THE UNIVERSITY OF CALGARY Stochastic Models for Natural Gas and Electricity Prices by Guanghui Quan A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE DEPARTMENT OF MATHEMATICS AND STATISTICS CALGARY, ALBERTA DECEMBER, 2006 c Guanghui Quan 2006
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THE UNIVERSITY OF CALGARY
Stochastic Models for Natural Gas and Electricity Prices
by
Guanghui Quan
A THESIS
SUBMITTED TO THE FACULTY OF GRADUATE STUDIES
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
A Transform Analysis for Affine Jump-diffusion Models 100
B Some Integrals 101
C Parameter Estimates for Model 1C and Model 2C 104
D Solving the Riccati Equations 106
Bibliography 111
vi
List of Tables
2.1 Descriptive statistics of APP Hourly EP . . . . . . . . . . . . . . . . 92.2 Descriptive statistics of AECO Daily NGL . . . . . . . . . . . . . . . 92.3 Descriptive statistics of APP Daily EP . . . . . . . . . . . . . . . . . 92.4 Descriptive statistics of AECO Daily NG . . . . . . . . . . . . . . . . 102.5 Statistics for Log-returns of NG and EP . . . . . . . . . . . . . . . . 142.6 Asymptotic Acceptance Limits for the KS Test . . . . . . . . . . . . . 142.7 Drift Test Applied to Log-returns of NG and EP . . . . . . . . . . . . 152.8 Mean-Reversion Rate for Log-returns of NG and EP . . . . . . . . . . 172.9 t-statistic Critical Values . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.1 APP Hourly EP Parameter Estimates for Model 1A . . . . . . . . . . 474.2 AECO Daily NG Parameter Estimates for Model 1A . . . . . . . . . 474.3 APP Hourly EP Parameter Estimates for Model 1B . . . . . . . . . . 524.4 AECO Daily NG Parameter Estimates for Model 1B . . . . . . . . . 524.5 Daily NG/EP Parameter Estimates for Model 2A . . . . . . . . . . . 604.6 Daily NG/One-Peak EP Parameter Estimates for Model 2A . . . . . 604.7 Daily NG/EP Parameter Estimates for Model 2B . . . . . . . . . . . 654.8 Daily NG/On-peak EP Parameter Estimates for Model 2B . . . . . . 654.9 Daily NG/EP Parameter Estimates for Model 3A . . . . . . . . . . . 734.10 Daily NG/EP Parameter Estimates for Model 3B . . . . . . . . . . . 78
5.1 Comparison of Simulation Results Fitting on Hourly EP . . . . . . . 875.2 Comparison of Simulation Results Fitting on Daily NG/EP . . . . . . 885.3 Criteria Statistics for 1-factor Jump-Diffusion Models . . . . . . . . . 905.4 Criteria Statistics for 2-factor Jump-Diffusion Models . . . . . . . . . 915.5 Statistics for M-1C fitting on Hourly EP . . . . . . . . . . . . . . . . 925.6 Effects of Sample Size for M-1C Fitting on Hourly EP . . . . . . . . . 945.7 Effects of Sample Size for M-2C Fitting on Daily NG/EP . . . . . . . 95
C.1 Alberta Power Pool Hourly EP Parameters Estimates for M-1C . . . 104C.2 AECO Daily NG Parameters Estimates for M-1C . . . . . . . . . . . 104C.3 Daily NG/EP Parameter Estimates for M-2C . . . . . . . . . . . . . 105C.4 Daily NG/On-peak EP Parameter Estimates for M-2C . . . . . . . . 105
vii
List of Figures
2.1 QQ Plot for Log-returns of NG and EP vs. Standard Normal . . . . 112.2 Left Tail of c.d.f for Log-returns of NG and EP vs. Standard Normal 122.3 Log-returns of NG and EP Prices . . . . . . . . . . . . . . . . . . . . 162.4 Historical Volatility for Log-returns of NG and EP . . . . . . . . . . . 192.5 Correlation between Daily NG/EP Prices . . . . . . . . . . . . . . . . 21
4.1 PDF and CDF of Double Exponential Distribution . . . . . . . . . . 444.2 PDF of Double Exponential Distributions . . . . . . . . . . . . . . . 454.3 PDF and Peak-finding Results for Model 1A fitting on APP Hourly EP 484.4 PDF and Peak-finding Results for Model 1A fitting on AECO Daily
NG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.5 PDF and CDF of Gamma Distribution . . . . . . . . . . . . . . . . . 504.6 PDF and Peak-finding Results for Model 1B fitting on APP Hourly EP 534.7 PDF and Peak-finding Results for Model 1B fitting on AECO Daily
NG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.8 Daily NG/EP Histogram Plot and Density Plot for Model 2A . . . . 614.9 Daily NG/On-peak EP Histogram Plot and Density Plot for Model 2A 614.10 Peak-finding Results for Model 2A fitting on Daily NG/EP . . . . . . 624.11 Peak-finding Results for Model 2A fitting on Daily NG/On-peak EP . 624.12 Daily NG/EP Histogram Plot and Density Plot for Model 2B . . . . 664.13 Daily NG/On-Peak EP Histogram Plot and Density Plot for Model 2B 664.14 Peak-finding Results for Model 2B fitting on Daily NG/EP . . . . . . 674.15 Peak-finding Results for Model 2B fitting on Daily NG/On-peak EP . 674.16 Daily NG/EP Histogram Plot and Density Plot for Model 3A . . . . 744.17 Peak-finding Results for Model 3A fitting on Daily NG/EP . . . . . . 744.18 Daily NG/EP Histogram Plot and Density Plot for Model 3B . . . . 794.19 Peak-finding Results for Model 3B fitting on Daily NG/EP . . . . . . 79
5.1 APP Hourly EP Simulated Prices and Real Prices for Model 1A . . . 825.2 AECO Daily NG Simulated Prices and Real Prices for Model 1A . . . 825.3 APP Hourly EP Simulated Prices and Real Prices for Model 1B . . . 835.4 AECO Daily NG Simulated Prices and Real Prices for Model 1B . . . 835.5 Daily NG/EP Simulated Prices and Real Prices for Model 2A . . . . 845.6 Daily NG/On-peak EP Simulated Prices and Real Prices for Model 2A 845.7 Daily NG/EP Simulated Prices and Real Prices for Model 2B . . . . 855.8 Daily NG/On-peak EP Simulated Prices and Real Prices for Model 2B 85
viii
5.9 daily NG and daily EP Simulated Prices and Real Prices for Model 3A 865.10 daily NG and daily EP Simulated Prices and Real Prices for Model 3B 865.11 Parameter Estimates vs. Sample Size for M-1C fitting on Hourly EP 925.12 Comparison of Parameter Estimates for M-1C fitting on Hourly EP . 93
ix
Chapter 1
Introduction
Natural gas, as one of the cleanest burning fuels that accounts for nearly a quar-
ter of all the energy consumed in North America, is increasingly used for heating,
cooling, by industry and for electrical generation. Alberta produces nearly 5 trillion
cubic feet (tcf) and consumes 1.36 tcf of natural gas per year1. Natural gas prices
rose dramatically after 2000, raising concerns about the potential for natural gas
in North America economy. Unlike natural gas that is a fluid and can be stored
in tanks or pipelines, electricity power2 has no shelf life and has to be produced in
real time as customers demand it. Since demand fluctuates, it has to be continually
reviewed and anticipated to ensure enough electricity is steadily available to meet
the needs of consumers. Compared to other commodities, natural gas and electricity
prices have proved to be very volatile and are influenced by many variables such as
supply and demand, storage loadings and withdrawals, weather and region patterns,
pricing, market participants’ view of the future and so on. Therefore, modeling gas
and electricity prices, especially in deregulated markets, is quite different from the
standard approaches adopted in a regulated environment.
In this thesis, we focus on the peculiar behavior of natural gas and electricity
prices. We are devoted to discovering proper models that are capable of capturing
1This data is cited from http://www.directenergy.com2According to the context in this thesis, we sometimes use the terms “gas” and “electricity” to
represent “Natural Gas” (NG) and “Electricity Power” (EP), use the term “price” to present “spotprice”.
1
2
the essential behavior of gas and electricity prices as seen from the analysis of empir-
ical data. While 1-factor jump-diffusion models are employed for modeling a single
prices series, 2-factor jump-diffusion models are used to model correlated gas and
electricity prices.
In a Nexen project (2003) carried out by the Finance Lab at the University of
Calgary, Tony Ware, his students Lei Xiong and James Xu purposed mean-reverting
jump-diffusion processes to model natural gas and electricity prices (see [10]). In the
1-factor mean-reverting jump-diffusion model, they considered two types of distri-
butions for the jump amplitudes. One is normal distributed jump amplitudes, and
the other is exponentially distributed with a Bernoulli random variable assigned to
the sign of the jumps3. In the 2-factor model, they assumed that the logarithm of
natural gas prices follows a mean-reverting diffusion process without jumps, while
the logarithm of electricity prices follows a mean-reverting jump-diffusion process
with exponentially distributed jump amplitudes. They also assumed that there ex-
ist correlations between the noise components of the two processes4. Meanwhile,
they studied how the 1-factor and 2-factor models could match the underlying spot
prices by comparing the empirical distributions of the spot prices with the analytical
probability distributions generated from the models, and found that the shape of
the analytical distributions did match that of the the empirical distributions quali-
tatively, but were not quantitatively very accurate.
