the university of calgary stochastic models...

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

THE UNIVERSITY OF CALGARY

FACULTY OF GRADUATE STUDIES

The undersigned certify that they have read, and recommend to the Faculty of

Graduate Studies for acceptance, a thesis entitled “’Stochastic Models for Natural

Gas and Electricity Prices’ submitted by Guanghui Quan in partial fulfillment of the

requirements for the degree of MASTER OF SCIENCE.

Supervisor Dr. A. SwishchukDepartment of Mathematics and Statistics

Dr. L. P. BosDepartment of Mathematics and Statistics

Dr. C.-J. U. OsakweHaskayne School of Business

Date

ii

Abstract

This thesis focuses on modeling natural gas and electricity prices by affine jump-

diffusion processes. We start from investigating the peculiar behavior of natural gas

and electricity prices such as distribution, mean reversion, volatility and correlation.

Then we utilize maximum likelihood estimation to calibrate several mean-reverting

jump-diffusion models fitting on gas and electricity prices. While 1-factor jump-

diffusion models are employed for modeling a single prices series, 2-factor jump-

diffusion models attend to discover possible correlations between gas and electricity

prices. Meanwhile, we examine how different distributions of jump amplitudes affect

model calibration. We finally conduct model comparison based on empirical data

and simulation.

iii

Acknowledgements

The work on this thesis has been an inspiring, challenging, but always interesting

experience. It has been made impossible without many people who have supported

me in the past two years.

In the first place I would like to thank my former supervisor, professor Tony

Ware, for his supervision, advice and guidance. Above all and the most needed, he

provided me constant encouragement and support in various ways. His passions of

science exceptionally inspire me to improve my work. I am grateful in every possible

way and hope to keep up our collaboration in the future.

I would like to record my gratitude to professor Anatoliy Swishchuk for his con-

structive suggestions and precise comments on this thesis. I am very thankful that

he accepted to be my current supervisor and responsible for organizing my defence

committee.

Many thanks go in particular to professor Len Bos, who greatly enriched my

knowledge of mathematical finance with his excellent teaching and instructions.

I convey special thanks to Dr. Miro Powojowski, whose indispensable help ex-

traordinarily encourages me to carry out my research.

I would also like to acknowledge Z. Xu and L. Xiong for their contributions to

the Nexen project. I am indebted so much to their sincere assistance.

Finally by most importantly, I would like to thank my parents and my wife Jessy.

Their thoughtful support and persistent confidence in me, have made this thesis an

achievement of all the family members.

iv

Table of Contents

Approval Page ii

Abstract iii

Acknowledgements iv

Table of Contents v

1 Introduction 1

2 Natural Gas and Electricity Prices 72.1 Descriptive Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Distribution Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Mean Reversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.4 Volatility Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.5 Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3 Stochastic Models 223.1 Modeling Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2 Geometric Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . 253.3 Jump-Diffusion Models . . . . . . . . . . . . . . . . . . . . . . . . . . 283.4 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4 Model Calibration 354.1 Parameter Estimation Methods . . . . . . . . . . . . . . . . . . . . . 35

4.1.1 Maximum Likelihood estimation . . . . . . . . . . . . . . . . . 354.1.2 Method of Moments Estimation . . . . . . . . . . . . . . . . . 38

4.2 Model 1A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.2.1 Double Exponential Jump Amplitude . . . . . . . . . . . . . . 434.2.2 Calibration Results for Model 1A . . . . . . . . . . . . . . . . 47

4.3 Model 1B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.3.1 Double Gamma Jump Amplitude . . . . . . . . . . . . . . . . 494.3.2 Calibration Results for Model 1B . . . . . . . . . . . . . . . . 52

4.4 Model 2A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.4.1 Double Exponential Jump amplitude . . . . . . . . . . . . . . 584.4.2 Calibration Results for Model 2A . . . . . . . . . . . . . . . . 60

4.5 Model 2B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.5.1 Double Gamma Jump Amplitude . . . . . . . . . . . . . . . . 63

v

4.5.2 Calibration Results for Model 2B . . . . . . . . . . . . . . . . 654.6 Model 3A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.6.1 Independent Jump Amplitude with Same Arrive Rate . . . . . 694.6.2 Calibration Results for Model 3A . . . . . . . . . . . . . . . . 73

