second cycle (d.m. 270/2004) in economics | models and
TRANSCRIPT
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Ca’ FoscariDorsoduro 324630123 Venezia
UniversitàCa’FoscariVenezia Master’s Degree programme —
Second Cycle (D.M. 270/2004)
in Economics — Models and Methods of Quanti-
tative Economics
Final Thesis
A model-combination approach
to pricing weather derivatives
Supervisor
Ch. Prof. Roberto Casarin
Ch. Prof. Martina Nardon
Graduand
Iuliia Lipanova
Matriculation number 855735
Academic Year
2015/2016
Abstract
This study provides an empirical example of fitting a model that can explain and
predict the behavior of weather time series. Different models of GARCH-type are
fitted on the series of the average temperature. Autoregressive, seasonal and trend
components are included in mean and variance equations. The seasonal components
are constructed as a combination of simple harmonic functions, the frequencies of
which are found with Fast Fourier Transform. Forecast combination approach is
applied to define the best model and thus, this model is used as an underlying to
price weather derivatives. Numerical examples of pricing some insurance contracts
are given, found with Monte Carlo simulations.
Keywords: Weather derivatives, ARMA-GARCH, Fast Fourier Transform, Monte
Carlo, Insurance premium calculation
Acknowledgment
I would like to express my deepest gratitude to my thesis advisor Professor
Roberto Casarin for his support of this Master’s thesis. His profound knowledge
helped me to organize the econometrics part of this thesis, while his personal interest
in the field was a real motivation to me. I would also like to thank my co-advisor
Professor Martina Nardon, for her guidance on finance part of this thesis. She
was always open to communication and has significantly broadened my understand-
ing of the whole field. What is more, I would like to express my gratitude to QEM
Master’s program for providing me with the aid to gain such a unique and life-
changing experience studying in two the most beautiful cities, Paris and Venice.
I place on record, my sincere thank you to my dearest colleagues of this course
Laura Hurtado Moreno, Erdem Yenerdag and Kedar Kulkarni. It was a real
pleasure to work on several course projects with Laura and Erdem. Not only their
contribution was important for the final grade, but more importantly, I significantly
enriched my knowledge in various fields of economics thanks to their contribution.
Finally, I must express my gratitude to my parents for being always supportive
in any decisions I have made in my life so far. I am exceptionally grateful to my
mother Iryna Lipanova for encouraging me to push my boundaries and to my
father, Vyacheslav Lipanov for his constant concern. Special thanks go to my
dearest friends Daniele Genovese and Ksenia Nepomnyashaya. It would have
been much harder for me to adjust to the life abroad without encouragement and
guidance from Daniele and his family.
i
Contents
List of Tables iv
List of Figures v
1 Introduction to weather derivatives 3
1.1 Weather derivatives market . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Premuim calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Asian-style insurance policy . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Parisian-style insurance policy . . . . . . . . . . . . . . . . . . . . . . 7
1.5 Insurance policy based on weather index . . . . . . . . . . . . . . . . 8
2 Models for temperature 10
2.1 Fourier analysis of the temperature . . . . . . . . . . . . . . . . . . . 10
2.2 GARCH model with seasonal components . . . . . . . . . . . . . . . 13
2.3 GARCH in mean model . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4 Exponential GARCH model . . . . . . . . . . . . . . . . . . . . . . . 14
3 Empirical results 16
3.1 Data and descriptive statistics . . . . . . . . . . . . . . . . . . . . . . 16
3.2 Spectral analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.3 Model estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.4 Model selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.5 Forecast combination . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.6 Pricing weather insurance contracts . . . . . . . . . . . . . . . . . . . 32
ii
Appendices 39
Appendix A Discrete Fourier transformation, MATLAB function 40
Appendix B An example of Fourier analysis on simulated series, MAT-
LAB script 41
Appendix C Spectral analysis,MATLAB script 43
Appendix D Diagnostics of the residuals 45
Bibliography 52
iii
List of Tables
3.1 Summary of the Series . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Augmented Dickey-Fuller Test, Results . . . . . . . . . . . . . . . . . 17
3.3 Ljung-Box Test, Results . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.4 Fourier Transformation, Results . . . . . . . . . . . . . . . . . . . . . 22
3.5 The components inclueded in the models . . . . . . . . . . . . . . . . 25
3.6 Jarque-Bera test on residuals, Results . . . . . . . . . . . . . . . . . . 25
3.7 Goodness of forecast, Results . . . . . . . . . . . . . . . . . . . . . . 28
3.8 Weights for the model combination . . . . . . . . . . . . . . . . . . . 31
3.9 Goodness of forecast, Result of Model combination . . . . . . . . . . 31
3.10 Description of the contracts . . . . . . . . . . . . . . . . . . . . . . . 34
3.11 Prices of Asian-style contracts . . . . . . . . . . . . . . . . . . . . . . 35
3.12 Prices of Parisian-style contracts . . . . . . . . . . . . . . . . . . . . . 36
3.13 Prices of Index-based contracts . . . . . . . . . . . . . . . . . . . . . 36
D.1 White test, Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
D.2 Estimation output . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
iv
List of Figures
2.1 The magnitude spectrum of the observations versus the frequencies . 12
3.1 The graph of the observations and the histogram of the distribution . 17
3.2 ACF and PACF of the observations . . . . . . . . . . . . . . . . . . . 18
3.3 Fourier analysis of the observations a) Magnitude spectrum; b)Log
transformation of the magnitude spectrum . . . . . . . . . . . . . . . 20
3.4 Graph of the observations and the harmonic functions . . . . . . . . . 21
3.5 Fourier analysis of the squared observations a) Magnitude spectrum;
b)Log transformation of the magnitude spectrum . . . . . . . . . . . 22
3.6 Graph of the squared observations and the harmonic functions . . . . 23
3.7 Q-Q plots: model residuals against Student’s t quantiles . . . . . . . 26
3.8 Actual,fitted,residual of studied models . . . . . . . . . . . . . . . . . 27
3.9 Ex post forecast and credible intervals . . . . . . . . . . . . . . . . . 29
3.10 Forecast combination, Result . . . . . . . . . . . . . . . . . . . . . . . 32
3.11 Simulated trajectories of the combined model over the time . . . . . . 33
D.1 Box-Pierce P-values . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
v
Introduction
Weather changes have a huge impact on the different branches of business activity.
Nowadays, these changes are not only represented by some periodical patterns (such
as climate oscillations, El Nino pattern, monsoons) or by low probable hazards (as
volcanic activity, earthquakes, wildfires etc.), but, mainly, by the global warming.
The increase of the temperature has direct and indirect effects on such industries
as agriculture, energy, tourism. One of the latest destructive impacts of the climate
change is bleaching of Great Barrier Reef caused both by warm streams of El Nino
and global temperature increase. The result of the bleaching will probably be the
loss of countless marine species, a decrease of scuba-diving tourism and tourism on
the adjacent Australian shore cost. Obviously, in such case, no financial instrument
would neither cover the losses nor fix the problem. However, in other situations,
short-term derivative instruments might cover the possible risks, so that companies
would be able to sustain unpredictable changes. In a long-term the reconsideration
of the technologies is required that is out of the scope of this study. We would like
to construct and analyze insurance contracts that may be used to hedge against
weather-related losses. In the literature, similar empirical case studies are conducted
by Carolyn W. Chang, Jack S. K. Chang and Min-Teh Yu (1996); Calum G. Turvey
(2001).
The insurance contracts on weather are similar to the derivatives. However, due
to specifics of the underlying and its market, these contracts cannot be evaluated
using assumptions of the Black-Scholes model. The first issue that arises is the
need to construct a model for the underlying. The Black-Scholes model assumes
1
that the prices of the underlying follow a geometric Brownian motion process with
Markov’s property that is interpreted into Efficient Market Hypothesis that actual
price of a stock already includes all available market information. Consequently, its
expected value does not depend on historical values thus, it can be explained only
by today’s value. However, Efficient Market Hypothesis cannot be valid for weather
patterns that depend on the seasonality. Another reason why the standard valuation
approaches cannot be used is the lack of evidence towards the assumption of the
risk-neutral valuation principle, as the weather is a non-traded asset and there is no
corresponding spot market. Even though there are some traded weather derivatives,
mainly presented by Chicago Mercantile Exchange, the market is not liquid. All it
calls to consider not a standard derivative instrument on weather, but an insurance
contract that requires premium calculation in order to determine its price.