Thereafter, Lei Xiong and James Xu continued the research in their masters’
3The calibration results with exponentially distributed jump amplitude are presented in Ap-pendix C.1. In this thesis, we call this type of 1-factor mean-reverting jump-diffusion model asModel 1C.
4The calibration results for this 2-factor model are presented in Appendix C.2. In this thesis,we call this 2-factor model as Model 2C.
3
theses. Lei Xiong (2004) proposed several 1-factor mean-reverting jump-diffusion
models for modeling electricity prices. She examined the possibility of specifying the
long-term mean as a time varying function. She also investigated the combination
of upward jumps and downward jumps that are independent and exponentially dis-
tributed. She showed that the two-jump version of mean-reverting jump-diffusion
model with time varying long-term mean is generally superior to the other 1-factor
models she presented (see [20]). James Xu (2004) focused on modeling natural gas
prices using mean-reverting diffusion processes without jumps. He considered the
seasonality features in his 1-factor and 2-factor models. In the 2-factor models, he
specified the long-term mean as another mean-reverting diffusion process. He ex-
amined how those models match the futures prices, and found that 2-factor models
with seasonality performed the best among all models he presented. However, the
analytical distributions generated from his models are not very consistent with the
distributions of the empirical gas spot prices.
The weakness of Lei and James’ models is that they did not consider gas and
electricity prices as a whole. We believe that gas and electricity prices have some
kind of correlations that nevertheless are not easy to be observed. As a result, this
thesis is mostly based on what Tony Ware et al have done in the Nexen project, but
with improvements in the follow aspects:
Firstly, this thesis is devote to investigating whether double exponential distri-
bution with a location parameter or double gamma distribution with the shape pa-
rameter equals to 2 are more capable of representing the jump amplitudes of natural
gas or electricity prices. Compared with the 1-factor model calibrated in the Nexen
project, we found that double gamma distributed jump amplitude is a bit superior
4
to the exponentially distributed one in capturing the high spikes of electricity prices.
Secondly, 2-factor jump-diffusion models with correlation on jumps are examined
in this thesis to discover more essential relationship between natural gas and elec-
tricity prices. Compared with the 2-factor model in the Nexen project, we found
that by assigning simultaneously correlated jumps between gas and electricity prices
we can achieve better representation of the evolution of gas and electricity prices.
Lastly, numerical methods and the use of multi-dimensional Fourier transform for
the computation of probability density function are developed in this thesis, which
extend our capability of calibrating affine type jump-diffusion models5.
This thesis is composed of six chapters. In Chapter 1, we briefly review some
previous works, and then give an outline of what we plan to do in this thesis.
In Chapter 2, we begin by examining the empirical behavior of gas and electric-
ity prices. From the distribution test we realize that the normality assumption of
log-returns are not consistent with empirical observations. So in practice we should
consider more realistic models other than geometric Brownian motion to achieve
better representations of gas and electricity prices. A simple test is provided to dis-
cover possible evidence of mean-reversion for gas and electricity prices. The volatility
structure and correlation issues are also investigated to reveal relationships between
gas and electricity prices.
In Chapter 3, we start with an outline of a typical modeling process. Our re-
search is actually expanded along this process, and further research could be given
according to this routine. Thereafter, geometric Brownian motion is reviewed as a
5The numerical method and multi-dimensional are basically reflected in our programs. Compareto the programs for Nexen project, the optimization procedures are also improved.
5
foundation for expending upon mean-reverting process and jump process. Finally,
we concentrate on so-called jump-diffusion models that will be calibrated in the next
chapter.
In Chapter 4, we calibrate six jump-diffusion models specified as the following:
• M-1A: 1-factor Mean-Reverting Jump-Diffusion model with double exponen-
tially distributed jump amplitude of gas or electricity price.
• M-1B: 1-factor Mean-Reverting Jump-Diffusion model with double gamma dis-
tributed jump amplitude of gas or electricity price.
• M-2A: 2-factor Mean-Reverting Jump-Diffusion model with correlation be-
tween noises, and the jump amplitude of electricity price is double exponentially
distributed . There is no jump for the gas price.
• M-2B: 2-factor Mean-Reverting Jump-Diffusion model with correlation be-
tween noises, and the jump amplitude of electricity price is double gamma
distributed. There is no jump for the gas price.
• M-3A: 2-factor Mean-Reverting Jump-Diffusion model with correlation be-
tween jumps, and the two exponentially distributed jump amplitudes of gas
and electricity prices have the same arrive rate.
• M-3B: 2-factor Mean-Reverting Jump-Diffusion model with correlation be-
tween jumps, and the simultaneous jumps of gas and electricity prices are
normally distributed.
The parameter estimation we used is the maximum likelihood (ML) method. Since
all these models submit to the framework of affine jump-diffusion processes, it is
6
sometimes possible to obtain analytic solutions for the conditional probability density
functions (PDF). If there is no explicit solution for the PDF, we utilize numerical
methods to do the inverse Fourier transforms. The calibration results are presented
in th form of tables and figures.
Chapter 5 focuses on model testing and model comparison. Firstly, simulated
prices are presented to compare with the empirical prices. Then we present two
criterion tests on models we calibrated in Chapter 4, and try to figure out which one
has the best goodness-of-fit. We also perform tests on several models to investigate
the effects of the sample size on calibration results.
We come to our conclusions in Chapter 6.
In the Appendix, we present some computational issues that are referred to the
text of this thesis, including the transform analysis for affine jump-diffusion models,
the integrals for jump transforms and the solutions for Riccati equations. We also
present some calibration results of the Nexen project for model comparison.
In the bibliography, we only list references that are cited in this thesis, although
the scope of our references are actually beyond this list.
Chapter 2
Natural Gas and Electricity Prices
Analysis of the available data is perhaps one of the most important steps in under-
standing and quantifying the essential features of natural gas and electricity prices.
In this chapter, we perform statistical tests on four data sets -Alberta Power Pool
(APP) hourly electricity spot price (HEP) from Jan 1, 2002 to May 30, 2003 with
12383 observations, AECO1 daily natural gas (DNGL) spot price from May 1, 1990
to May 31, 2001 with 2782 observations, correlated Alberta Power Pool daily elec-
tricity (DEP) spot price and AECO daily natural gas (DNG) spot price both from
May 1, 1998 to May 8, 2000 with 739 observations.
Descriptive statistics are given in the first section. Then we apply distribution
tests to investigate whether the log-returns of underlying prices are consistent with
the normality hypothesis. Another test is implemented to discover available evidence
that energy prices do somewhat exhibit mean reversion. Subsequently, we examine
the volatility structure of gas and electricity prices, and try to figure out possible
correlations between gas and electricity prices. Finally, we briefly investigate the
futures prices of gas and electricity.
1The AECO spot price is the Alberta gas trading price, which has become one of North Americasleading price-setting benchmarks. It is closely tied to the Henry Hub natural gas price. Seehttp://www.energy.gov.ab.ca.
7
8
2.1 Descriptive Statistics
Mean, Variance and high order moments are clearly among the most frequently used
statistical quantities. Let us first review these concepts.
Given a sample size M , the estimate of the population mean and variance are
given by the following expressions:2
X =1
M
M∑i=1
Xi (2.1)
s2x =
1
M − 1
M∑i=1
(Xi −X)2 (2.2)
The 95% confidence interval is given by:[X − 1.96sx√
M,X +
1.96sx√M
]. (2.3)
The estimates of Kth moments are give by the following expression 3:
X(k)
= E(X −X)k =1
M
M∑i=1
(Xi −X)k (2.4)
A summary of statistics for APP hourly electricity prices is presented in Table
2.1. Statistics reported in this table are the spot prices (P), the change of spot prices
(dP), the logarithmic spot prices (ln(P)), the log-returns of spot prices (dln(P)),
the deseasonalized logarithm of spot prices (ln(DP)), and the deseasonalized log-
returns of spot prices (dln(DP))4. Similar summaries are reported in Table 2.2-2.4
for APP daily electricity prices and AECO daily natural gas prices. We observe from
2The ratio sx√M
is often referred as the standard error.3The third and fourth moments around the mean are called Skewness and Kurtosis, namely
skewness = E[X−E[X]]3
[var[X]]1.5 , kurtosis = E[X−E[X]]4
[var[X]]2 . For a standard normal distribution, skewness isequal to 0 and kurtosis is equal to 3.
4The statistical properties of undeseasonalized electricity price is hard to be captured. In theremainder of this thesis, we always use the deseasonalized hourly EP.
9
HEP Mean Std.Dev Skewness Kurtosis Minimum Maximum
Table 2.4: Descriptive statistics of AECO Daily NG
2.2 Distribution Test
Understanding the distribution properties of underlying variables is essential for mod-
eling any stochastic process. As a matter of fact, modeling methodologies, model
parameter estimation and finally, model testing are all based on the choices of proper
distributions. As a principle, if the model distributions are obviously different from
the empirical ones, we should modify the model to achieve better agreement.