4.7 Model 3B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.7.1 Simultaneously Correlated Jump Amplitudes . . . . . . . . . . 764.7.2 Calibration Results for Model 3B . . . . . . . . . . . . . . . . 78

5 Model Tests 805.1 Monte Carol Simulation . . . . . . . . . . . . . . . . . . . . . . . . . 805.2 Criterion Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875.3 Effects of Sample Size . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6 Conclusion 96

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

3.1 A Typical Modeling Process . . . . . . . . . . . . . . . . . . . . . . . 23

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

P 52.1140 70.1055 5.8779 53.2367 0.0100 999.9900dP -0.0012 65.2659 -0.4622 45.9535 -862.1000 826.3900

ln(P) 3.5395 0.8675 0.1668 4.3806 -4.6052 6.9077dln(P) -0.0001 0.5451 -0.0031 10.0143 -6.0426 6.0234ln(DP) 0.0000 0.7575 0.4159 5.4669 -7.6523 3.6708dln(DP) -0.0001 0.5625 -0.0484 10.5352 -5.9802 5.7431

Table 2.1: Descriptive statistics of APP Hourly EP

DNGL Mean Std.Dev Skewness Kurtosis Minimum Maximum

P 2.3711 1.1653 2.9038 14.0326 1.0460 9.9780dP 0.0008 0.1171 -1.4495 35.5970 -1.4110 1.1010

ln(P) 0.7835 0.3693 1.1970 5.0733 0.0450 2.3004dln(P) 0.0003 0.0355 -0.3753 12.1076 -0.3757 0.2462

Table 2.2: Descriptive statistics of AECO Daily NGL

these tables that the standard deviations, skewness and kurtosis of natural gas and

electricity prices are pretty large. Correspondingly, their behaviors are difficult to

be handled. However, the standard deviation, as well as the skewness and kurtosis

are decreased significantly after taking the logarithm. Therefore we will use the

logarithmic prices rather than the raw spot prices in our model calibration. So the

log-returns are the main statistics that are tested in the following sections.

DEP Mean Std.Dev Skewness Kurtosis Minimum Maximum

P 44.2894 36.5783 4.0346 22.0515 13.3546 308.7258dP 0.0047 37.2572 0.1170 18.3032 -273.1254 223.7000

ln(P) 3.6322 0.4839 1.7302 6.8181 2.5919 5.7325dln(P) 0.0001 0.4291 0.1401 8.0037 -2.2816 1.9844

Table 2.3: Descriptive statistics of APP Daily EP

10

DNG Mean Std.Dev Skewness Kurtosis Minimum Maximum

P 2.6362 0.608 0.1913 2.4496 0.7600 3.9800dP 0.0025 0.1049 -0.5975 17.2041 -0.8700 0.6500

ln(P) 0.9417 0.2391 -0.3996 3.2065 -0.2744 1.3813dln(P) 0.0009 0.0578 -1.5655 66.3801 -0.7630 0.6180

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].

14

Skewness Kurtosis JS Statistic KS Statistic KS 5% ALHEP -0.0484 10.5352 29195.8320 0.1239 0.0122

DNGL -0.3753 12.1076 9441.0854 0.0724 0.0258DNG -1.5655 66.3801 123498.4492 0.1849 0.0498DEP 0.1401 8.0037 788.1870 0.1247 0.0498

Table 2.5: Statistics for Log-returns of NG and EP

Significance level: .20 .15 .10 .05 .01Acceptance Limits (AL): 1.07√

n1.14√

n1.22√

n1.36√

n1.63√

n

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

DNGL 2782 0.0003 0.0355 0.4896DNG 739 0.0009 0.0578 0.4266DEP 739 0.0001 0.4291 0.0092

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.

Strong seasonality, extreme price spikes implying fat tails of price distributions, non-

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:

L({Xt}Nt=1, θ) = fD(X1|θ)fD(X2|θ)...fD(XN |θ). (4.1)

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):


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 =

lnSt, t = 0, 1, ..., n, can be constructed as:

θ = argmaxθ

n−1∑t=1

ln f(Xt+1, θ|Xt) (4.21)

47

4.2.2 Calibration Results for Model 1A

Estimate Std. T-ratioκ 0.1042 0.0047 22.1702α -0.2392 0.0311 -7.6913σ2 0.0137 0.0009 15.2222ω 1.1432 0.0387 29.5401µ 0.0163 0.0053 3.0755γ 0.3398 0.0072 47.1944