Models that can possibly explain the behavior of the weather are discussed in the
literature by Sean D. Campbell and Francis X. Diebolt (2005),Timothy J. Richards,
Mark R. Manfredo, Dwight R. Sanders (2004). In this thesis, we use a model sim-
ilar to the one introduced by Campbell (2005) and we consider other specifications
from the family of ARMA-GARCH models. In order to obtain a model with better
forecasting properties, the model combination approach is applied as it is shown by
A.Timmermann (2005). The evaluation of the insurance contract is done by cal-
culating the premiums based on the values of average temperature simulated with
Monte-Carlo method.
The structure of the paper is as follows. Section 2 presents a theoretical analysis
of the weather derivative market. It also defines how the premiums are calculated
and what the possible representations of the loss function are. In Section 3 univariate
structural time series models on temperature are discussed. The empirical results
are given in Section 4.
2
Chapter 1
Introduction to weather
derivatives
In the following section, the market of weather derivatives is described and the place
of insurance contracts on this market. We present different approaches to define the
premium of an insurance contract and we also give representations of possible loss
functions. For a potential claimant, there are several ways to include the expected
loss caused by weather changes. In this thesis, we focus on the insurance policies
that account for some threshold, crossing which may lead to losses. This logic is
similar to barrier options and, indeed, insurance policies constructed in this thesis
have some features of this type of derivatives.
1.1 Weather derivatives market
The market for weather derivatives appeared around 20 years ago that is relatively
short term compared to other markets of financial instruments. Initially weather
as an asset appeared in individually constructed over-the-counter (OTC) contracts
mainly between some companies of the energy sector. The main purpose of such
negotiations was risk-management and so hedging against possible losses. This reason
still can be considered as the main driver of the markets growth.
3
The increase of the demand for the OTC contracts on weather spurred the need to
create standardized derivatives as well as to establish a market that would provide
both some liquidity and regulation. Consequently, Chicago Mercantile Exchange
(CME) introduced weather derivatives in 1999. Nowadays, CME is the biggest
market that offers such types of contracts 1 mainly on CDD and HDD indexes 2.
Derivatives currently traded on CME are Monthly Futures, Seasonal Strip Futures,
Monthly Options, Seasonal Strip Options.
The contracts mentioned above are built on the meteorological data of a quite
limited number of cities. Precisely, there are weather derivatives for New York,
Chicago, Atlanta, Cincinnati, Dallas, Sacramento, Las Vegas, Minneapolis, London
and Amsterdam. In fact, the market for such instruments is rather illiquid and
cannot cover the needs of all the weather dependent sectors.
An alternative solution to cover risks potentially caused by climate changes is
presented by insurance contracts on weather (that we consider as a specific type of
weather derivatives). It has to mention, that the nature of such weather derivatives
is far from standard insurance instruments. For standard insurance contracts, an
issuer has a portfolio of clients insured against the same risk and some historical
data on the claims. To evaluate such contracts, an issuer can use actuarial approach
taking into account statistical distributions of claims and payments. From the point
of view of an individual company the probability that a risky event occurs is rather
low. Yet, in this thesis we consider the contracts uniquely constructed for the needs
of the specific sector and the underlying for which is not an asset, but the weather.
Insurance contracts on weather are historically on demand in agriculture. Weather
changes have a direct and indirect impact on crop yields, need of fertilizers, livestock,
storage life and quality of the production. Events against which farmers usually
consider the purchase of the insurance contracts are temperature changes, extreme
rainfalls, draughts, the number of sunny days etc. Other sectors may be interested
1According to the Weather Product State on April 2016 from CME group web-site:
http://www.cmegroup.com/trading/weather/files/updated-weather-product-slate.pdf2See chapter 2.5 for a detailed explanation about the weather indexes
4
in the derivatives of this type are energy and tourism.
1.2 Premuim calculation
There are various issues that prevent spread usage of these contracts. For instance,
if a farmer requires the creation of unique derivative that covers risks exceptionally
present in his enterprise, an issuer may not have a portfolio of similar cases. Conse-
quently, the premium of such a contract will be quite the same as expected losses.
The farmer may anyway stick to the insurance contract when his own expectation
of the losses is higher than those of the issuer.
Evaluation of the price for such an insurance contract is rather different from
the approach generally used to price non-life insurance contract. As it is mentioned
above, an issuer usually attributes an incoming contract to some portfolio of similar,
but independent contracts. The price of the derivative depends on the equivalence
premium that accounts for the historical number of claims, the time-pattern of a
claim, the amount of the reimbursement, i.e. the loss of an issuer. The issuer also
takes into account competition on the market expressed by services and prices offered
by other insurance companies.
The price of an insurance contract that policyholder has to pay is the premium.
In this thesis, we will apply three different premium calculation principles on the
contracts: the pure premium principle, the standard deviation principle and the
risk-adjusted premium principle.
The pure premium principle is the expectation of losses Y of a policyholder:
Πpure = E[Y ] (1.1)
In order to take into account the variation of the expected loses the standard
deviation principle can be applied:
Πdev = E[Y ] + ϑ1(V ar[Y ])1/2 (1.2)
5
where ϑ1 > 0 is a safety loading that accounts for the volatility risk.
The safety loading can be also applied to adjust the distribution of the expected
loss. It may be useful when the historical data is not available or when there is any
other reason to assume that the real data-generating process has heavier tails. In
such cases the safety loading is directly applied on the probability:
Πrisk.adj =
∫ ∞0
[Pr(Y > y)]1/ϑdy =
∫ ∞0
[1− F (Y )]1/ϑ2dy (1.3)
where ϑ2 > 1. Equation 1.3 can be alternatively shown in discreet form as:
Πrisk.adj =∞∑i=0
p1/ϑ2Yi
Yi (1.4)
In the next sections, we will apply these 3 methods to calculate premiums for
different insurance policies.
1.3 Asian-style insurance policy
In order to create an insurance contract, it is important to make some assump-
tions about a possible loss function. Dealing with temperature as an underlying of
derivatives, losses are assumed to appear when the real grade is rather different from
the expected one. An example of exotic derivatives is barrier options that may be
activated when the price of an underlying crosses some barrier. Asian option is a
particular type of exotic options that compares the average temperature during the
maturity of the contract to some fixed level of the temperature.
In the literature, Asian options are discussed among others by L.C.G. Rogers and
Z. Shi (1995), J. Aase Nielsen and Klaus Sandmann (2003) etc. In this thesis, we
implement a loss function similar to the payoff function of an Asian option.
Let us consider a situation when a policyholder is sensitive to the average tem-
perature. The loss function l(XT ) is assumed to depend on how much the average
temperature XT during the time interval [0,T] exceeds some predefined level of tem-
perature K,
l(XT ) = L ·max(XT −K, 0) (1.5)
6
where L is the loss per 1 ◦C of temperature increase and XT is computed
Analogically, in case a policyholder suffers losses on the decreasing temperature,
the loss function is:
l(XT ) = L ·max(K − XT , 0). (1.6)
A contract built with this approach can be of interest in cheese making, viticulture,
sausage making and other agriculture spheres that have a maturity stage in the
production cycle. In fact, deviations from the average temperature are unlikely to
destroy the production, but it can significantly affect the quality of the output. For
instance, the sort of wine is dependent on the temperature during most of the stages
of wine making process such as harvesting, crushing and pressing, fermentation,
clarification, aging and bottling. Relatively small fluctuations can affect the amount
of sugar in grapes and life of bacteria vital for the wine fermentation. Not only a
producer may not obtain expected quantity, but also the ready product may not
meet its quality standards.
1.4 Parisian-style insurance policy
Another example of a loss function with a barrier is implemented in Parisian
options. The payoff of a Parisian exotic option does not only depend on the fact
whether an underlying crosses some predefined barrier, but it also takes into account
the duration over or under this barrier. In the literature, such derivatives are de-
scribed by Marc Chesney, Monique Jeanblanc-Picque and Marc Yor (1997), Angelos
Dassios and Shanle Wu (2011), etc.
Let us consider a case when a company tries to minimize the costs that could
be potentially caused by extreme temperatures so that either too high or too low
temperature may lead to major loss of the production. For instance, in agriculture,
there may occur a situation such that a proportion of the total crops will be destroyed
each day out of maximum D days that the temperature stays above (or below)
some value H. Time of excursion is the duration of the period of time when the
7
temperature is above the barrier. It can be shown as a function D(Xt, t):
D(Xt, t) =
0, if Xt < H
t− gt, if Xt ≥ H
(1.7)
where gt is the time when temperature Xt reaches barrier H.
gt = sup {τ ≤ t|Xτ = H} (1.8)
Taking into account how far from the barrier the temperature may get, the loss
l(XT , D) can be defined as:
l(Xt, D) =
L ·max(Xt −K, 0), if D(Xt, t) ≤ D,D ≤ t ≤ T
0, otherwise
(1.9)
where L is the loss per 1 ◦C of temperature increase, t ∈ [0, T ].