A popular assumption states that the log-returns of energy prices usually follow
the normal distribution. However, this normality assumption needs to be carefully
examined before we build it into our stochastic models for modeling gas and electric-
ity prices. As we observe from the descriptive statistics in the previous section, the
skewness and kurtosis of log-returns are greater than that of the normal distribution.
Hence, the null hypothesis5 that the underlying log-returns follow normal distribu-
tion should be tested seriously. In this section, we perform several distribution tests
to discover available evidence for rejecting the normality hypothesis.
We start from a Quantile-quantile plot6 for the log-returns of gas and electricity
5In statistics, a null hypothesis is a hypothesis set up to be nullified or refuted in order to supportan alternative hypothesis. When used, the null hypothesis is presumed true until statistical evidencein the form of a hypothesis test indicates otherwise. See [12].
6QQ plot is a plot of the sample quantiles of X vs. theoretical quantiles from a normal distri-bution. The plot will be close to a straight line if the distribution of X is normal. See [36].
11
prices that are adjusted by the mean and variance of the data7. From the plots in
Figure 2.1 we can see that none of these log returns is strictly consistent with the
standard normal distribution, although the log-return of AECO daily natural gas is
seemingly normal in a very broad acceptance level.
Figure 2.1: QQ Plot for Log-returns of NG and EP vs. Standard Normal
Note: The four plots in Figure 2.1 have the sample data displayed with the plot
symbol ’+’. Superimposed on the plot is a line that is extrapolated out to the ends
of the sample to help evaluate the linearity of the data.
7All programs in this thesis are developed in MatLab and ready for request.
12
In Figure 2.2, we present the plots of corresponding cumulative distribution func-
tions (c.d.f) of the log-returns versus the c.d.f of a normal distribution. We find that
all these distributions of log-returns have fat tails that can be deduced from the
magnitude of the kurtosis. As pointed by Alexander Eydeland (see [1]), the distri-
bution kurtosis (or spikiness) of energy prices is the main cause of non-normality,
although the third distribution moment, skewness, can also be quite different from
the skewness of the normal distribution.
Although Figure 2.2 provide us qualitative analysis of the non-normality for the
Figure 2.2: Left Tail of c.d.f for Log-returns of NG and EP vs. Standard Normal
13
log-returns of gas and power prices, we want to do some quantitative tests to confirm
our judgment. The first test we present is so-called Jarque-Bera (JB) test8. Given a
sample size M , the statistic of a JB test is given by:
JB = M
[S2
6+
(K − 3)2
24
](2.5)
where S and K are skewness and kurtosis respectively. For the standard normal
distribution, it is easy to verify that S = 0 and K = 3, hence JB = 0. If the JB
statistic is greater than 6.0 at the 5% singnificance level, the nomalility hypothesis
should be rejected.
On the other hand, the Kolmogorov-Smirnov (KS) test is more general and allows
one to test any distribution hypothesis9. Mathematically, let X1, X2, ..., Xn be a
random sample. The empirical distribution function Fn(x) is a function of x, which
equals the fraction of Xi that are less than or equal to x for each x,
Fn(x) =1
n
n∑i=1
1Xi<x, (−∞ < x <∞)
Suppose the hypothesized distribution function is F (x), then the Kolmogorov-Smirnov
test statistics are given by:
Dn = supx|F (x)− Fn(x)| (2.6)
Table 2.5 is a summary of the JS and KS tests for the log-returns of gas and
electricity prices. The acceptance limits 10 for Kolmogorov-Smirnov test is also listed
8In statistics, the Jarque-Bera test is a goodness-of-fit measure that is used to verify whether asample is drawn from a normal distribution or not. See [13].
9The KS test compares the empirical distribution function with the cumulative distributionfunction specified by the null hypothesis. See [11].
10In table 2.6, we give only the asymptotic acceptance limits for large values of the sample sizen. See [14].
Table 2.6: Asymptotic Acceptance Limits for the KS Test
in Table 2.6. If the KS statistic exceeds the acceptance limit in a given significance
level11, then null hypothesis should be rejected. From Table 2.5 we can see that all the
JS values are extremly big, and all the KS statistics have exceeded the acceptance
limit in a 5% significance level. Therefore, the normality hypothesis for the log-
returns of natural gas and electricity prices should be rejected.
2.3 Mean Reversion
In energy markets where technology changes rapidly, it is not easy to identify long-
term means. But in the medium-term, we expect the log-returns of energy prices
to somewhat exhibit mean reversion. Let us first look at the drift test presented in
Table 2.7. As is seen form the last column in Table 2.7, none of the t-statistics12 is
greater than 2.0 in a 95% condifence level. Therefore, we have no evdience to say
that the log-returns of natural gas and electricity prices exhibit drifts. However, no
11the significance level of a test is the maximum probability that the observed statistic wouldbe observed under the null hypothesis that is considered consistent with chance variation, andtherefore with the truth of null hypothesis. A result which is significant at the 1% level is moresignificant than a result which is significant at the 5% level. See [23]
12the expression |√n− 1 X
sx| is called t-statistic.
15
Observations Mean Std.Dev. t-statisticHEP 12383 -0.0001 0.5625 -0.0177
Table 2.7: Drift Test Applied to Log-returns of NG and EP
drfit for log-returns of gas and electricity prices doesn’t necessarily mean that there
is no mean reversion. So in the next step we look at possible evidentce of mean
reversion for the log-returns of natural gas and electricity prices.
The fluctuations of logarithmic natural gas and electricity prices are shown in
Figure 2.3 . We are actually unable to identify any presence of mean reversion from
this figure.
The mean reversion test we present here is based on a simple model of constant
volatility and no autocorrelation. The reason why we do not offer a test on some
more general mean-reverting, stochastic-volatility or jump-diffusion model is for the
consideration of simplicity and applicability. We find that this simple model gives
roughly the results as we expected. Furthermore, it is hard to say that a model in
terms of jumps or stochastic volatility is a natural description of the evolution of
natural gas and electricity prices.
The simple regression model we run in this section has the following form:
∆Xt = α+ βXt−1 + σεt. (2.7)
We perform a statistical test to verify if the coefficient β is negative, as required by
the assumption of mean reversion. A summary of the standard t-test is reported in
Table 2.8. As we can see, all the mean reversion coefficients are negative. Moreover,
16
Figure 2.3: Log-returns of NG and EP Prices
by comparing the t-statistics in the last column with the critical value in Table 2.913,
we find that mean reversion for the log-returns of electricity prices is very strong.
However, this result is very questionable because of the presence of spikes14 in the
electricity prices evolution. It can be explained that our estimation just reflects
the spiky nature of electricity prices. Generally, identifying the presence of mean-
reversion is difficult in terms of natural gas and electricity prices, for which the mean
reversion tests depend on a variety of variables, and can be affected significantly by
13For samples of gas and electricity data, the critical values should be empirically higher.14spikes are characterized by significant upward moves followed closely by sharp drops.
17
Time Period MR Coefficient Mean Value t-statisticHEP Jan 1,2002-May 30,2003 -0.0001 0.2415 41.1851
DNGL May 1,1990-May 31,2001 -0.0043 0.0037 2.3712DNG May 1,1998-May 8, 2000 -0.0283 0.0275 3.1890DEP May 1,1998-May 8, 2000 -0.3927 1.4267 13.3905
Table 2.8: Mean-Reversion Rate for Log-returns of NG and EP
Significance critical value1% 3.45% 2.8810% 2.56
Table 2.9: t-statistic Critical Values
the presence of varying volatility. Therefore, we must be very careful interpreting
our test results. One can refer to Escribano et al (2001) and Boswijk (2000) (see
[15], [16]) for a profound analysis of this issue.
2.4 Volatility Structure
As a measure of the randomness of price changes, volatility is commonly associated
with the standard deviation of the distribution of prices. This association is based
on the assumption that price changes are log-normally distributed and independent.
However, from the distribution test we have realized that the log-returns of natural
gas and electricity prices are not normal. They have higher peaks and fatter tails
than predicted by a normal distribution. Despite these words of caution15, we still
assume that the volatility for the log-returns of gas and electricity is constant in a
relatively short period of time, and then we can estimate volatility from the time
15Further discussion about this issue can be found in [1].
18
series of historical prices16.
Historical Volatility
Let Pi be a time series of historical prices at time ti, i = 0, 1, ...M . Assume that
the log-returns ln Pi
Pi−1are independent and normally distributed, then the return
volatility, namely, historical volatility, denoted by σ, can be estimated using the
following formula:
σ∗ =
√√√√ 1
M − 1
M∑i=1
(Xi −1
M
M∑i=1
Xi)2 (2.8)
where
Xi =1√
ti − ti−1
lnPi
Pi−1
If the normality assumption does not hold, for example, the standard derivation
σ changes with time, we assume again that volatility is constant in a relatively short
period of time. Then we use the so-called moving window method17 to estimate
volatility at a given time tk by the expression:
σ∗(tk) =
√√√√ 1
m− 1
k∑i=k−m+1
(Xi −1
m
k∑i=k−m+1
Xi)2. (2.9)
Here m is a specified number of observations preceding the given time tk.