Log-Likelihood -6701.7003

Table 4.1: APP Hourly EP Parameter Estimates for Model 1A

Estimate Std. T-ratioκ 0.0002 0.0011 0.1818α 6.3023 37.2891 0.1690σ2 0.0003 0.0000 10.7143ω 0.6250 0.1136 5.5018µ -0.0008 0.0017 -0.4706γ 0.0264 0.0023 11.4783

Log-Likelihood 5667.5633

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:


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 -

4.27), we get (see Appendix D.2)

A(s, t, T, θ) = iαs(1− e−κ(T−t))− σ2s2

4κ(1− e−2κ(T−t))

+ω

2κln

1 + γ2s2e−2κ(T−t)

1 + γ2s2

+iω(1− 2ψ)

κ(tan−1(γse−κ(T−t))− tan−1(γs))

+ω

κ

((1− 2ψ)iγse−κ(T−t) − 1

1 + γ2s2e−2κ(T−t)− (1− 2ψ)iγs− 1

1 + γ2s2

)B(s, t, T, θ) = ise−κ(T−t). (4.30)

Correspondingly, the CDF of Xt+1 given Xt is

f(Xt+1, θ|Xt) =1

2π

∫ ∞

−∞Φθ(s,Xt+1|Xt)e

−isXt+1ds

=1

π

∫ ∞

−∞R(e−isτh(θ, s))ds, (4.31)

where R(A) denotes the real part of A, and

τ = (Xt+1 − α)− e−κ(Xt − α) (4.32)

h(θ, s) = exp(− σ2s2

4κ(1− e−2κ) +

ω

2κln

1 + γ2s2e−2κ

1 + γ2s2

+iω(1− 2ψ)

κ(tan−1(γse−κ)− tan−1(γs))

+ω

κ

((1− 2ψ)iγse−κ − 1

1 + γ2s2e−2κ− (1− 2ψ)iγs− 1

1 + γ2s2

).

(4.33)

52

The maximum likelihood estimators θ is

θ = argmaxθ

n−1∑t=1

ln f(Xt+1, θ|Xt) (4.34)

4.3.2 Calibration Results for Model 1B

Estimate Std. T-ratioκ 0.1038 0.0046 22.5652α -0.2521 0.0272 -9.2684σ2 0.0177 0.0013 13.6154ω 0.7745 0.0605 12.8017ψ 0.5306 0.0074 71.7027γ 0.2291 0.0182 12.5879


Table 4.3: APP Hourly EP Parameter Estimates for Model 1B

Estimate Std. T-ratioκ 0.0003 0.0003 1.0000α 3.5111 1.0096 3.4777σ2 0.0004 0.0001 4.0000ω 0.4001 0.6153 0.6503ψ 0.4859 0.0569 8.5395γ 0.0187 0.0410 0.4561


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),

A(s1, s2, t, T, θ) = is1e−κ1(T−t),

B(s1, s2, t, T, θ) = is2e−κ2(T−t),

C(s1, s2, t, T, θ) = −is1α1(e−κ1(T−t) − 1)− is2α2(e

−κ2(T−t) − 1)

+σ2

1s21

4κ1

(e−κ1(T−t) − 1) +σ2

2s22

4κ2

(e−κ2(T−t) − 1)

+ρσ1σ2s1s2

κ1 + κ2

(e−(κ1+κ2)(T−t) − 1)

+ω

∫(1− eµis2e−κ2(T−t)

1 + γ2s22e

−2κ2(T−t))dt.

(4.42)

Suppose that t = 0 and T = 1, then we can compute the joint conditional PDF of

59

X1t+1, X

2t+1 given X1

t , X2t by taking Fourier transform:

f(X1t+1, X

2t+1|X1

t , X2t ) =

1

(2π)2

∫R2

e−i(s1X1t+1+s2X2

t+1)eΦ(X1t+1,X2

t+1|X1t ,X2

t )ds1ds2

=1

(2π)2

∫R2

e−is1X1t+1−is2X2

t+1+AX1t +BX2

t +Cds1ds2

=1

(2π)2

∫ ∞

−∞e−iT1u+Du2

du

∫ ∞

−∞e−iT2v+(E−F2

4D)v2+H(v)dv

=1

4π2

√π

−Dexp (

T 21

4D)