The loss function described are used to price Parisian-style insurance contracts.
1.5 Insurance policy based on weather index
One of the questions that arises with the need of accounting for weather-related
risks is what exactly should (or even could) be used as an underlying asset. This
problem was solved by the introduction of indexes on the weather. Such indexes can
be built on snowfall, rain, wind speed, etc. Yet the most widely spread are contracts
built on the temperature indexes such as CDD and HDD.
Heating degree days (HDD) is the difference between the temperature measured
at some day and some base temperature (usually 18 ◦C). Cumulative Heating degree
days is the sum of all the deviations from the base level during certain period of time
and it can be represented by the following:
HDD =T∑t=1
max{Xt − 18, 0} (1.10)
Cooling degree days (CDD) is an analog index for the days during which daily
average temperature Xi was below some certain level. And it is defined by the
8
following:
CDD =T∑t=1
max{18−Xt, 0} (1.11)
The loss function can also take into account the deviations from some fixed thresh-
old K instead of using predefined threshold of 18 ◦C. In this thesis, we calculate
expected values of CDD and HDD indexes from simulated series. The loss function
of an indexed-based insurance contract is the same as the payoff function of a stan-
dard option. In order to price such contracts, time to maturity and strike have to be
specified.
Let L be some loss that accrues when the index crosses the threshold, let CDDstrike
and HDDstrike be strikes, than the loss functions based on CDD and HDD indexed
are shown:
lCDD = L ·max(CDD − CDDstrike, 0) (1.12)
lHDD = L ·max(HDDstrike −HDD, 0) (1.13)
In fact, it is hard to imagine practical usage of insurance contracts on weather
indexes, as a potential insurance holder may not easily interpret their value. However,
we create such insurance contracts as alternative instruments to weather derivatives
traded on CME.
9
Chapter 2
Models for temperature
In a univariate structural time series models, the dynamics is decomposed in four
factors: trend, seasonality, cyclicality, and residual. A trend represents a general
tendency in the series. This tendency is present throughout all the time interval of
the observations. Seasonality has a repetitive nature that lasts for some limited time
period. Cyclicity is similar to seasonality, yet it may not exhibit fixed period pattern.
The final components of the structural time series is residual that serves to explain
all the irregularities that other components failed to capture. In case of weather time
series the proper construction of the seasonal component is particularly important.
In the following, the specification of some GARCH type models are studied in
order to apply them to weather time series analysis. We also present the theoretical
notion of Fourier analysis used to detect the seasonal components in the time series.
2.1 Fourier analysis of the temperature
One of the methods used to detect the seasonal components in time series is
Fourier analysis. More specifically, the discrete Fourier transformation allows
a researcher to find the most relevant frequencies that would explain the periodic
behavior of the series. The transformation provides the frequency domain represen-
tation of time series instead of the time domain representation which is commonly
10
used in time series analysis. As a result, it allows to determine at what time certain
frequency occurs. To apply Fourier transformation the power spectrum function has
to be constructed as following:
s(fj) =1
2π
∞∑τ=−∞
γje−ifjτ (2.1)
where γj is the autocovariate of the order j; i=√−1 and the Fourier frequency :
fj =2πj
T, j = 0, 1, ..., T − 1 (2.2)
Applying the theorem of De Moivre one can show that :
e−ifjτ = cos(fjτ)− isin(fjτ) (2.3)
Substituting the result (2.3) into the equation (2.1) and the stationarity condition
of the time series: γj = γ−j one can obtain and the power spectrum:
s(fj) =1
2π
∞∑τ=−∞
γj (cos(fjτ)− isin(fjτ))
=1
2π
(γ0 +
∞∑τ=−∞
γj [cos(fjτ) + cos(−fjτ)− isin(fjτ)− isin(−fjτ)]
) (2.4)
due to the properties of the trigonometric functions cos(θ) = cos(−θ) and −sin(θ) =
sin(−θ) :
s(fj) = 12π
(γ0 + 2
∞∑τ=−∞
γjcos(fjτ)
)(2.5)
The sample power spectrum is used at Fourier transformation, instead of the
theoretical power spectrum one. The sample spectrum I(fj) is found in a similar
way as to find the power spectrum. The difference is that in order to obtain the
sample power spectrum one should use the autocovariances γj of the series:
I(fj) = 12π
(γ0 + 2
T−1∑τ=1
γjcos(fjτ)
)(2.6)
11
After the frequencies are calculated, the models seasonal component can be given as
the combination of the following harmonic functions :
ωt =K∑k=1
(λ1k cos(fkt) + λ2k sin(fkt)) (2.7)
where K is the number of picks in the magnitude spectrum, i.e. the appropriate
number of combinations of the harmonic functions (see Appendix A for the MatLab
function).
As an example, let us simulate a time series that includes an autoregressive part
and the combination of simple harmonic functions:
yt = 0.7yt−1 − 0.36yt−2 + (0.7cos(f1t) + 1.2sin(f1t))+
+(−5.6cos(f2t) + 0.09sin(f2t)) + εt, εt ∼ N(0, 1)(2.8)
for t ∈ [1,1500].The frequencies (f1 and f2) of the harmonic components of the time
series are f1=0.02 and f2=0.075.
Assuming that the true data generating process is unknown, the discrete Fourier
transformation is used in order to determine the frequencies of the harmonic compo-
nent(see Appendix A for the MatLab code). From the graph of magnitude spectrum
( Figure ?? ) one can see that there are present two pics at the frequency bins 5 and
19. The values corresponding to the frequencies bins found are 0.0168 and 0.0754.
These values approximately equal to the frequencies of the harmonic components
specified in (2.8).
Figure 2.1: The magnitude spectrum of the observations versus the frequencies
12
2.2 GARCH model with seasonal components
Due to the nature of the underlying asset i.e. the average temperature, periodic
(or seasonal) components and possibly a nonlinear trend should be considered as
explanatory variables. A similar model was specified by Campbell and Diebold
(2002).
Let Xt for t=1,...,N be the average daily temperature. The mean equation is:
Xt = ωt + µt +I∑i=1
αiXt−i + σtεt; εtiid∼ (0, σ2
ε) (2.9)
The autoregressive part is included in the mean model, while the variables µt and
ωt represent trend and seasonal components, respectively. The trend is given in the
form of M-th order polynomial as following:
µt =M∑m=1
γmtm (2.10)
The seasonal component is assumed to be present due to the nature of the weather.
One of the possible ways to capture it in time series is by constructing external
variables both in the equation for mean and variance. The external variables are
well defined by the simple harmonic functions as it is shown in (2.7).
The model for variance is a periodic GARCH(P,Q) that includes the external
regressor $t that is the component of seasonal nature driving volatility.The spectral
analysis of the squared values of the underlying was performed in order to determine
the frequencies fvar present in the model for the volatility. fvar has the same structure
as the frequency of the mean model specified in (2.2). So the variance model is given
by:
σ2t = ρ+$t +
P∑p=1
βpσ2t−p +
Q∑q=1
ϕq(σt−qεt−q)2 (2.11)
In the study two models of this type are analysed:
Model I: I=14, Kmean = 1, M=1, P=6,Q=0,Kvar = 1;εt ∼ N(0, 1);
Model II: I=14, Kmean = 1, M=1, P=6,Q=2,Kvar = 1;εt ∼ St(ν).
13
2.3 GARCH in mean model
GARCH in mean model (GARCH-M) is one the extensions of classical generalized
autoregressive conditional heteroskedasticity model (Bollerslev 1986).This model un-
derlines a possible correlation between a variance and a mean. It allows not only the
effect of the conditional variance on the mean but also the direct effect of variance.
So that the variance appears two times in the model specification: the direct effect is
represented in terms of an additional covariate in the mean equation, and standard
deviation is present in the error term.
The motivation to use GARCH-M model comes from the assumption about the
dependence between a variance and a mean mentioned above.
GARCH-M (P,Q) has the following representation :
Xt = ωt + µt +I∑i=1
αiXt−i + δσ2t + σtεt (2.12)
σ2t = ρ+
P∑p=1
βpσ2t−p +
Q∑q=1
ϕq(σt−qεt−q)2; εt
iid∼ (0, σ2ε) (2.13)
The specifications of the fitted models of this type are:
Model III: I=14, Kmean = 1, M=1, P=3,Q=0,εt ∼ N(0, 1);
Model IV: I=14, Kmean = 1, M=0, P=1,Q=0,εt ∼ St(ν).