We present the historical volatilities for the log-returns of natural gas and elec-
tricity prices in Figure 2.4. The graphs indicate that the historical volatilities for
gas and electricity prices are not constant. The non-constant volatilities can be ex-
plained by either jumps or stochastic volatility. Correspondingly, to represent gas
16Volatility based on historical prices is commonly called historical volatility, which thereforemakes it impossible to capture information about future price movement. On the contrary, impliedvolatility can be used to describe anticipated volatility of prices in the future .
17For any time t, the volatility estimates (2.9) uses only a specified number of observations attimes that precede t and fall into the window of constant volatility. This approach is often calledthe moving window method. See [1].
19
and electricity prices more precisely, we should consider to present jumps in the ge-
ometric Brownian motion, or specify the volatility itself as a stochastic process.
Note: In Figure 2.4, the length of the moving windows for daily NG and daily
Figure 2.4: Historical Volatility for Log-returns of NG and EP
EP are 30 days. The length of the moving windows for hourly EP is 15 days. We
observe that the volatilities for natural gas and electricity prices are not constant.
Therefore, we need to add a jump component to the geometric Brownian motion or
specify the volatilities being stochastic to achieve a more realistic representation.
20
2.5 Correlations
In energy markets, joint behavior of various prices is an important characteristic
that should be considered. A popular measure of dependence between prices is
linear correlation. Mathematically, correlation is a proper measure of the dependence
between two random variables for a limited set of joint distributions, including joint
normal, log-normal, t, χ2 and other joint distributions. In this section, we will
examine the correlations between natural gas and electricity prices.
Let first review some concepts about correlation. Suppose X and Y are two
random variables, a linear correlation is given by the following formula:
ρX,Y =cov(X, Y )√var(X)var(Y )
=E[XY ]− E[X]E[Y ]√
E[X2]− (E[X])2√E[Y 2]− (E[Y ])2
(2.10)
The range of possible values covers the interval [−1, 1].
The standard estimator of correlation between X and Y is give by the following
formula:
ρ(x, y) =1N
∑Ni=1 (xi − x)(yi − y)
σxσy
(2.11)
σ2x =
1
N − 1
N∑i=1
(xi − x)2
σ2y =
1
N − 1
N∑i=1
(yi − y)2
Before examining possible correlations between gas and electricity, we should be
careful about the range of the applicability of correlation. We know that the possible
values of correlation depend on the distributions of the variables we try to correlate.
21
If two random variables X and Y are independent, then the correlation ρX,Y = 0.
But ρX,Y = 0 does not necessarily means that X and Y are independent. It is
important to understand, therefore, low correlations do not necessarily imply weak
dependence. This reflects the fact that joint distribution usually carries more infor-
mation that can not be simply described by correlation. Therefore, to get the whole
picture of relations between two specified random variables, we have to understand
their marginal distributions and possible transformations. Nevertheless, all of these
are challenges in natural gas and electricity markets, since the correlated log-returns
could follow a variety of stochastic processes.
As serious as these may sound, however, we still ignore them and estimate cor-
relation directly (see [18]). We find that the simple procedure of estimating the
correlation is likely to work well. We present the correlation between log-returns of
daily natural gas and electricity prices in Figure 2.5. From this figure, however, it is
hard to say that there is a clear correlation between daily gas and electricity.
Figure 2.5: Correlation between Daily NG/EP Prices
Note: We still use the moving window method introduced in previous section
to estimate the correlation. In Figure 2.5, the length of the moving window for
correlated daily NG and daily EP is 30 days.
Chapter 3
Stochastic Models
Depending on the purpose of modeling, we might be able to adopt a specified model
that is validated to be the most efficient among possible selections. However, there
are many answers regarding the question of what model in the energy markets is
the best. As is seen from the analysis of empirical data in Chapter 2, natural
gas and electricity prices in general, and electricity prices in particular, exhibit be-
haviors significantly different from other energy commodities or financial products.
stationarity of correlations and non-constant volatility structure, on the other hand,
make modeling the evolution of natural gas and electricity prices even more difficult
and challenging. So far, there is still a shortage of satisfying pricing models for en-
ergy products in practice, which makes the model calibration and model comparison
for natural gas and electricity prices even harder.
This chapter concentrates on some fundamental issues related to mean-reverting
jump-diffusion models. We start with a sketch of an idealized modeling process, then
we introduce the development of geometric Brownian motion and jump-diffusion pro-
cess for modeling natural gas and electricity prices.
22
23
3.1 Modeling Process
A typical modeling process is given in Figure 3.1:
Figure 3.1: A Typical Modeling Process
In the energy market, however, this standard process is often impossible to follow.
24
The lack of available historical data makes model calibration and parameter estima-
tion very difficult. Moreover, most stochastic models are not capable of capturing
non-price information embodied in traded contracts, weakening their availabilities in
practice.
Model Selection
As we have mentioned above, it is hard to say what specific model in the energy
market is the best. However, we do have some general principles to determine if a
model is “not good” (see [1]) . The applicability of a model could be significantly
damaged if the model displays problems as the following:
1. Several parameters that have a dramatic effect on the value of prices are not
stable in the model.
2. The model has stable parameters but cannot be fit in with the market quotes.
3. The model has stable estimates of parameters and is also able to match market
quotes through calibration, but this calibration cannot deal with market changes.
On the contrary, a good model in general is capable of meeting the following
criteria:
1. It is a good fit to the historical data.
2. It is able to recover the out-of-sample historical distribution.
3. It matches the current prices of liquid contracts.
4. It has stable parameters that can be estimated efficiently.
5. It is efficient in the sense of using parameters as few as possible.
6. It is time-efficient in the evaluation of prices and hedges.
25
Model Calibration
Calibration of complex models with a large set of parameters is a challenging job.
The most popular technique for parameter estimating is the likelihood methods,
which makes inferences about parameters of the underlying probability distribution
of a given data set. A broad alternative is the method of moments, which compares
some moments of the sample distribution with the theoretical distribution1. However,
both approaches require an adjustment to arrive at the risk-neutral process. Another
consideration is that we may recover the parameters directly from the prices of liquid
products such as forwards, options and so on. This method is simple and universal
but the computation is complex and has to be done numerically.
3.2 Geometric Brownian Motion
A geometric Brownian motion (GBM) is a continuous-time stochastic process in
which the logarithm of the randomly varying quantity follows a Brownian motion,
perhaps more precisely, a Wiener process (see [5]). It is useful for the mathematical
modeling of some phenomena in financial markets, particularly in the field of option
pricing. GBM generates only positive random numbers, which is a critical property
for financial applications when the numbers represent prices2.
The use of GBM for modeling price evolution goes back to the work of Samuelson
(1965). In 1970’s, Fisher Black, Myron Scholes and Robert Merton made a milestone
in the pricing of stock options and developed the well-known Black-Scholes Model.
1More discussion about these two methods is presented in the Chapter 4.2A quantity that follows a GBM is required to take any value strictly greater than zero, which
is precisely the nature of a stock price.
26
From then on, the Black-Scholes formula has been pervasive used in financial markets,
and actually has become an integral part of market conventions. The heart of the
Black-Scholes model is the Black-Scholes partial differential equation (PDE), which
is derived based on a no-arbitrage (or delta-hedging) argument. The underlying asset
in the Black-Scholes model is assumed to follow a geometric Brownian motion.
In the most standard form, a stochastic process St is said to follow a GBM if it
satisfies the following stochastic differential equation (SDE):
dSt = µStdt+ σStdWt (3.1)
where dSt is the random movement of the prices over a time interval [t, t + dt], the
constants µ and σ are the percentage drift and the percentage volatility, and dWt
denotes the increments of a Wiener process that is defined as a time-continuous pro-
cess with the following properties:
1. W0 = 0 a.s..
2. Wt is almost always continuous in t, and any realization of Wt is a continuous
function of t.
3. Wt −Ws follows the N(0, t− s) distribution for all t ≥ s ≥ 0.
4. for all times 0 < t1 < t2 < ... < tn, the random variables Wt1 ,Wt2 −
Wt1 , ...,Wtn −Wtn−1 are independent.
Given an arbitrary initial value S0, the SDE has an analytic solution:3
St = S0e(µ−σ2
2)t+σWt . (3.2)
3Define Xt = lnSt, then by Ito’s formula we have dXt = (µ − σ2/2)dt + σdWt, which impliesthat Xt ∼ N(X0 + (µ− σ2/2)t, σ2t). The log-normality of St reflects the fact that increments of aGBM are normal relative to the current price. That is why the process has the name “geometric”.
27
Mean-Reverting GBM
In practice, GBM in its standard form does not perform well in energy markets. We
may modify the GBM process to contain some essential effects such as mean rever-
sion and seasonality for achieving more realistic representations of price evolutions
in natural gas and electricity markets.
Assume that spot prices on average travel toward a long-term mean, then the evo-
lution of spot prices can be represented by the so-called Ornstein-Uhlenbeck model :
dSt
St
= κ(L− St)dt+ σdWt (3.3)
where L is the long-term mean of spot prices, and κ is the strength of mean reversion.
