to all
members of the Institute for Statistics at Humboldt University, C.A.S.E. and CRC 649
for friendly atmosphere and encouragement. I gratefully acknowledge the financial
support from NaFOEG - Promotionsfoerderung and the Deutsche Forschungsgemein-
schaft via CRC 649 Oekonomisches Risiko, Humboldt-Universitaet zu Berlin.
Last but certainly not least I am deeply indebted to my family for their constant sup-
port.
Berlin, March 16, 2010.
Brenda López CabreraContents
Acknowledgement iv
1 Introduction 1
2 Theoretical Background 6
2.1 Stochastic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 price modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Pricing futures on the spot market . . . . . . . . . . . . . . . . . . . . . . 13
3 Catastrophe (CAT) Bonds 17
3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Calibrating a Mexican Parametric CAT Bond . . . . . . . . . . . . . . . . 19
3.2.1 Calibration in the Reinsurance Market . . . . . . . . . . . . . . . . 22
3.2.2 in the Capital Market . . . . . . . . . . . . . . . . . . . 23
3.2.3 via Historical data . . . . . . . . . . . . . . . . . . . . 24
3.3 Pricing modelled-index CAT bonds for Mexican earthquakes . . . . . . . 28
3.3.1 Severity of Mexican earthquakes . . . . . . . . . . . . . . . . . . . 29
3.3.2 Frequency of . . . . . . . . . . . . . . . . . . 34
3.3.3 Pricing modelled-Index CAT bonds . . . . . . . . . . . . . . . . . 35
3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4 Weather Derivatives 44
4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.2 Modelling Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.2.1 Properties of temperature data . . . . . . . . . . . . . . . . . . . . 47
4.2.2 An Ornstein-Uhlenbeck driven by a Fractional Brownian Motion 48
4.2.3 An Model driven by a Brownian motion . . 49
4.2.4 An by a Lévy Process . . . . . 49
4.2.5 Empirical Analysis of Temperature Dynamics . . . . . . . . . . . 50
4.2.6 Localizing temperature residuals . . . . . . . . . . . . . . . . . . . 61
4.3 Stochastic Pricing model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.4 The implied market price of weather risk . . . . . . . . . . . . . . . . . . 74
4.4.1 Constant market price of risk for different daily contract . . . . . 74
4.4.2 price of risk per trading day . . . . . . . . . . . 75
4.4.3 Two constant market prices of risk per trading day . . . . . . . . 75
4.4.4 General form of the market price of risk per trading day . . . . . 76
4.4.5 Bootstrapping the price of risk . . . . . . . . . . . . . . . . 77
vContents
4.4.6 Smoothing the market price of risk over time . . . . . . . . . . . . 79
4.4.7 Statistical and economical insights of the MPR . . . . . . . . . . . 80
4.4.8 Pricing CAT-HDD-CDD futures . . . . . . . . . . . . . . . . . . . 85
4.5 The risk premium and the market price of weather risk . . . . . . . . . . 85
4.6 Temperature baskets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.6.1 Basket indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.6.2 Stochastic modelling for Basket temperatures . . . . . . . . . . . . 91
4.6.3 Pricing of Basket temperatures . . . . . . . . . . . . . . . . . . . . 92
4.7 Conclusions and further research . . . . . . . . . . . . . . . . . . . . . . . 95
Bibliography 97
viList of Figures
3.1 Cash flows diagram of a CAT bond . . . . . . . . . . . . . . . . . . . . . . 18
3.2 Number of Mexican earthquakes occurred during 1900-2003 . . . . . . . 20
3.3 Map of seismic regions in Mexico. . . . . . . . . . . . . . . . . . . . . . . 21
3.4 The cash flows diagram for the Mexican CAT bond . . . . . . . . . . . . 22
3.5 Magnitude of trigger events . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.6 Historical and modelled losses of Mexican earthquakes (in million dol-
lars) occurred in Mexico during 1900-2003 and without outliers of the
earthquakes in 1985 and 1999 . . . . . . . . . . . . . . . . . . . . . . . . . 31
n o
3.7 The log of the empirical mean excess function log eˆ for the mod-n(x)
elled loss data with and without the outlier of the earthquake in 1985. . . 32
n o
ˆ3.8 The log of the empirical limited expected value function log l andn(x)
log(l ) for the log-normal, Pareto, Burr, Weibull and Gamma distribu-x
tions for the modelled loss with and without the outlier of the 1985 earth-
quake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
n o
3.9 The log of the empirical mean excess function log eˆ for the earth-n(t)
qu