2.4 Exponential GARCH model
Another way to model the correlation between a mean and a variance is by
exponential GARCH model (EGARCH). Precisely, it takes into account a possible
asymmetric relationship between the value of the mean model and the magnitude of
the volatility. It preserves the conditions of the non-negativity of the variance and
zt = σtεt is modeled in the way such that it includes both the magnitude and the
sign effect of the variance:
g(zt) = θzt + λ(|zt|−E(zt)) (2.14)
14
where zt corresponds to the sign effect and the expression (|zt|−E(zt)) for the mag-
nitude.
So the EGARCH(P,Q) model is the following:
Xt = ωt +I∑i=1
αiXt−i + δlog(σ2t ) + zt (2.15)
where log(σ2t ) is logarithm of the GARCH in mean component;
log(σ2t ) = ρ+$t +
P∑p=1
βplog(σ2t−p) +
Q∑q=1
ϕqg(zt−q); εtiid∼ (0, σ2
ε) (2.16)
The specifications of the fitted models of this type are:
Model V: I=6, Kmean = 1, P=2,Q=0,Kvar = 1, εt ∼ N(0, 1);
Model VI: I=6, Kmean = 2, P=1,Q=0,Kvar = 1, εt ∼ St(ν).
15
Chapter 3
Empirical results
In the following, we present the results of fitting the average temperature on the
models described in the previous chapter. The preliminary descriptive data analysis
and spectral analysis are given as well as the goodness of fit is evaluated. In the end,
we make ex-post forecasts for each of the GARCH models and a forecast based on a
model combination approach.
3.1 Data and descriptive statistics
The data used in this thesis is the average daily temperature in London and it
has been extracted from the real-time weather information provider Weather Under-
ground. The decision to use the observations from London is driven by the fact that
there are some weather derivatives traded on Chicago Mercantile Exchange written
on the temperature from this city. So that it may be possible to price derivatives
similar to those currently traded in order to evaluate the adequateness of our pricing
methodology.
The sample consists of the observation from 01/01/2010 till 02/02/2016 i.e.
2224 observed days. The part of the sample from 01/01/2010 till 02/02/2015 is used
for the fitting purpose, while another part from 02/02/2015 till 02/02/2016 is used
for evaluating the goodness of forecast and selecting the best fit model.
16
Figure 3.1: The graph of the observations and the histogram of the distribution
Table 3.1: Summary of the Series
Mean Med Max Min Std.Dev Skew Kurt JB P-Val. Num
Xt 11.233 11.000 29.000 -4.000 5.684 -0.108 2.368 41.320 0.000 2224
Table 3.2: Augmented Dickey-Fuller Test, Results
Variable T-Statistic %1 C.V. %5 C.V. %10 C.V P-Value
Xt -5.078086 -3.433102 -2.862642 -2.567402 0.0000
From the graph of the observations (see Figure 3.1) one can clearly see the
seasonal nature of the time series. The main descriptive statistic is given in Table 3.2
and the histogram of the distribution are shown in Figure 3.1. According to Jarque-
Bera test statistics, we cannot accept the null hypothesis that the data is normally
distributed. From the histogram of the distribution one can conclude multimodality,
precisely, three modes are observed.
17
(a) ACF of the observations (b) PACF of the observations
(c) ACF of the squared observations (d) PACF of the squared observations
Figure 3.2: ACF and PACF of the observations
The stationarity of the given series is checked by applying Augmented Dickey-
Fuller unit root test (the results are shown in Table 3.2). The hypothesis of the unit
root is rejected that means that the series is stationary.
In order to justify the possible use of the AR part in the mean equation of
GARCH model, the presence of serial autocorrelation has to be controlled. The
serial autocorrelation is tested with the Ljung-Box-Pierce Q-Test (see Table 3.3 and
Figure 3.2 ). Graphs of ACF and PACF functions show the pattern typical for the
autoregressive process and the results of the Ljung-Box-Pierce Test prove that the
null hypothesis of independence in a given time series is rejected. Similarly, the ACF
and PACF functions of the squared observations exhibit the behavior common for AR
models. Consequently, all it calls for the use of models that includes autocovariates
both equations for mean and variance.
18
Table 3.3: Ljung-Box Test, Results
Lags Statistic df P-Value
5 3535.101 5 0
10 6946.427 10 0
15 10235.461 15 0
20 13403.745 20 0
25 16452.874 25 0
30 19384.499 30 0
3.2 Spectral analysis
Spectral analysis helps us to determine the frequency of the harmonic functions
with which the seasonality can be possibly explained. It is done by applying the
discrete Fourier transformation on the series. The procedure is formulated in (2.1) -
(2.6).
In order to interpret the result of the Fourier transformation it is important to
remember that the transformation rescales all the data from the time domain into
the domain in radians such that the whole cycle 2π is divided by the total number
of the observations in the sample as it is given in equation (2.6). For the sake of
simplicity the rescaling in terms of years ( the sampling frequency) is also used, so
that the result obtained in radians can be easily interpreted :
fyear =j
fsamp, j = 1, ..., T (3.1)
where fyear is in terms of the sampling frequency, fsamp is the sampling frequency
i.e. the duration of a year in days and T is the total number of the observations in
the sample.
The frequency given in terms of years is interpreted as the effect of the harmonic
function occurs every fyear years. Then the frequency per year (that shows how often
19
the effect of the harmonic function occurs in 1 year) can be also defined:
dcycle =T
fyearfsamp(3.2)
fp.y =dcycle365
(3.3)
where dcycle is the duration of 1 cycle in days; fp.y is the frequency per year.
In order to define the seasonal component in the mean and in the variance, the
spectral analysis is performed on the values of the sample and its squared values
respectively (see Appendix B for the MatLab code).
(a) Magnitude spectrum (b) Log transformation of the magnitude spec-
trum
Figure 3.3: Fourier analysis of the observations a) Magnitude spectrum; b)Log trans-
formation of the magnitude spectrum
Analyzing the magnitude spectrum of the observations (see Figure 3.3) one can
conclude that the magnitude effects are present on the frequency bin 7 and on the
frequency bin 14. The magnitude of the last one is less significant, though.
The transformation of the results obtained in the bins of the frequency into the
frequency in terms of the frequency per year and in radians is given in Table 3.4.
According to the results obtained, the seasonal component can be expressed with two
simple harmonic functions such that the whole cycle of one is of one year and another
is twice in a year. The graphical representation of sine functions built with the both
20
frequencies found and sine function that is the combination of those mentioned above
is given in Figure 3.4. The combination of both frequencies may be able to explain
multimodality present in the distribution of the observations (see Figure 3.1).
Figure 3.4: Graph of the observations and the harmonic functions
Analogically, the frequencies present in the seasonal component for the equation
of the variance is studied. In this case, discrete Fourier transformation is applied to
the squared observations. The magnitude spectrum is given in Figure 3.5.
21
(a) Magnitude spectrum (b) Log transformation of the magnitude
spectrum
Figure 3.5: Fourier analysis of the squared observations a) Magnitude spectrum;
b)Log transformation of the magnitude spectrum
Table 3.4: Fourier Transformation, Results
bin number frequency fp.y f(radians) duration of the cycle dcycle
fr1 7 1.0155 0.0170 370.6667
fr2 14 0.4687 0.0367 171.0769
fr3 13 0.5078 0.0339 185.033
The results are similar to the previously obtained: as before the seasonal
component can be expressed with two simple harmonic functions one with the cycle
of one year and another with half a year cycle. The last weather component though
has slightly smaller frequency fr3 (see Table 3.4). The graphical representation of
sine functions built with the both frequencies found and sine function that is the
combination of those mentioned above is given in Figure 3.6.
22
Figure 3.6: Graph of the squared observations and the harmonic functions
3.3 Model estimation
In order to estimate a coefficient on the explanatory variables the quasi-maximum
likelihood (QML) method is applied.
In the models I, III, V the residuals are assumed to follow a standard Gaussian
distribution. Let us denote the vector of unknown parameters as
θ = (λ11, ...λ1K , λ21, ...λ2K , γ1, ...γ1M , α1, ...αI ,
ρ, λ31, ...λ3Kvar , λ41, ...λ4Kvar , β1, ...βP , ϕ1, ...ϕQ)(3.4)
where K,M,I,Kvar,P,Q are given by model specification.