The first term on the right side represents a deterministic drift of prices. It will drift
up when prices are below the long-term mean L and vice versa. The parameter κ
determines how the long-term mean attracts prices. The lager the value of κ, the
faster the prices move toward L.
A mean-reverting process of GBM could be also expressed as the following Schwartz-
Ross model :
dSt
St
= κ(l − ln(St))dt+ σdWt (3.4)
Here l− 12κσ2 turns to be the long-term mean of logarithmic prices after some simple
calculation. The advantage of (3.4) is that it allows a closed-form solution for the
distributions of the logarithmic prices. Suppose Xt = lnSt, and Yt = eκtXt then we
have:
dYt = κeκtXt + eκtdXt = κ(l − 1
2κσ2
)eκtdt+ σeκtdWt
Then we obtain that the random variable Xt also follows a normal distribution.
28
Seasonality
As we can see from the analysis of empirical data, seasonality is an important charac-
teristic of natural gas and electricity prices, especially natural gas prices. A natural
thinking is that we consider the spot prices as a sum of a seasonal term and an
unseasonalized mean-reverting term. For example, we can specify the spot prices Xt
as:
Xt = f(t) + St (3.5)
where
f(t) = at+M∑i=1
αi cos(2πit
P) + βi sin(
2πit
P)
represents the seasonal effect, a, αi, βi are all constants, M is a positive number, P
is the number of trading days in one year, and St follows a mean-reverting GBM as
defined in (3.3)4.
3.3 Jump-Diffusion Models
There is plenty of evidence that the logarithmic gas and electricity prices do not fol-
low normal distribution, which as we know is the basic assumption for the underlying
asset prices in GBM. As we have seen from the Chapter 2, those pronounced char-
acteristics of gas and electricity prices can only be explained by some non-normal
distributions. In other words, geometric Brownian motion is incorrect for model-
ing gas and electricity prices. Therefore, several theories have been put forward for
the non-normality of the empirical distribution of natural gas and electricity prices.
4Z. Xu (2004) calibrated a mean-reverting GBM model that is similar to (3.5) but with timedependent volatility in the mean-reverting term St. See [19].
29
Among them stochastic volatility and the presence of jumps in the price process are
undoubtedly the most popular ones that are able to recover the empirical price distri-
butions. Since the empirical characteristics of log-returns can be explained either by
stochastic volatility or jump models, can we somehow infer from the available data
to differentiate stochastic volatility from jumps? The following are some references,
although in practice we can encounter both effects at the same time (see [1]):
- For stochastic volatility, the excess kurtosis increases with the scale of the re-
turn. But for jump models, the excess kurtosis decreases with the scale of the return.
- For stochastic volatility, the smile is flat for the near contracts and increases
for the later contracts. But for jump modes, the smile is the steepest for prompt
contracts, and then gradually flattens out.
In this thesis, we adopt a jump-diffusion process to describe the log-returns of gas
and electricity. As we have observed, there are many more spikes in the evolution
of electricity spot prices than expected from a normal distribution. On the shortest
timescales the prices looks discontinuous and have occasional jumps now and then.
This obvious behavior inspires researchers to think of presenting jumps in the price
process. Since the pioneering work of Robert Merton (see [22]), jump-diffusion pro-
cesses (JDP) have been widely used for modeling price evolution in financial markets
(see [2],[3]). In recent years these processes started appearing frequently in energy
applications for capturing the notable price spikes in the energy markets. It has
been proved that JDP can capture the fat tails of energy price distributions quite
well, suggesting that the jump-diffusion model has good performance in recovering
the price distributions of energy prices.
A jump-diffusion process is composed of a diffusion component and a jump com-
30
ponent. The diffusion part usually takes the form of the standard GBM, while the
jump part is expressed by a Poisson process. Although many other processes can be
used to represent discontinuous jumps, the Poisson process is chosen more frequently.
Affine Jump-Diffusion Process
In mathematical terms, an affine jump-diffusion process (AJD) is a jump process for
which the drift vector, volatility matrix, and jump intensity vector are all affine on
some state space D ⊂ Rn (see [7]). More precisely, suppose that Xt is a Markov
process5 on D, satisfying the stochastic differential equation:
dXt = µ(Xt)dt+ σ(Xt)dWt + dZt (3.6)
where Wt is a standard Brownian motion in Rn, µ : D → Rn, σ : D → Rn×n, Zt
is a jump process whose jumps have a fixed probability distribution v on Rn, and
arrived intensity {λ(Xt) : t ≥ 0} for some λ : D → [0,∞). Then Xt is an affine
jump-diffusion process if
µ(x, t) = K0(t) +K1(t)x,
σ(x, t)σ(x, t)′
= H0(t) +n∑
k=1
Hk1 (t)xk,
λ(x, t) = l0(t) + l1(t)x,
where for each 0 ≤ t < ∞, K0(t) ∈ Rn, K1(t) ∈ Rn×n, H0(t) ∈ Rn×n and symmetric,
H1(t) ∈ Rn×n×n and, for k = 1, ..., n,H(k)1 (t) is in Rn×n and is symmetric. The Fourier
transform ofXt is known in closed form up to an ordinary differential equation(ODE).
5A stochastic process has the Markov property if the conditional probability distribution offuture states of the process, given the present state and all past states, depends only upon thecurrent state and not on any past states. A process with the Markov property is usually called aMarkov process. See [6].
31
Consequently, the distribution of Xt can be recovered by inverting this transform (see
[7]).
Poisson Process
A Poisson process Pt with arrive intensity λ is characterized by the following prop-
erties (see [23]):
1. The number of change occurring in any two non-overlapping intervals are in-
dependent.
2. The probability of one change in a sufficiently short interval of length t is
approximately λt.
3. The probability of two or more changes in a sufficiently short interval is un-
likely.
The mean and variance of Pt are given by:
E(Pt) = λt, var(Pt) = λt
An important relation between Poisson distribution and exponential distribution
is that if T1, T2, ..., Tn are the arrival times of jumps, then the random variables
Xi = Ti+1 − Ti, the lengths of time intervals between jumps, are independent and
have exponentially distributed with parameter λ.
Jump-Diffusion Process with Mean Reversion
Jump-diffusion processes are capable of modeling sudden discontinuous in the price
evolution, but once the jumps occur and the prices move to a new level, the price
tends to stay in that level until a new jump arrives. This is definitely not a behavior
32
of energy prices, especially electricity prices. As we have observed that the electricity
price returns quickly to the normal level after jumps. Therefore, we can consider to
combine a mean reversion effect in the jump-diffusion model as the following6:
dSt = κ(α∗ − lnSt)dt+ σStdWt + JtStdPt (3.7)
where St = S0 for t = 0, St > 0, the parameters κ is the mean reversion rate, Pt is
a discontinuous, one dimensional standard Poisson process with arrival rate ω and
associated jump amplitude Jt7. Then from Ito’s formula, we get a SDE for the log-
returns Xt = lnSt:
dXt = κ(α−Xt)dt+ σdWt +QtdPt (3.8)
where α = α∗ − 12κσ2 is the long-term mean of Xt, Qt = ln(1 + Jt). Notice that this
equation still fits the affine jump diffusion framework, and the distribution of Xt can
be obtained by taking the inverse Fourier transform of Xt.
Jump diffusion models with mean reversion (MRJ) can capture all the essential
features of energy prices. However, too many parameters need to be estimated,
making the calibration results unreliable under the restriction of insufficient data.
Distributions of jump amplitude
Jump amplitude distribution is one research point in this thesis. More interpretation
about this issue is given in Chapter 4 when we calibrate those specified 1-factor and
2-factor jump-diffusion models. In the Nexen project (see [10]), Dr. Tony Ware
6In the context of this type of energy models, please refer to Deng (1998). See [24].7We assume that the Brownian motion, Poisson process and jump amplitude all have the Markov
property and are pairwise-independent. In this thesis, the associated jump amplitude Jt eitherfollows a double exponential distribution or a double gamma distribution. See Chapter 4.
33
and his students made two exponential distributions back-to-back for describing the
distribution of the log-returns of gas and electricity prices. At the same time, a
Bernoulli random variable was assigned to the sign of the jump amplitudes. They
calibrated a 1-factor jump-diffusion model (Model 1C) and a 2-factor jump-diffusion
model (Model 2C) with this kind of jump amplitude distribution, and the corre-
sponding calibration results are shown in Appendix C. This kind of exponential
distribution proved to fit the distribution of empirical data very well. Meanwhile,
they also examined normally distributed jump amplitudes. In this thesis, we will
consider double exponential and double gamma distributed jump amplitudes. We
will compare our calibration results with that in Appendix C, and see whether the
double exponential and double gamma distributions could make any improvement
for modeling natural gas and electricity prices.
3.4 Extensions
Models introduced in this section may be used for possible future work.