∫ ∞


4D)v2+H(v)dv (4.43)

where u = s1 + F2Ds2, v = s2 and

D :=σ2

1

4κ1

(e−κ1 − 1)

E :=σ2

2

4κ2

(e−κ2 − 1)

F :=ρσ1σ2

κ1 + κ2

(e−(κ1+κ2) − 1)

H := ω

∫ 1

0

(1− eµis2e−κ2τ

1 + γ2s22e

−2κ2τ)dτ

T1 := (X2t+1 − α2)− e−κ2(X2

t − α2)

T2 := (X1t+1 − α1)− eκ1(X1

t − α1)−F

2D

((X1

t+1 − α2)− e−κ2(X1t − α2)

).(4.44)

The integral H can be computed numerically as we have done in Model 1A. Then

the maximum likelihood estimators are obtained by:

θ = argmaxθ

n−1∑t=1

ln f(X1t+1, X

2t+1|X1

t , X2t ). (4.45)

The calibration results for Model 2A fitted to correlated daily natural gas and

electricity spot prices are given in Table 4.5 - 4.6.

60


Estimate Std. T-ratioκ1 0.0279 0.0092 3.0326α1 0.9741 0.0775 12.5690σ2

1 0.0581 0.0015 38.7333κ2 0.5007 0.0387 12.9380α2 3.3788 0.0263 128.4715σ2

2 0.1777 0.0143 12.4266ω 0.5027 0.1001 5.0220µ 0.2325 0.0549 4.2350γ 0.4247 0.0502 8.4602ρ -0.0309 0.0345 -0.8957


Table 4.5: Daily NG/EP Parameter Estimates for Model 2A


1 0.0581 0.0015 38.7333κ2 0.5690 0.0490 11.6122α2 3.5035 0.0263 133.2129σ2

2 0.1767 0.0164 10.7744ω 0.5816 0.1117 5.2068µ 0.2752 0.0543 5.0681γ 0.4481 0.0509 8.8035ρ -0.0462 0.0435 -1.0621


Table 4.6: Daily NG/One-Peak EP Parameter Estimates for Model 2A

Note: Table 4.5 - 4.6 are the estimations for Model 2A (4.37) fitted to correlated

AECO daily natural gas spot prices and Alberta Power Pool daily (and on-peak

Daily) electricity spot prices (739 observations). The second column gives the pa-

rameter estimates. The third column gives the standard error of the estimates. The

fourth column is the t-ratio of the second column to the third column.

61

Figure 4.8: Daily NG/EP Histogram Plot and Density Plot for Model 2A

Figure 4.9: Daily NG/On-peak EP Histogram Plot and Density Plot for Model 2A

Note: Figure 4.8 - 4.9 are comparisons of the histograms and the densities for

Model 2A (4.37) fitted to daily natural gas and daily (and on-peak daily) electricity

spot prices. We observe that the histogram plots are similar to the density plots.

62

Figure 4.10: Peak-finding Results for Model 2A fitting on Daily NG/EP

Figure 4.11: Peak-finding Results for Model 2A fitting on Daily NG/On-peak EP

Note: Figure 4.10 - 4.11 report Peak-finding results for Model 2A (4.37) fitted to

correlated AECO daily natural gas spot prices and Alberta Power Pool daily (and

on-peak daily) electricity spot prices. All the estimates are at the peaks of the curves,

which implies that we have found the maximized likelihood points.

63

4.5 Model 2B

In Model 2B, we consider the same model as defined in (4.37) but with a double

gamma distributed jump amplitude for logarithmic electricity prices. Again, we

assume that the state vector process [X1t X

2t ]′ is under the risk-neutral measure. The

conditional characteristic function has the same form as in (4.38), and A(·), B(·) and

C(·) in (4.38) satisfy the Riccati equations system (4.39) with boundary conditions

(4.40).