Conditionally on all the initial valuesX0, ..., Xt−I , σ2t−P , ..., σ
20, εt−Q, ...ε0,the Gaus-
23
sian quasi-likelyhood function is:
L(θ) =T∏t=1
1√2πσ2
t
exp
(− ε2
t
2σ2t
)(3.5)
To make computation easier the log-likelyhood function can be used :
lnL(θ) = −T∑t=1
1
2
(log(2π) + log(σ2
t ) +ε2t
σ2t
)(3.6)
Residuals of the models II,IV,VI are assumed to be of Student’s t distribution. In
this case, we have to use the relevant conditional quasi-likelihood function. Precisely,
it was defined by Bollerslev (1987) in the following form:
Ln(θ) =T∏t=1
Γ[12(υ + 1)]
π12 Γ[1
2υ]
[(υ − 2)σ2
t
]− 12
[1 +
(εtσt)2
(υ − 2)σ2t
]− 12
(υ+1)
(3.7)
where υ is the degrees of freedom parameter.
In order to find the coefficients θ the following maximization problem has to be
solved :
θQMLE = arg maxθ∈Θ
lnLn(θ) (3.8)
3.4 Model selection
For this study, 6 models of GARCH type are fitted on the weather time series.
Table 3.5 gives the detailed information about the external components specified for
each of the models and the assumption on the error terms. The symbol * indicates the
presence of the component in the model, µt corresponds to the trend component in
the mean (given in (2.10)), ωt and $t are seasonal components in mean and variance
respectively (see (2.7)), δ shows whether the variance is present in the specification
of the mean (GARCH-M model).
Having estimated the parameters of the models (see Table D.2 in Appendix D),
the qualitative analysis of the models explanatory and forecasting strength has been
done (see the Diagnostics of the Residuals, Appendix D).
24
Table 3.5: The components inclueded in the models
Model µt ωt $t δ Error distribution
Model I * * * Normal
Model II * * * Student’s t
Model III * * * Normal
Model IV * * Student’s t
Model V * * * Normal
Model VI * * * Student’s t
Jarque-Bera test on normality evidences that the residuals of the models I,III,V
are not normal (see Table 3.6). These results violate the assumptions of GARCH
models.
Table 3.6: Jarque-Bera test on residuals, Results
Model I III V
Jarque-Bera statistics 21.22133 14.21267 15.57638
P-value 0.000025 0.000820 0.000415
In order to avoid the presence of heavy tails of the residuals distributions, the
error terms of the models II,IV,VI are assumed to be of the Student’s t distribution.
However residuals with Student’s t distribution have less heavy tails, it is shown on
Q-Q plots (see Figure 3.7) that some outliers are still present.
Despite the rejection of the null hypothesis about the normality of the residuals,
models show good statistical properties, so it is possible to perform ex-post forecast.
The part of the sample from 02/02/2015 till 02/02/2016 is used for evaluating the
goodness of the forecast and selecting the best fit model.
25
(a) Model II (b) Model IV
(c) Model VI
Figure 3.7: Q-Q plots: model residuals against Student’s t quantiles
26
(a) Model I (b) Model II
(c) Model III (d) Model IV
(e) Model V (f) Model VI
Figure 3.8: Actual,fitted,residual of studied models
Root mean square error (RMSE) is used to compare the forecasts made for the
same time series, but with different fitted models. The general rule to distinguish
the better model in terms of RMSE values is the smaller the RMSE the better the
27
predicting ability.
Theil coefficient is one of the forecast accuracy measures. Its values can possibly
lay between 0 and 1 where the closer to 0 the better the forecast. Theil coefficient
consists of three parts that represent the bias proportion, the variance proportion
and the covariance proportion. From the first two components, one can determine
the systematic error of the forecast and the accuracy of the prediction of the variance,
respectively.
The values of the coefficients mentioned above are listed in Table 3.7. Model
I and Model II are the models with the weakest forecasting accuracy in terms of
RSME and Theil coefficient.However, these models capture the maximum values of
the series (see Figure 3.9). Model III can be considered as the best model as it has
the smallest RMSE and Theil coefficient and shows the smallest systematic error.
Its ex-post forecast explains the seasonal behavior in the series but fails to catch
extreme values.
Table 3.7: Goodness of forecast, Results
Model I II III IV V VI
RMSE 5.906904 5.872250 2.964946 2.999786 2.988514 2.966667
Theil coeficient 0.239421 0.242125 0.118669 0.120621 0.120321 0.119474
Bias proportion 0.097271 0.123517 0.000014 0.001347 0.000889 0.002101
Variance Proportion 0.094629 0.058407 0.044618 0.052574 0.054530 0.042252
28
(a) Model I (b) Model II
(c) Model III (d) Model IV
(e) Model V (f) Model VI
Figure 3.9: Ex post forecast and credible intervals
29
3.5 Forecast combination
The analysis of the estimated models shows that the average value of the observation
is well forecasted. Model I and II are able to catch the maximum picks, while model
IV and V have the widest amplitudes. In the end, it is not possible to state that
there is a model that is definitely better compared to others. All it calls for the
consideration of a forecast combination.
In the literature the advantages and disadvantages of the forecast combination
approach have been widely discussed. One of the arguments against it is that by
combining different models a researcher somewhat diminished the importance of
single models. In this case, there may arise a controversy on whether it is better to
improve a single model or to use a composition of them.
On the contrary, the forecast combination may be a nice method to exploit all
the desired qualities of each single model. In fact, Bates and Granger (1969) showed
that the composite forecast of the models built on the separate information sets may
have higher forecasting accuracy. Moreover, Chan, Stock, Watson (1999) argued
that the improvements are possible even for the models that information sets are not
necessarily exclusive. Hsiao and Wanz (2011) affirmed that by combining different
forecasts, one increases the robustness against possible misspecification and errors.
The reason to combine the models in this thesis is to obtain a model with better
forecasting qualities. The new model is expected to be good in explaining the general
behavior of the series as well as the maximum values and the trend.
One of the ways to perform the forecast combination is by the construction of
a weighted sum of the other models. The most crucial part of this approach is the
assignment of weights. Stock and Watson (2001) use MSE as a defining parameter
to choose the appropriate weights. Precisely, let Xt+h,i be a forecast for h periods
ahead, performed by model i. In our case i ∈ [1,6]. Let υi be a weight assigned to
the model i and Xt+h be the forecast combination. The forecast combination is the
following:
30
Xt+h =6∑i=1
υiXt+h,i, (3.9)
υi =1/MSEi
6∑i=1
1/MSEi
(3.10)
The weights that can be assigned are given in Table 3.8.
Table 3.8: Weights for the model combination
Model Model I Model II Model III Model IV Model V Model VI
MSE 34.89151 34.48332 8.79090 8.99872 8.93122 8.80111
Weight 0.05640 0.05699 0.22356 0.21840 0.22005 0.22330
The new model has average forecast accuracy compared to previously forecasted
models (see Table 3.7 and Table 3.9). Analyzing the graphical representation of the
simulated series (see Figure 3.10), it captures the main behavior of the series and a
slight trend. The trend of this model seems to catch better the behavior of the series
compared to the trends in the Model I and II.
Table 3.9: Goodness of forecast, Result of Model combination
Model RMSE Theil coeficient Bias Proportion Variance proportion
Forcast combination 2.99372 0.12102 0.00990 0.039297
31
Figure 3.10: Forecast combination, Result
3.6 Pricing weather insurance contracts
Let us consider an agriculture company that is exposed to the losses related to
the change of temperature. The expected loss l per 1◦C for temperature increase
and decrease is taken the same as the value of a contract unit of weather derivatives
traded on CME i.e. L = 20 euro. In order to cover some losses, a policyholder can
also specify the number of contracts. In this thesis, we show the prices for a single
contract. Types of the contracts and their time to maturity are listed in Table 3.10.
In order to price contracts we used 30000 trajectories of the series simulated with
Monte Carlo method. The terms of possible insurance policies are the following:
Asian-style insurance policy would insure the losses if the average temperature
during the life of the contract reaches no more than the average temperature
of a specified strike volume during the same period of time. The model built
with model combination approach is used as an underlying model and Model
IV is used as a barrier for both call and put type of contracts.
32
Parisian-style insurance policy can insure the policyholder against extreme tem-
peratures. We use the model built with model combination approach as an
underlying model for all types of contracts. A model for the barrier depends
on whether the company wants to be issued against extreme temperature in-
crease or decrease. In other words, we use different barriers for short and long
contracts. Instead of using some fixed over the time strike volume, we take the
performance of a fitted to the series model as an upper or lower bound of the
contract. Specifically, Model II is assumed to be a proper upper bound and
Model VI is used as a lower bound. In such way, we assure that the barriers
are not constant over the time. A Parisian style insurance contract can cover
occurred losses provided if the underlying crosses the barrier and stays above
or below it for at least five consecutive days.
Insurance policy based on weather index would insure payments with respect
to HDD and CDD indexes on the daily average temperature, such that the
temperature base level is 18 ◦C. The indexes are calculated on the simulated
series of the model combination approach.