Stochastic Volatility Jump-Diffusion Models
We know that estimation of jumps with maximum-likelihood methods causes fre-
quent, small amplitude jumps that can be easily explained by stochastic volatility
effects. It is worthwhile, then, to investigate if these effects can be added to the
jump-diffusion model for stabilizing its parameters. The following stochastic volatil-
34
ity jump-diffusion model is put forward for this purpose:
dSt
St
= κ(α− lnSt)dt+√VtdW
1t + JtdPt (3.9)
dVt
Vt
= θ(ω − lnVt)dt+ η√VtdW
2t
dW 1t dW
2t = ρdt
where the volatility of St is Vt, which follows another mean-reverting jump-diffusion
process, ρ is the correlation coefficient between the noises of St and Vt, and the other
parameters in (3.9) are similar to those in (3.7). The estimation of these kinds of
stochastic volatility jump-diffusion models can be found in [25].
Chapter 4
Model Calibration
In the first section of this chapter, we introduce two parameter estimation methods,
namely, maximum likelihood (ML) estimation and the method of moments (MM) es-
timation. Then we use the ML method to calibrate six mean-reverting jump-diffusion
models: Model 1A is a 1-factor model with double exponentially distributed jump
amplitude, Model 1B is a 1-factor model with double gamma distributed jump am-
plitude, Model 2A is a 2-factor model with correlation between noises and a double
exponentially distributed jump amplitude, Model 2B is a 2-factor model with correla-
tion between noises and a double gamma distributed jump amplitude, Model 3A is a
2-factor model with correlation between jumps and the two exponentially distributed
jump amplitudes have the same arrive rate, Model 3B is a 2-factor model with cor-
relation between jumps and the simultaneous jumps are normally distributed. For
each model, we present the parameter estimates, a comparison between the density
function and the empirically expected values, and the Peak-finding results fitting to
underlying natural gas or electricity prices.
4.1 Parameter Estimation Methods
4.1.1 Maximum Likelihood estimation
Maximum likelihood estimation is a popular statistical method used to estimate the
parameters of a given model when the underlying probability distribution of the data
35
36
originating from this model can be written down analytically. The principle of ML
estimation is to find values of the parameters that maximize the likelihood of the
data occurring.
Given a probability distribution D, associated with either a known probability
density function (PDF) or a known probability mass function (PMF) denoted as fD,
and distributional parameter θ, we may draw a sample {X1, X2, ..., XN} of N values
from this distribution. Then using fD we may compute the probability associated
with our observed data:
P (X1, X2, ..., Xn) = fD(X1, X2, ..., Xn|θ).
However, we don’t know the value of the parameter θ despite knowing that our
data comes from the distribution D. Since ML estimation seeks the most likely
value of the parameter θ, we maximize the likelihood of the observed data set over
all possible values of θ. Mathematically, we define the likelihood function as:
then the value that maximizes the likelihood is known as the maximum likelihood
estimator for θ.
In practice, instead of using the likelihood function, we usually use the log-
likelihood function, that is, the maximum likelihood estimates are given by θ such
that1,
θ = argmaxθ
(lnL({Xt}N
t=1, θ))
= argmaxθ
( N∑i=1
ln fD(Xi|θ))
(4.2)
1In mathematics, ′argmax′ stands for the argument of the maximum, that is, the value of thegiven argument for which the value of the given expression attains its maximum value.
37
If the distribution function satisfies certain regularity conditions, the solution of
(4.2) is given by the first order condition:
∂ log(L({Xt}Nt=1, θ))
∂θ= 0 (4.3)
ML estimation for Affine Jump-diffusion Models
It is essential for the ML estimation to have an analytical form PDF of the stochastic
variable. For the affine jump-diffusion models, it is possible to derive a closed-form
expression by using the affine jump-diffusion process transform (see [7],[8]).
Suppose that {Xt}t>0 is an affine jump-diffusion process as defined in (3.10).
Moreover, suppose again that R(Xt, t) is a stochastic function such as “discount
rate”, v0 and v1 are scalars that may be real or complex. Then we can derive a
closed-form expression for the transform
Et
[exp
(−
∫ T
t
R(Xs, s)ds)(v0 + v1XT )euXT
](4.4)
An important application is that, by setting u = is, v0 = 1, v1 = 0 and R(Xt, t) =
0, we obtain a closed-form expression for the conditional characteristic function of
XT with respect to Xt:
φθ(s, t, T,Xt) = Eθ[exp(is ·XT )|Xt
]= Φθ(is, t, T,Xt), i =
√−1 (4.5)
Because knowledge of the conditional characteristic function is equivalent to
knowledge of the conditional density function of XT , we can apply the inverse Fourier
transform to recover the probability density function of XT conditional on Xt (CDF):
f(θ,XT |Xt) =1
(2π)N
∫RN
e−is·XTφθ(s, t, T,Xt)ds. (4.6)
A more complete introduction to the transform analysis for affine jump-diffusion
models can be found in Appendix A.
38
4.1.2 Method of Moments Estimation
ML estimation requires a complete specification of the model and its probability
distribution. However, full knowledge of the specification and strong assumption of
the distribution are not easy to get in practice. An alternative to this likelihood-type
estimation is the method of moments (MM) estimation. The idea of MM estimation
is that, instead of comparing the whole distribution, we compare only some moments
of the sample distribution to the theoretical one.
Consider the theoretical distribution function of f(Xt|θ), and a set of moments
that are functions of the model parameters:
µ1f (θ), µ
2f (θ), ..., µ
Nf (θ).
If we can calculate the analogous moments from the sample:
µ1f (θ), µ
2f (θ), ..., µ
Nf (θ), (4.7)
then we can choose parameters such that the theoretical moments are equal or close
to the sample moments. In mathematical terms, we define a N -dimensional vector
of moment functions m(Xt|θ), such that the following moment conditions hold
E[m(Xt|θ)] = 0. (4.8)
If we can get as many moment conditions as parameters to be estimated, we
may solve this system of equations to obtain an estimate of the parameters. If the
number of moment conditions is smaller than the number of parameters, we cannot
identity the parameters. If the number of moment conditions is bigger than the
number of parameters, we will be involved in an over-determined problem. In such
39
case, we can utilize the so-called generalized method of moments (GMM) that uses
a quadratic objective function with an appropriate weighting matrix to yield consis-
tent and asymptotically normal estimates (see [26]).
The significant benefit of MM estimation is that it uses only a subset of all the
possible restrictions that full knowledge of the underlying distributions implies. Take
the affine jump-diffusion models for example, in the case that the likelihood func-
tions is difficult to construct analytically, MM estimation, on the other hand, has
a computational advantage. Since many members of the class of AJD models have
an explicit expression for the characteristic function of the distribution, and know-
ing the characteristic function is equivalent to knowing explicit expressions for the
moments of the distribution, we may compute the moments by using approximation
or simulation methods that is more efficient than ML estimation. More references
about MM estimation can be find in [7] and [26].
40
4.2 Model 1A
In 1965, P. Samuelson first proposed geometric Brownian motion to model the ran-
dom behavior of the underlying stock. Based upon GBM, he modeled the random
value of the option at exercise (see [27]). In the 1970’s, Fischer Black, Myron Scholes
and Robert Merton made a milestone in the pricing of stock options by developing
the famous Black-Scholes (BS) model. The BS formula assumes the underlying
stock price follows a geometric Brownian motion with constant volatility. The BS
model described a general framework for pricing derivative instruments, and actually,
launched the field of financial engineering in the next decades. In energy markets,
however, empirical tests have suggested that the geometric Brownian motion is in-
sufficient to represent the exact nature of energy prices (see [1],[28],[29]). Therefore,
researchers considered more realistic processes for modeling energy prices.
Ornstein-Uhlenbeck process is the most basic mean-reverting process (see [32])
that is given by the following SDE:
dXt = κ(α−Xt)dt+ σdWt
where Wt is a standard Brownian motion, κ and α are all positive constant numbers.
Vasicek (1977) proposed this process for modeling financial time series (see [33]).
Thereafter, this kind of mean-reverting diffusion models are also called Vasicek type
models. Cox, Ingersoll and Ross (1985) used a similar process to model interest
rate. The so-called Cox-Ingersoll-Ross model is the same as above SDE, except
the volatility term is specified as σ√Xt. Hull and White (1990) extended Vasicek
model by specifying the long-term mean α as a function of time t. Nowadays, mean-
reverting models have been widely used in energy markets, especially in modeling
41
natural gas prices (see [1],[29],[31]). But in the case of modeling electricity prices,
mean-reverting diffusion models are inadequate to capture the price “spikes”, which
is a distinguished feature of electricity. Therefore, researchers managed to present
jumps in mean-reverting diffusion models.
Jump models were introduced by R.C. Merton (1976). In his paper, an option
pricing formula is derived for the more-general case when the underlying stock returns
are generated by a mixture of both continuous and jump processes (see [22]), that
is, the posited stock price returns can be written as a stochastic differential equation
(conditional on S(t) = S) as
dS(t)
S(t)= (α− λκ)dt+ σdW (t) + dq(t),
where α is the instantaneous expected return on the stock, σ2 is the instantaneous
variance of the return, conditional on no arrivals of important new information, dw(t)
is a standard Wiener process, q(t) is the independent Poisson process. λ is the mean
number of arrivals per unit time, κ is the expectation of the percentage change in
the stock price if the Poisson event occurs.