4.5.1 Double Gamma Jump Amplitude

The jump transform of double gamma jump amplitude is given as the following:

ϕ(A(·), B(·)) =

∫ ∞

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. (4.46)

Thus, we have a solution for A(·), B(·) and C(·) in the CCF as following:

A(s1, s2, t, T, θ) = is1e−κ1(T−t), B(s1, s2, t, T, θ) = is2e

−κ2(T−t),

C(s1, s2, t, T, θ) = −is1α1(e−κ1(T−t) − 1)− is2α2(e

−κ2(T−t) − 1)

+σ2

1s21

4κ1

(e−κ1(T−t) − 1) +σ2

2s22

4κ2

(e−κ2(T−t) − 1)

+ρσ1σ2s1s2

κ1 + κ2

(e−(κ1+κ2)(T−t) − 1)

+ω

2κ2

ln1 + γ2s2

2e−2κ2(T−t)

1 + γ2s22

+iω(1− 2ψ)

κ2

(tan−1(γs2e−κ2(T−t))− tan−1(γs2))

+ω

κ2

((1− 2ψ)iγs2e−κ2(T−t) − 1

1 + γ2s22e

−2κ2(T−t)− (1− 2ψ)iγs2 − 1

1 + γ2s22

).

(4.47)

64

The joint conditional PDF of X1t+1, X

2t+1 given X1

t , X2t can be computed by taking

the Fourier transform:

f(X1t+1, X

2t+1|X1

t , X2t ) =

1

(2π)2

∫R2

e−i(s1X1t+1+s2X2

t+1)eΦ(X1t+1,X2

t+1|X1t ,X2

t )ds1ds2

=1

4π2

√π

−Dexp (

T 21

4D)

∫ ∞


4D)v2+H(v)dv(4.48)

where u = s1 + F2Ds2, v = s2 and

D :=σ2

1

4κ1

(e−κ1 − 1)

E :=σ2

2

4κ2

(e−κ2 − 1)

F :=ρσ1σ2

κ1 + κ2

(e−(κ1+κ2) − 1)

H :=ω

2κ2

ln1 + γ2s2

2e−2κ2

1 + γ2s22

+iω(1− 2ψ)

κ2

(tan−1(γs2e−κ2)− tan−1(γs2))

+ω

κ2

((1− 2ψ)iγs2e−κ2 − 1

1 + γ2s22e

−2κ2− (1− 2ψ)iγs2 − 1

1 + γ2s22

)T1 := (X2

t+1 − α2)− e−κ2(X2t − α2)

T2 := (X1t+1 − α1)− eκ1(X1

t − α1)−F

2D

((X1

t+1 − α2)− e−κ2(X1t − α2)

).(4.49)

Then the maximum likelihood estimators are obtained by:

θ = argmaxθ

n−1∑t=1

ln f(X1t+1, X

2t+1|X1

t , X2t ). (4.50)

The calibration results of Model 2A for correlated daily natural gas and electric-

ity spot prices are given in Table 4.7 - 4.8.

65



1 0.0581 0.0015 38.7333κ2 0.4940 0.0375 13.1733α2 3.3919 0.0160 211.9938σ2

2 0.1891 0.0197 9.5990ω 0.3586 0.1669 2.1486ψ 0.7518 0.0413 18.2034γ 0.3018 0.1467 2.0573ρ -0.0318 0.0236 -1.3475


Table 4.7: Daily NG/EP Parameter Estimates for Model 2B


1 0.0581 0.0015 38.7333κ2 0.5580 0.0458 12.1834α2 3.5109 0.0126 278.6429σ2

2 0.1869 0.0152 12.2961ω 0.4535 0.1431 3.1691ψ 0.7774 0.0363 21.4160γ 0.3048 0.0956 3.1883ρ -0.0454 0.0328 -1.3841


Table 4.8: Daily NG/On-peak EP Parameter Estimates for Model 2B

Note: Table 4.7 - 4.8 are the parameter estimates for Model 2B (4.37) fitted to

correlated AECO daily natural gas spot prices and Alberta Power Pool daily (and

on-peak daily) electricity spot prices (739 observations). The second column gives

the parameter estimates. The third column gives the standard error of the estimates.

66

The fourth column is the t-ratio of the second column to the third column.

Figure 4.12: Daily NG/EP Histogram Plot and Density Plot for Model 2B

Figure 4.13: Daily NG/On-Peak EP Histogram Plot and Density Plot for Model 2B

Note: Figure 4.12 -4.13 are comparisons of the histograms and the densities for

Model 2B (4.37) fitted to AECO daily natural gas and Alberta Power Pool daily

(and on-peak daily) electricity spot price. We notice that the histogram plots are

similar to the density plots.