Figure 3.11: Simulated trajectories of the combined model over the time
33
Table 3.10: Description of the contracts
Name Type Maturity Start and end dates
March 2015 long 1 month [01.03.2015 - 31.03.2015]
July 2015 long 1 month [01.07.2015 - 31.07.2015]
August 2015 long 1 month [01.08.2015 - 31.08.2015]
September 2015 long 1 month [01.09.2015 - 30.09.2015]
December 2015 short 1 month [01.12.2015 - 31.12.2015]
February 2016 short 1 month [01.02.2016 - 29.02.2016]
March 2016 long 1 month [01.03.2016 - 31.03.2016]
Summer 2015 long 3 months [21.06.2015 - 20.09.2015]
Summer 2016 long 3 months [21.06.2016 - 20.09.2016]
Winter 2016-2017 short 3 months [21.12.2016 - 20.03.2017]
Results of pricing Asian-style contracts are shown in Table 3.11 and the prices
for Parisian-style contracts are listed in Table 3.12. One can see whether a contract
covers the expected losses of a policy holder by analyzing profit of a contract Y :
Y = l(Xt)− Π (3.11)
where l(Xt) is a loss function described in (1.9) and (1.5) (it depends on an insurance
policy applied to the contract) and Π is premium paid by insured party. In order
to calculate final profit or loss of the policyholder, we take the minimum among the
premiums as a price of insurance contracts. Negative profit of a contract such that
Y < 0 shows that the event that potentially could have caused the losses, did not
occur. Using the historical data, the payoffs and profits for the contracts up to May
2016 were calculated.
As one can see the prices of Asian-style contracts are usually more expensive
than those of Parisian-style contracts. It is a logical consequence as there are more
trajectories that can cross an average barrier than those that can cross some upper
or lower bound.
34
Table 3.11: Prices of Asian-style contracts
Πpure Πdev Πrisk.adj Payoff Profit/loss
March 2015 14.171 19.545 23.151 42.471 28.301
July 2015 46.500 55.539 75.971 0 -46.500
August 2015 51.866 61.121 84.738 0 -51.866
September 2015 55.966 65.401 91.436 0 -55.966
December 2015 0.727 2.045 1.187 71.317 70.591
February 2016 0.802 2.222 1.311 31.248 30.446
March 2016 66.177 77.071 108.119 0 -66.177
Summer 2015 49.522 57.903 80.908 0 -49.522
Summer 2016 87.313 96.347 142.651 n/a n/a
Winter 2016-2017 0.027 0.240 0.044 n/a n/a
Index-based contracts have quite different construction of a loss function, so we
consider them separately from other two types of the derivatives. CDD and HDD
indexes are publicly available on CME and some standard derivatives are traded on
the indexes. We, in fact, represent an alternative weather derivative contract of this
type. As an example, we priced contracts on the hottest on the coolest months in
a year that are July and February respectively. The result of pricing is shown in
Table 3.13.
Prices of Index-based contract not only depend on the temperature performance
and the base level of the temperature (18◦C), but also on a strike price. This makes
these insurance contracts more similar to standard derivatives. The value of indexes
may not be intuitively clear to an agriculture company, as neither one-time variation
from some temperature K nor the cumulative number of such variations may cause
any significant loss. However, the indexes are important to determining the need for
heating or cooling, so this type of contracts are more applicable in the energy sector.
35
Table 3.12: Prices of Parisian-style contracts
Πpure Πdev Πrisk.adj Payoff Profit/loss
March 2015 20.167 21.080 32.948 41.043 20.876
July 2015 5.790 6.596 9.458 0 -5.790
August 2015 3.789 4.656 6.190 0 -3.789
September 2015 2.556 3.510 4.175 0 -2.556
December 2015 4.108 5.280 6.710 0 -4.108
February 2016 1.281 2.482 2.094 1.281 10.272
March 2016 1.377 2.516 2.250 0 -1.377
Summer 2015 5.650 6.617 9.231 0 -5.650
Summer 2016 0.253 1.282 0.414 n/a n/a
Winter 2016-2017 0.221 1.359 0.361 n/a n/a
Table 3.13: Prices of Index-based contracts
February 2016 CDD call February 2016 CDD put
Strike Πpure Πdev Πrisk.adj payoff Πpure Πdev Πrisk.adj payoff
300 27.287 37.926 44.582 0 17.210 25.772 28.118 55
350 7.480 13.089 12.221 0 47.403 60.659 77.447 5
370 3.816 7.743 6.234 15 63.739 78.292 104.135 0
390 1.781 4.382 2.909 35 81.703 97.132 133.486 0
410 0.760 2.388 1.242 55 100.683 116.639 164.495 0
July 2015 HDD call July 2015 HDD put
Strike Πpure Πdev Πrisk.adj payoff Πpure Πdev Πrisk.adj payoff
30 22.193 31.369 36.258 0 11.425 16.235 18.666 8
70 12.983 20.129 21.212 32 22.216 29.288 36.296 0
90 6.942 12.128 11.341 52 36.174 45.168 59.101 0
110 3.348 6.859 5.471 72 52.581 63.020 85.906 0
130 1.443 3.661 2.357 92 70.675 82.065 115.469 0
36
Conclusion
Over the last few years, the average temperature has been increasing. If this trend
continues, many economic sectors will become exposed to additional costs and losses.
Such companies as agricultural may even reconsider changes to its production tech-
nologies. In a short term, though, purchasing some insurance contracts on weather
could cover losses.
In this thesis we created insurance contracts on the temperature that include
some hybrid characteristics of both standard derivatives (such as exotic options) and
insurance contracts (in terms of the premium calculation). First of all, we modeled
the underlying (in our case it is average daily temperature) so that it would be
possible to simulate future trajectories of the series. However, instead of geometric
Brownian motion process that is often used to price derivatives, our models are of
ARMA-GARCH type. Consequently, pricing such contracts requires an approach
alternative to Blach-Scholes.
Some underlying models we used are similar to the one presented by Sean D.
Campbell and Francis X. Diebolt (2005), as well as other specifications of ARMA-
GARCH models, are studied to explain and simulate the series. Model II is a GARCH
model with seasonal components in mean and variance with students’t distribution
of errors is the best model to catch the extreme picks. For that reason, it is used in
pricing Parisian-style insurance contracts. The model build with model combination
approach has an average performance both in term of simulated values so that we
use it as a model for the underlying.
Asian-style insurance contracts may be the most interesting for a potential policy
37
holder as exposure to the risk cased by the average temperature increase is increasing
every year. What is more, this contracts will also account for the extreme jumps
in temperature. Parisian-style insurance contract has a more limited application,
however, it is a cheaper alternative to Asian-style contracts.