Since the most noticeable features of electricity prices are mean reversion and the
presence of price jumps, Deng (1998), Barz (1999), Clewlow and Strickland (1999)
managed to model the behavior of electricity price by adding mean reversion and
jumps in GBM model (see [24],[30],[34]). D. Duffie et al (2000) proposed transform
analysis for the affine type jump-diffusion processes. In the setting of affine jump-
diffusion state processes, an analytical treatment of a class of transforms is provided.
The Fourier transform, as a special case, allows an analytical treatment of a range
of valuations and applications.
42
Now let us begin by calibrating a 1-factor mean-reverting jump-diffusion model.
Suppose the logarithmic gas or electricity prices satisfies the following stochastic
differential equation (SDE):
dXt = κ(α−Xt)dt+ σdWt +QtdPt (4.9)
where Wt is a standard Brownian motion, Pt is a discontinuous, one dimensional
standard Poisson process with arrive rate ω and associated jump amplitude Qt.
Notice that (4.9) fits in the framework of an affine process with
K0 = κα, K1 = κ, H0 = σ2, H1 = 0, l0 = ω, l1 = 0.
Thus the conditional characteristic function Φ(s,XT |Xt) is given by:
Φ(s,XT |Xt) = E[exp(isXT )|Xt]
= exp(A(s, t, T, θ) +B(s, t, T, θ)Xt
)(4.10)
where A(·) and B(·) satisfy the following complex-valued ordinary differential equa-
tions:
∂A(s, t, T, θ)
∂t= −καB2(s, t, T, θ)− 1
2σ2B(s, t, T, θ)
−ω(ϕ(B(s, t, T, θ))− 1) (4.11)
∂B(s, t, T, θ)
∂t= κB(s, t, T, θ) (4.12)
with boundary conditions
A(s, T, T ) = 0 (4.13)
B(s, T, T ) = is. (4.14)
43
Before solving these so-called Riccati Equations with respect to A(·) and B(·), we
need to specify the jump transform ϕ(B(s, t, T, θ)).
4.2.1 Double Exponential Jump Amplitude
In probability theory and statistics, the double exponential distribution is a continu-
ous probability distribution that can be thought of as two exponential distributions
(with an additional location parameter) spliced together back-to-back. It is also
known as the Laplace distribution named after Pierre-Simon Laplace. A random
variable X has a double exponential distribution if its probability density function
is
f(x) =1
2γexp
(−|x− µ|γ
)=
1
2γ
exp(−µ−xγ
), x < µ
exp(−x−µγ
), x ≥ µ
where µ is a location parameter and γ > 0 is a scale parameter. If µ = 0, the positive
half-line is exactly an exponential distribution scaled by 12. The probability density
function and cumulative distribution function of double exponential distribution are
plotted in Figure 4.12.
We observe that the double exponential distribution is similar to the normal
distribution. However, whereas the normal distribution is expressed in terms of the
squared difference from the mean µ, the double exponential density is expressed in
terms of the absolute difference from the mean. Consequently the double exponential
2Figure 4.1 is cited from the free encyclopedia website: http://en.wikipedia.org
44
Figure 4.1: PDF and CDF of Double Exponential Distribution
distribution has fatter tails than the normal distribution.
In the Nexen technical report (see [10]), the jump magnitude is specified as ex-
ponential distribution with mean γ, whose probability density function is given by:
f(x) =
1γ
exp (−xγ), x ≥ 0
0, x < 0
and the sign of jumps is defined as a Bernoulli random variable with parameter ψ. For
example, if there are more upward jumps than downward jumps (ψ > 0.5), then the
probability density function is shown as PDF One in Figure 4.2. However, instead of
specifying two non-symmetric exponential distributions back-to-back, on can assign
a location parameter µ, with respect to which the two exponential distributions are
symmetric. Correspondingly, there are more upward jumps if µ > 0, or vice versa.
PDF Two in Figure 4.2 is the probability density function with location parameter
µ = 2.
In Model 1A, we suppose the jump amplitude is double exponentially distributed
with a location parameter µ that could be either positive or negative, and a scale
parameter γ > 0. We let θ = (κ, α, σ2, ω, µ, γ) denotes the true parameters needed
45
Figure 4.2: PDF of Double Exponential Distributions
to be estimated. Then the jump transform ϕ(·) in (4.11) is given by (see Appendix
B.1):
ϕ(B(s, t, T, θ)) =
∫ ∞
µ
eB(s,t,T,θ)xe−x−µ
γ
2γdx+
∫ µ
−∞
eB(s,t,T,θ)xex−µ
γ
2γdx
=eµB(s,t,T,θ)
1− γ2B2(s, t, T, θ). (4.15)
Solving the system of equations (4.11 - 4.12) with the boundary conditions (4.13 -
4.14), we get (see Appendix D.1)
A(s, t, T, θ) = iαs(1− e−κ(T−t))− σ2s2
4κ(1− e−2κ(T−t))
+ω
∫(1− eµise−κ(T−t)
1 + γ2s2e−2κ(T−t))dt (4.16)
B(s, t, T, θ) = ise−κ(T−t). (4.17)
Suppose that t = 0 and T = 1, then the density function of Xt+1 conditional on Xt
(CDF) is given by:
f(Xt+1, θ|Xt) =1
2π
∫ ∞
−∞Φθ(s,Xt+1|Xt)e
−isXt+1ds
=1
π
∫ ∞
0
R(e−isτh(θ, s))ds (4.18)
46
where R(A) denotes the real part of A, and
τ = (Xt+1 − α)− e−κ(Xt − α) (4.19)
h(θ, s) = exp(− σ2s2
4κ(1− e−2κ) + ω
∫ 1
0
(1− eµise−κτ
1 + γ2s2e−2κτ
)dτ. (4.20)
Notice that the CDF (4.18) involves an infinite integration. However, we can evaluate
it by truncating it to a reasonable finite interval outside which the integrand is
negligibly small. Then this integral can be computed on a suitable grid of s values by
the fast Fourier transform (FFT) algorithm. Since a FFT is a an efficient algorithm
to compute the discrete Fourier transform or its inverse, we are able to compute any
integral in h(θ, s) numerically when there is no explicit formula for h(θ, s). Therefore,
the maximum likelihood (ML) estimators θ, given a sequence of observations Xt =
Table 4.2: AECO Daily NG Parameter Estimates for Model 1A
Note: Tables 4.1 - 4.2 report the parameter estimates for Model 1A (4.9) fitting
Alberta Power Pool hourly electricity spot prices (12383 observations) and AECO
daily natural gas spot prices (2782 observations). The second column is the param-
eter estimates. The third column is the standard error of the estimates. The fourth
column is the t-ratio of the second column to the third column.
48
Figure 4.3: PDF and Peak-finding Results for Model 1A fitting on APP Hourly EP
Figure 4.4: PDF and Peak-finding Results for Model 1A fitting on AECO Daily NG
Note: Figure 4.3 - 4.4 are the estimation results for Model 1A fitting APP hourly
EP and AECO daily NG. The histograms of deviations of the expected values are
very similar to the plots of the density functions, which indicates that Model 1A fits
Hourly EP and Daily NG very well. Moreover, all the parameter estimates are at the
peak of the curves, which implies that we have found the maximized likelihood points.
49
4.3 Model 1B
In Model 1B, we still suppose the logarithmic gas or electricity prices satisfies the
same SDE as in Model 1A:
dXt = κ(α−Xt)dt+ σdWt +QtdPt (4.22)
As a result, the CCF Φ(·) is still the same as computed in Model 1A:
Φ(s,XT |Xt) = E[exp(isXT )|Xt]
= exp(A(s, t, T, θ) +B(s, t, T, θ)Xt
)(4.23)
where A(·) and B(·) satisfy the following ODEs:
∂A(s, t, T, θ)
∂t= −καB(s, t, T, θ)− 1
2σ2B(s, t, T, θ)2
−ω(ϕ(B(s, t, T, θ))− 1) (4.24)
∂B(s, t, T, θ)
∂t= κB(s, t, T, θ) (4.25)
with the same boundary conditions
A(s, T, T ) = 0 (4.26)
B(s, T, T ) = is. (4.27)
In Model 1B, however, the jump amplitudes of the logarithmic gas and electricity
prices are assumed to satisfy a gamma distribution.
4.3.1 Double Gamma Jump Amplitude
A random variable X has a Gamma distribution if its probability density function
is given by:
f(x) =1
Γ(κ)θκxκ−1e−
xθ (4.28)
50
where κ > 0 is the shape parameter and θ > 0 is the scale parameter of the gamma
distribution. The probability density function and cumulative distribution function
of gamma distribution is plotted in Figure 4.53. Notice that exponential distribution
Figure 4.5: PDF and CDF of Gamma Distribution
is a special case of gamma distribution, corresponding to setting κ = 1 for the PDF
of gamma distribution. The Gamma distribution has good performances when the
exponential distribution is not enough to capture the high jump amplitudes. For
example, electricity prices usually exhibit more higher jumps than that of many
other commodities. In this case, we may apply the gamma distribution to describe
the jump amplitudes of electricity.