67

Figure 4.14: Peak-finding Results for Model 2B fitting on Daily NG/EP

Figure 4.15: Peak-finding Results for Model 2B fitting on Daily NG/On-peak EP

Note: Figure 4.14 - 4.15 report Peak-finding results for Model 2B (4.37) fitted

to correlated daily natural gas and Alberta Power Pool daily (and On-peak daily

electricity spot price. All the estimates are at the peak of the curves, which implies

that we have found the maximized likelihood points.

68

4.6 Model 3A

As we have mentioned previously, the correlation between two spot price series could

be between noises or between jumps. In Model 2A and Model 2B, we have used

correlation between noises. Now we investigate correlation between jumps. We start

with a weak type of correlation between jumps, saying, the two jump components

are generated from two independent Poisson processes but with the same arrived

rate. Let us consider the following 2-factor jump-diffusion model:

dX1t = κ1(α1 −X1

t )dt+ σ1dW1 +Q1tdP

1t (4.51)

dX2t = κ2(α2 −X2

t )dt+ σ2dW2 +Q2tdP

2t (4.52)

Again, here κ1 and κ2 are the mean-reverting intensities, α1 and α2 are the long-term

means, W 1t and W 2

t are two independent standard Brownian motions. Q1t and Q2

t are

the logarithmic jump amplitudes, P 1t and P 2

t are two discontinuous, one dimensional

Poisson processes with the same constant arrival rate 2ω. Then 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

0 σ2

dW1

dW2

+

Q1tdP

1t

Q2tdP

2t

(4.53)

Here we assume that the state vector process [X1t X2

t ]′ is under the risk-neutral

measure. Notice that (4.53) is affine where we have:

K0 =

κ1α1

κ2α2

, K1 =

−κ1 0

0 κ2

,

69

H0 =

σ21 0

0 σ22

, H(1)1 = H

(2)1 =

0 0

0 0

,

l0 = 2ω, l1 =

0

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)X2t + C(s1, s2, t, T, θ)

)(4.54)

where A(·), B(·) and C(·) satisfy the following equations system:

∂A(·)∂t

= κ1A(·)

∂B(·)∂t

= κ2B(·)

∂C

∂t= −κ1α1A(·)− κ2α2B(·)− 1

2σ2

1A(·)2 − 1

2σ2

2B(·)2

−2ω(ϕ(A(·) B(·))− 1) (4.55)


A(s1, s2, T, T, θ) = is1

B(s1, s2, T, T, θ) = is2

C(s1, s2, T, T, θ) = 0. (4.56)

4.6.1 Independent Jump Amplitude with Same Arrive Rate

In his paper on the transform analysis for Affine Jump-Diffusion processes (see [7]),

Duffie (2000) provided explicit transforms for a 2-dimensional affine jump-diffusion

70

model that have correlation between jumps. In that mode, the 2-dimensional jump

transform ϕ is defined as:

ϕ(c1, c2) = ω−1(ωaϕa(c1) + ωbϕb(c2) + ωcϕc(c1, c2)) (4.57)

where ω = ωa + ωb + ωc, ϕa and ϕb are individual jumps in each dimension, ϕc is

simultaneous correlated jumps between the two dimensions, ωa and ωb are the arrival

intensities of jumps in each dimension, ωc is the arrival intensity of the simultaneous

correlated jumps.

In Model 3A, we assume that the arrived rates for P 1t and P 2

t are equal. We

don’t consider the simultaneous correlated jumps, so that ωa = ωb = 2ω2

= ω.

We suppose again that the jump amplitude distributions are double exponentially

distributed with means γ1 and γ2, and the signs of jump amplitudes Q1t and Q2

t are

two independent Bernoulli random variables with parameter ψ1 and ψ2. Then the

jump transform is given as the following:

ϕ(A(·), B(·)) =1

2

[ ψ1

1− A(·)γ1

+(1− ψ1)

1 + A(·)γ1

+ψ2

1−B(·)γ2

+(1− ψ2)

1 +B(·)γ2

]. (4.58)

71

Solving the equation system (4.55) with the boundary conditions (4.56), we get

A(·) = is1e−κ1(T−t)

B(·) = is2e−κ2(T−t)

C(·) = is1α1(1− e−κ1(T−t))− σ21s

21

4κ1

(1− e−2κ1(T−t))

+is2α2(1− e−κ2(T−t))− σ22s

22

4κ2

(1− e−2κ2(T−t))

+ω

2κ1

ln1 + γ2

1s21e

−2κ1(T−t)

1 + γ21s

21

+iω(1− 2ψ1)

κ1

(tan−1(γ1s1e−κ1(T−t))− tan−1(γ1s1))

+ω

2κ2

ln1 + γ2

2s22e

−2κ2(T−t)

1 + γ22s

22

+iω(1− 2ψ2)

κ2

(tan−1(γ2s2e−κ2(T−t))− tan−1(γ2s2)).