38
Appendices
39
Appendix A
Discrete Fourier transformation,
MATLAB function
%% f f t c o v c o s i s the d i s c r e t e Four ie r transform (DFT)
% of au tocova r i a t e s o f vec tor X
func t i on [X,X mag , X phase , k ,w]= f f t c o v c o s (x )
T=length (x ) ;
w=0:(1/T)∗2∗ pi :2∗ pi ;
k=length (w) ;
%cons t ruc t c ova r i a t e s
f o r i i =1:T 2 % at T 1 we have only cov (num1 , num2)=0 , so no need
cov matr=cov (x ( 1 :T i i ) , x(1+ i i :T) ) ;
autocov ( i i )=cov matr ( 1 , 2 ) ;
end
f o r l =1:k %number o f f requency bin
X sum=0;
f o r m=1:T 2 %number o f the autocov
X temp=autocov (m)∗ cos (w( l )∗m) ;
X sum=X sum+X temp ;
end
X( l )=(1/2∗ pi )∗ ( var (x)+2∗X sum ) ;
end
X mag=abs (X) ;
X phase=angle (X) ;
40
Appendix B
An example of Fourier analysis on
simulated series, MATLAB script
c l e a r
c l c
t =1:1500;
T=length ( t ) ;
w=l i n spa c e (0 ,2∗ pi ,T) ; %frequency in rad iance
%s imulat i on o f AR proce s s with harmonic component
f1 =0.02;
f 2 =0.075;
a0=5.3;
a1=0.7;
a2=1.2;
a3 = 5 . 6 ;
a4=0.09;
ar=arima ( ’ Constant ’ , 0 . 5 , ’AR’ , { 0 . 7 , 0 . 3 6 } , ’ Variance ’ , . 1 ) ;
x0=s imulate ( ar , 1 5 0 0 ) ’ ;
x=a0∗x0+(a1∗ cos ( f1 ∗ t)+a2∗ s i n ( f1 ∗ t ))+( a3∗ cos ( f2 ∗ t)+a4∗ s i n ( f2 ∗ t ) ) ;
X=f f t c o v c o s (x ) ;
X mag=abs (X) ;
f i g u r e (1 )
p lo t (X mag ) ;
t i t l e ( ’ Magnitude spectrum of the observat ions ’ ) ;
bin1=6; % from the graph o f the magnitude spuctrum
bin2=19;
f r 1=w( bin1 ) ;
f r 2=w( bin2 ) ;
harm1=cos ( f1 ∗ t)+ s in ( f1 ∗ t ) ;
harm2=cos ( f2 ∗ t)+ s in ( f2 ∗ t ) ;
harm3=harm1+harm2 ;
f i g u r e (2)
subplot (4 , 1 , 1 )
p lo t (x , ’ blue ’ )
t i t l e ( ’ Simulated time s e r i e s ’ )
41
hold on
subplot (4 , 1 , 2 )
p lo t (harm1 , ’ red ’ )
t i t l e ( ’ Harmonic component 1 ’ )
hold on
subplot (4 , 1 , 3 )
p lo t (harm2 , ’ red ’ )
t i t l e ( ’ Harmonic component 2 ’ )
hold on
subplot (4 , 1 , 4 )
p lo t (harm3 , ’ red ’ )
t i t l e ( ’The combination o f the harmonic components ’ )
42
Appendix C
Spectral analysis,MATLAB script
%% SPECTRAL ANALYSIS OF THE OBSERVATIONS
c l e a r
c l c
f i l e p a t h 1 = ’D:\ un i v e r s i t y \ !\ Thes is\Data\dataLondon . csv ’ ;
data1=load data ( f i l e p a t h 1 ) ;
T=length ( data1 ) ;
s f =365; %sampling f requency
w=l i n spa c e (0 ,2∗ pi ,T) ; %frequency in rad iance
f=1/ s f : 1/ s f :T/ s f ; %r e s c a l e data wrt to sampling frequency
t=1:T;
f r=w( t ) ;
[X,X mag , X phase , k ,w]= f f t c o v c o s ( data1 ) ;
f i g u r e (1 )
p lo t (X mag)
%t i t l e ( ’ Magnitude spectrum of the observat ions ’ ) ;
f i g u r e (2 )
X mag log=log (X mag ) ;
p l o t ( X mag log )
%t i t l e ( ’ Log t rans fo rmat ion o f the magnitude spectrum ’ ) ;
bin0=1;
bin1=7;% from the p lo t o f the magnitude spectrum
bin2=14;% from the p lo t o f the magnitude spectrum
cyc l e s 1=f ( bin1 )∗ s f ; % # of c y c l e s in the sample
durat ion=T/( f ( bin1 )∗ s f ) ; %durat ion o f 1 cy c l e
f r y e a r=durat ion /365 ; %frequency in year terms
f r 0=w( bin0 )∗ s f ;
f r 1=w( bin1 )∗ s f ; % frequency o f the harmonic s i n+cos func t i on
f r 2=w( bin2 )∗ s f ;
t e s t 1=cos ( f r 1 ∗ f )+ s in ( f r 1 ∗ f );% f o r s ea sona l component o f the from s in+cos
t e s t 2=cos ( f r 2 ∗ f )+ s in ( f r 2 ∗ f ) ; % f o r s ea sona l component o f the from s in+cos
t e s t 3=t e s t 1+t e s t 2 ;
f i g u r e (3 )
subplot ( 4 , 1 , 1 ) ;
p l o t ( data1 , ’ b ’ ) ;
t i t l e ( ’ Graph o f the observat ions ’ ) ;
ax i s ( [ 0 , 2204 , 5 , 3 0 ] ) ;
43
hold on
subplot (4 , 1 , 2 )
p lo t ( te s t1 , ’ r ’ ) ;
t i t l e ( ’ Harmonic component 1 ’ )
ax i s ( [ 0 , 2204 , 1 . 5 , 1 . 5 ] ) ;
hold on
subplot (4 , 1 , 3 )
p lo t ( te s t2 , ’ r ’ ) ;
t i t l e ( ’ Harmonic component 2 ’ )
ax i s ( [ 0 , 2204 , 1 . 5 , 1 . 5 ] ) ;
hold on
subplot (4 , 1 , 4 )
p lo t ( te s t3 , ’ r ’ ) ;
t i t l e ( ’ Combination o f the harmonic components ’ )
ax i s ( [ 0 , 2204 , 3 , 3 ] ) ;
% SPECTRAL ANALYSIS OF THE SQUARED OBSERVATIONS
x1 sq=data1 . ˆ 2 ;
[X2 , X mag2 , X phase2 , k2 ,w2]= f f t c o v c o s ( x1 sq ) ;
f i g u r e (4 )
p lo t (X mag2)
%t i t l e ( ’ Magnitude spectrum of the squared observat ions ’ ) ;
f i g u r e (5 )
X mag log2=log (X mag2 ) ;
p l o t ( X mag log2 )
bin11=7;% from the p lo t o f the magnitude spectrum
bin21=13;% from the p lo t o f the magnitude spectrum
cyc l e s 11=f ( bin11 )∗ s f ; % # of c y c l e s in the sample
durat ion11=T/( f ( bin11 )∗ s f ) ; %durat ion o f 1 cy c l e
f r y e a r 1 1=durat ion11 /365 ; %frequency in year terms
f r 11=w( bin11 )∗ s f ; % frequency o f the harmonic s i n / cos func t i on
f r 21=w( bin21 )∗ s f ;
t e s t 11=cos ( f r 11 ∗ f )+ s in ( f r 11 ∗ f );% f o r s ea sona l component o f the from s in+cos
t e s t 21=cos ( f r 21 ∗ f )+ s in ( f r 21 ∗ f ) ;
t e s t 31=t e s t 1+t e s t 2 ;
f i g u r e (6 )
subplot ( 4 , 1 , 1 ) ;
p l o t ( x1 sq , ’ b ’ ) ;
t i t l e ( ’ Graph o f the squared observat ions ’ ) ;
ax i s ( [ 0 , 2204 , 0 , 6 0 0 ] ) ;
hold on
subplot (4 , 1 , 2 )
p lo t ( test11 , ’ r ’ ) ;
t i t l e ( ’ Harmonic component 1 ’ )
ax i s ( [ 0 , 2204 , 1 . 5 , 1 . 5 ] ) ;
hold on
subplot (4 , 1 , 3 )
p lo t ( test21 , ’ r ’ ) ;
t i t l e ( ’ Harmonic component 2 ’ )
ax i s ( [ 0 , 2204 , 1 . 5 , 1 . 5 ] ) ;
hold on
subplot (4 , 1 , 4 )
p lo t ( test31 , ’ r ’ ) ;
t i t l e ( ’ Combination o f the harmonic components ’ )
ax i s ( [ 0 , 2204 , 2 . 5 , 3 ] ) ;
44
Appendix D
Diagnostics of the residuals
Additionally to Jack-Berra test on the residuals given in section 3.4, in the fol-
lowing more profound analysis is shown. Particularly the hypothesis tested are the
significance of the explanatory variables with Student’s t-test; the absence of the
serial autocorrelation with Box-Pierce test and the homoscedasticity of the residuals
with White test.
In Figure D.1 one can see that all of the models show the absence of the serial
autocorrelation in residuals: as p-values of Box-Pierce test are always greater than
0.05 we accept the null hypothesis of the residuals independence.
Table D.1: White test, Results
Model I Model II Model III
Stat P-val Stat P-val Stat P-val
F-stat 1.030 0.389 1.120 0.273 1.181 0.073
obs R2 156.281 0.389 49.175 0.274 176.936 0.081
Model IV Model V Model VI
Stat P-val Stat P-val Stat P-val
F-stat 1.212 0.055 1.025 0.153 1.048 0.101
obs R2 161.149 0.062 153.251 0.272 173.610 0.336
45
The results of White test on heteroskedasticity suggest to accept the null hypoth-
esis of absence of the heteroskedasticity (see Table D.1).
The output of the estimation is given in Table D.2. Regarding Student’s t-test, we
tried to include only statistically significant covariates, yet it was not always possible.
For example in autoregressive part, we included from 6 to 14 lags in order to avoid
serial autocorrelation. It appeared quite often that the coefficients corresponding to
the autocorrelation of lag higher than 10 were significant even though the covariates of
smaller lags were not significant. It can be explained as an another sign of seasonality
in the series. The insignificance of the coefficients was also accepted when one of the
parts of the seasonal component given in (2.7) is significant while another is not.
As long as discarding one part of seasonal component (2.7) will contradict Fourier
theorem, we left it as it is. In other cases, insignificant single coefficients show joint
significance.