In Model 1B, we suppose the jump magnitude of log-prices is gamma distributed
with shape parameter4 β = 2 and scale parameter γ > 0, the sign of the jump Qt is
a Bernoulli random variable with parameter ψ. In this case, the jump transform is
3Figure 4.5 is cited from the free encyclopedia website: http://en.wikipedia.org4In the remainder of this thesis, we always suppose the shape parameter equals 2 to get a
close-form formula for the jump transform.
51
given by (see Appendix B.2):
ϕ(B(s, t, T, θ)) =
∫ ∞
0
exp(B(s, t, T, θ)x
)( ψ
Γ(2)γ2x2−1e−
xγ +
1− ψ
Γ(2)(−γ)2x2−1e
xγ)dx
=ψ
(1− γB(s, t, T, θ))2+
1− ψ
(1 + γB(s, t, T, θ))2(4.29)
Solving the equation system (4.24 - 4.25) with the boundary conditions as in (4.26 -
Table 4.4: AECO Daily NG Parameter Estimates for Model 1B
Note: Table 4.3 - 4.4 report the parameter estimates for Model 1B (4.22) fit-
ting on Alberta Power Pool hourly electricity spot prices (12383 observations) and
AECO daily natural gas spot prices (2782 observations). The second column is the
parameter estimates. The third column is the standard error of the estimates. The
fourth column is the t-ratio of the second column to the third column.
53
Figure 4.6: PDF and Peak-finding Results for Model 1B fitting on APP Hourly EP
Figure 4.7: PDF and Peak-finding Results for Model 1B fitting on AECO Daily NG
Note: Figure 4.6 - 4.7 are the estimation results for Model 1B fitting APP hourly
EP and AECO daily NG. The histograms of the deviations of expected values are
very similar to the plots of the density functions, which indicates Model 1B fits
Hourly EP and Daily NG very well. Moreover, all the estimates are at the peak of
the curves, which implies that we have found the maximized likelihood points.
54
4.4 Model 2A
For 2-factor models, there are two random variables that can be specified as corre-
lated quantities, such as two spot price series, a spot price series and its volatilities, a
spot price series and its long-term means, or even two correlated future price series.
In energy markets, multi-factor models have been well documented in recent years,
because 1-factor jump-diffusion models are not adequate to recover the peculiar be-
havior of correlated energy prices. For example, the level of skewness in electricity
prices cannot easily be presented by a single 1-factor jump-diffusion model. It is
suggested to incorporate stochastic volatility to modeling electricity prices (see [28]).
Another class of 2-factor models are dedicated to discovering the correlation between
spot prices and their long-term means. Pilipovic (1998) represented a 2-factor mean-
reverting model where spot price is mean-reverting to a long-term mean that itself is
a log-normally distributed random variable. There are many works in the literature
involving 2-factor jump-diffusion models (see [1],[28],[29],[31]) that are impossible
to be fully discussed in this thesis. Instead, we only consider a pair of spot prices
such as gas and electricity spot price series, and investigate the possible correlation
between them.
The possible correlation of two spot price series could be between drifts, between
noises or between jumps. Generally, we can build a 2-factor model, in which the spot
prices are described by two jump-diffusion processes. Moreover, We assume that the
correlations are between both noises and jumps5. Before we jump into the calibra-
tion of this seemingly attractive general model, some issues have to be considered
5Here we do not consider the condition of correlation between drifts. We have done some researchon this topic, but these kinds of jump-diffusion models are very hard to handle.
55
carefully. First of all, we can not avoid to get involved in complicated computation
if there are two independent jump processes. Actually, no matter whether the jump
processes are independent or not, we have to deal with the conditional density func-
tion (CDF) that generally has no explicit solution. The calculation in this case is
expected to be very complex if we do not use numerical methods. Secondly, it is
hard to say whether a jump-diffusion model is better than a diffusion model without
jumps when the variation of the spot price series is not strong. Depending on the na-
ture of the spot price, it is possible to choose a jump-diffusion model without jumps
to achieve better calibration. For example, James Xu (2004) adopted mean-reverting
diffusion models to model natural gas spot prices, and the calibration results proved
to be consistent with empirical data. Since one random variable in our 2-factor mod-
els is the natural gas spot price, it is acceptable to employ a jump-diffusion model
without jumps to reduce the difficulties of calculation. Finally, we have to consider
the over-specification problem when the correlations are assumed to be between both
the noises and jumps.
On the other hand, the CDF of a 2-factor jump-diffusion model is more easily to
be computed if there is no correlation between noises and the jumps are also inde-
pendent. However, this makes no sense since the two random variables have nothing
related to each other. What we are doing is calibrating two independent models
at a time if we construct a 2-factor model like this. Therefore, if we consider the
correlation between jumps instead of correlation between noises, we should assume
that the jumps are somewhat correlated. In Model 3A and Model 3B, we will con-
sider correlation between jumps. However, in Model 2A and the subsequent Model
2B, we suppose that the correlation of gas and electricity spot prices are between
56
noises. Moreover, we suppose there is no jump for the gas price. Mathematically,
we suppose X1t and X2
t be logarithmic gas and electricity spot prices that satisfy the
following stochastic differential equations:
dX1t = κ1(α1 −X1
t )dt+ σ1dW1 (4.35)
dX2t = κ2(α2 −X2
t )dt+ ρσ2dW1 + σ2
√1− ρ2dW2 +QtdPt (4.36)
where κ1 and κ2 are the mean-reverting intensities, α1 and α2 are the long-term
means, W 1t and W 2
t are two standard Brownian motions, Qt is the logarithmic jump
amplitude for X2t , Pt is a discontinuous, one dimensional Poisson process with arrival
rate ω. The above 2-factor model can be written in matrix form as:
d
X1t
X2t
=
( κ1α1
κ2α2
+
−κ1 0
0 −κ2
X1
t
X2t
)dt
+
σ1 0
ρσ2
√1− ρ2σ2
dW1
dW2
+
0
QtdPt
. (4.37)
If we suppose that the state vector process [X1t X
2t ]′ given by (4.37) is under the
true measure6 and the risk premium7 associated with all state variables are linear
functions of the state variables, then there exists a risk-neutral probability measure8
Q over the state space represented by the state variables, such that the state vector
process has the same form as that of (4.37) under the risk-neutral measure, but with
different coefficients (see [24]). Here we choose to directly specify the state vector
6Under the true measure, the statistical properties of the underlying price process are observedfrom the real world. See [6].
7In financial markets, risk premium is a quantity subtracted from the mean return of financialasset in order to compensate the associated risks. See [6].
8A risk-neutral measure exists if no arbitrage condition holds. Under the risk-neutral measure,the price of a financial derivative is just the expected value of its discounted payoff. See [38].
57
process [X1t X
2t ]′ under the risk-neutral measure.
Notice that (4.37) fits in the framework of an affine jump-diffusion process where
we have:
K0 =
κ1α1
κ2α2
, K1 =
−κ1 0
0 −κ2
,
H0 =
σ21 ρσ1σ2
ρσ1σ2 σ22
, H11 = H2
1 =
0 0
0 0
,l0 = ω, l1 = 0
Consequently, the CCF is given by:
Φθ(s1, s2, X1t+1, X
2t+1|X1
t , X2t ) = exp
(A(s1, s2, t, T, θ)X
1t +B(s1, s2, t, T, θ)X
2t
+C(s1, s2, t, T, θ))
(4.38)
where A(·), B(·) and C(·) satisfy the following equations:
∂A
∂t= κ1A(·)
∂B
∂t= κ2B(·)
∂C
∂t= −κ1α1A(·)− κ2α2B(·)− 1
2σ2
1A(·)2 − 1
2σ2
2B(·)2
−ρσ1σ2A(·)B(·)− ω(ϕ(A(·), B(·))− 1) (4.39)
with the boundary conditions
A(s1, s2, T, T, θ) = is1,
B(s1, s2, T, T, θ) = is2,
C(s1, s2, T, T, θ) = 0. (4.40)
58
4.4.1 Double Exponential Jump amplitude
In the Nexen technical report (see [10]), the jump amplitude of the 2-factor model
is specified to be exponentially distributed with mean γ, and the sign of the jumps
is assumed to be a Bernoulli random variable with parameter ψ. In Model 2A,
just like what we have done in Model 1A, we suppose that the jump amplitude is
double exponentially distributed with a location parameter µ and a scale parameter
γ > 0. Again, we let θ = (κ, α, σ2, ω, µ, γ) denote the true parameters needed to be
estimated, then the jump transform ϕ(·) in (4.39) is given by:
ϕ(A(·), B(·)) =
∫ ∞
µ
eB(s,t,T,θ)xe−x−µ
γ
2γdx+
∫ µ
−∞
eB(s,t,T,θ)xex−µ
γ
2γdx
=eµB(s,t,T,θ)
1− γ2B2(s, t, T, θ)(4.41)
We have a solution for A(·), B(·) and C(·) in the CCF (4.38),
Table 5.3: Criteria Statistics for 1-factor Jump-Diffusion Models
3We may have noticed that, in general, the difference among these three models are not obvious.But we don’t have enough empirical data to confirm our statements.