(a) Model I (b) Model II (c) Model III
(d) Model IV (e) Model V (f) Model VI
Figure D.1: Box-Pierce P-values
46
Table D.2: Estimation output
Model I
Variable Coefficient Std. Error z-Statistic Prob.
α1 0.816331 0.024008 34.00203 0
α2 -0.074036 0.029262 -2.530106 0.0114
α3 0.037956 0.029853 1.271417 0.2036
α4 -0.00538 0.031392 -0.171394 0.8639
α5 0.029624 0.031944 0.927356 0.3537
α6 0.072276 0.031051 2.327686 0.0199
α7 0.010346 0.030288 0.341599 0.7327
α8 0.00205 0.030532 0.067147 0.9465
α9 0.019006 0.030491 0.62332 0.5331
α10 -0.03781 0.03068 -1.232419 0.2178
α11 0.054 0.030664 1.761022 0.0782
α12 -0.012978 0.030046 -0.431934 0.6658
α13 0.016889 0.029027 0.581844 0.5607
α14 0.047825 0.023087 2.071479 0.0383
λ11 -0.266298 0.090805 -2.93264 0.0034
λ12 0.300571 0.070714 4.250531 0
γ 0.000211 8.77E-05 2.402942 0.0163
Variance Equation
ρ 1.598102 0.074747 21.38005 0
β1 1.277811 0.004503 283.792 0
β2 -1.018661 0.00388 -262.5523 0
β3 -0.096077 0.002536 -37.89183 0
β4 0.531661 0.005297 100.3698 0
β5 -0.333428 0.003557 -93.74113 0
β6 0.228513 0.003153 72.47252 0
λ21 0.302219 0.077317 3.908848 0.0001
λ22 -0.090097 0.074289 -1.21279 0.2252
(a) Estimation output: Model I
47
Model II
Variable Coefficient Std. Error z-Statistic Prob.
α1 0.810341 0.023969 33.8079 0
α2 -0.077142 0.031124 -2.478536 0.0132
α3 0.052411 0.032014 1.637124 0.1016
α4 -0.018263 0.030886 -0.591302 0.5543
α5 0.033361 0.029311 1.138192 0.255
α6 0.067604 0.029434 2.296791 0.0216
α7 0.024325 0.031277 0.777718 0.4367
α8 -0.009702 0.031898 -0.304157 0.761
α9 0.024038 0.030216 0.795532 0.4263
α10 -0.032706 0.029915 -1.093301 0.2743
α11 0.059116 0.029609 1.996522 0.0459
α12 -0.017076 0.029275 -0.583312 0.5597
α13 0.011817 0.029379 0.402216 0.6875
α14 0.049594 0.023396 2.119797 0.034
λ11 -0.241109 0.092948 -2.594006 0.0095
λ12 0.274782 6.78E-02 4.051529 0.0001
γ 0.000193 8.84E-05 2.187391 0.0287
Variance Equation
ρ 2.203899 1.931892 1.140798 0.254
φ1 0.016164 0.018543 0.871671 0.3834
φ2 0.044892 0.019562 2.294804 0.0217
β1 0.62895 0.225795 2.785498 0.0053
β2 -0.778208 0.253479 -3.070109 0.0021
β3 -0.195731 0.317636 -0.616211 0.5378
β4 0.694454 0.328699 2.112737 0.0346
β5 -0.555604 0.25648 -2.166268 0.0303
β6 0.583307 0.183175 3.184432 0.0015
λ21 0.451797 0.435522 1.03737 0.2996
λ22 -0.222142 0.179655 -1.23649 0.2163
(b) Estimation output: Model II
48
Model III
Variable Coefficient Std. Error z-Statistic Prob.
δ 0.552785 0.073842 7.486094 0
α1 0.782545 0.023722 32.98835 0
α2 -0.08635 0.028944 -2.983384 0.0029
α3 0.032704 0.029873 1.094755 0.2736
α4 -0.018279 0.030173 -0.605826 0.5446
α5 0.021081 0.030556 0.689897 0.4903
α6 0.063528 0.029513 2.15252 0.0314
α7 0.007537 0.028681 0.262796 0.7927
α8 -0.019515 0.029192 -0.6685 0.5038
α9 0.019895 0.030165 0.659556 0.5095
α10 -0.043961 0.03019 -1.456135 0.1454
α11 0.037424 0.029874 1.252755 0.2103
α12 -0.023564 0.0289 -0.81535 0.4149
α13 0.009581 0.028342 0.338047 0.7353
α14 0.019924 0.023123 0.861647 0.3889
λ11 -1.405722 1.71E-01 -8.2063 0
λ12 -0.055451 8.11E-02 -0.683346 0.4944
γ 0.000131 8.99E-05 1.45397 0.146
Variance Equation
ρ 2.1241 1.015668 2.091332 0.0365
β1 0.73098 0.135747 5.384859 0
β2 -1.007305 0.005079 -198.3274 0
β3 0.711041 0.137096 5.186445 0
(c) Estimation output: Model III
49
Model IV
Variable Coefficient Std. Error z-Statistic Prob.
δ 0.546156 0.073497 7.430961 0
α1 0.783229 0.023717 33.02329 0
α2 -0.086278 0.0289 -2.985441 0.0028
α3 0.036149 0.029933 1.207646 0.2272
α4 -0.013872 0.030291 -0.457941 0.647
α5 0.016728 0.029106 0.574731 0.5655
α6 0.062351 0.024859 2.508159 0.0121
α7 0.01101 0.027809 0.3959 0.6922
α8 -0.015875 0.029196 -0.543758 0.5866
α9 0.015813 0.03018 0.52395 0.6003
α10 -0.044 0.02993 -1.470117 0.1415
α11 0.040415 0.028495 1.418324 0.1561
α12 -0.022481 0.028647 -0.784765 0.4326
α13 0.010763 0.028382 0.379235 0.7045
α14 0.020959 0.023047 0.909419 0.3631
λ11 -1.317104 1.71E-01 -7.687573 0
λ12 -0.045395 8.17E-02 -0.55581 0.5783
Variance Equation
ρ 0.871846 0.971893 0.897059 0.3697
β1 0.769045 0.25746 2.987045 0.0028
(d) Estimation output: Model IV
50
Model
VM
odel
VI
Var
iable
Coeffi
cien
tStd
.E
rror
z-Sta
tist
icP
rob.
Var
iable
Coeffi
cien
tStd
.E
rror
z-Sta
tist
icP
rob.
δ1.
6954
340.
1784
849.
4990
60
δ1.
7437
560.
1885
169.
2499
030
α1
0.78
3913
0.02
3595
33.2
2385
0α
10.
7802
430.
0237
4732
.856
830
α2
-0.0
8122
0.02
9005
-2.8
0023
80.
0051
α2
-0.0
8182
50.
0290
9-2
.812
859
0.00
49
α3
0.01
9394
0.02
933
0.66
1247
0.50
85α
30.
0241
270.
0296
010.
8150
690.
415
α4
-0.0
0796
20.
0301
3-0
.264
246
0.79
16α
4-0
.010
077
0.03
0505
-0.3
3032
60.
7412
α5
0.01
892
0.03
0525
0.61
982
0.53
54α
50.
0176
450.
0309
320.
5704
560.
5684
α6
0.06
8179
0.02
3502
2.90
1021
0.00
37α
60.
0631
150.
0237
532.
6571
750.
0079
λ11
-1.7
0702
80.
1986
28-8
.594
111
0λ
11
-1.8
3041
10.
2121
88-8
.626
366
0
λ12
0.01
2443
0.10
8199
0.11
5003
0.90
84λ
12
-0.0
1661
50.
1132
9-0
.146
659
0.88
34
λ21
-0.1
2028
30.
0646
1-1
.861
691
0.06
26
λ22
-0.0
623
0.06
4384
-0.9
6762
80.
3332
Vari
ance
Equati
on
Vari
ance
Equati
on
ρ0.
0524
170.
0018
528
.330
560
ρ0.
3734
060.
5782
060.
6458
010.
5184
β1
1.94
6724
0.00
6242
311.
8866
0β
10.
7170
480.
4378
451.
6376
760.
1015
β2
-0.9
8673
40.
0061
17-1
61.3
164
0λ
21
0.05
8346
0.09
3208
0.62
5979
0.53
13
λ21
0.00
7408
0.00
1839
4.02
8486
0.00
01λ
22
-0.0
1737
52.
57E
-02
-0.6
7535
90.
4994
λ22
-0.0
0228
90.
0018
74-1
.221
087
0.22
21
(e)
Est
imat
ion
outp
ut:
Model
V,
Model
VI
51
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