(4.59)

Then the joint conditional density function (CDF) of X1t+1, X

2t+1 given X1

t , X2t

can be separated into two single integrals

f(X1t+1, X

2t+1|X1

t , X2t ) =

1

(2π)2

∫R2

e−i(s1X1t+1+s2X2

t+1)eΦ(X1t+1,X2

t+1|X1t ,X2

t )ds1ds2

=1

(2π)2

∫R2


t+1+AX1t +BX2

t +Cds1ds2

=1

(π)2

∫ ∞

0

R(e−is1τ1h1(θ, s1))ds1

∫ ∞

0

R(e−is2τ2h2(θ, s2))ds2

(4.60)

where

τ1 = (X1t+1 − α)− e−κ1(X1

t − α1)

τ2 = (X2t+1 − α)− e−κ2(X2

t − α2) (4.61)

72

h1(θ, s1) = exp(− σ2

1s21

4κ1

(1− e−2κ1) +ω

2κ1

ln1 + γ2

1s21e

−2κ1

1 + γ21s

21

+iω(1− 2ψ1)

κ1

(tan−1(γ1s1e−κ1)− tan−1(γ1s1))

)h2(θ, s2) = exp

(− σ2

2s22

4κ2

(1− e−2κ2) +ω

2κ2

ln1 + γ2

2s22e

−2κ2

1 + γ22s

22

+iω(1− 2ψ2)

κ2

(tan−1(γ2s2e−κ2)− tan−1(γ2s2))

). (4.62)

Notice that each integral in the CDF can be computed by the fast Fourier transform.

Consequently, the maximum likelihood estimators are obtained by:

θ = argmaxθ

n−1∑t=1

ln f(X1t+1, X

2t+1|X1

t , X2t ). (4.63)

73



1 0.0002 0.0000 5.8575κ2 0.4948 0.0372 13.3011α2 3.3808 0.0264 128.0606σ2

2 0.0347 0.0045 7.7111ω 0.4763 0.0629 7.5723ψ1 0.4492 0.0451 9.9601γ1 0.0468 0.0046 10.1739ψ2 0.7523 0.0557 13.5063γ2 0.4698 0.0460 10.2130


Table 4.9: Daily NG/EP Parameter Estimates for Model 3A

Note: Table 4.9 gives the parameter estimates for Model 3A (4.53) fitted to cor-

related AECO daily natural gas spot prices and APP daily electricity spot prices

(739 observations). The second column gives the parameter estimates. The third

column gives the standard error of the estimates. The fourth column is the t-ratio

of the second column to the third column.

74

Figure 4.16: Daily NG/EP Histogram Plot and Density Plot for Model 3A

Figure 4.17: Peak-finding Results for Model 3A fitting on Daily NG/EP

Note: Figure 4.16 is a comparison of the histogram and the density for Model

3A (4.53) fitted to correlated AECO daily NG and APP daily EP prices. We can

see that the histogram plot is similar to the density plot. Figure 4.17 reports Peak-

finding results for Model 3A fitted to correlated AECO daily NG and APP daily

Electricity spot price. All the estimates are at the peaks of the curves, which implies

that we have found the maximized likelihood points.

75

4.7 Model 3B

In Model 3A, the correlation between jumps is weak. Basically, the two jump compo-

nents are generated from two independent Poisson processes, although we assumed

that the arrival rates of the two Poisson processes are the same. In Model 3B, we

consider a tight correlation between jumps, supposing that the two jump components

are simultaneously correlated.

Consider the following 2-factor jump-diffusion model:

dX1t = κ1(α1 −X1

t )dt+ σ1dW1 +Q1tdPt (4.64)

dX2t = κ2(α2 −X2

t )dt+ σ2dW2 +Q2tdPt. (4.65)

Here κ1 and κ2 are the mean-reverting intensities of the log prices, α1 and α2 are

the long-term means, W 1t and W 2

t are two independent standard Brownian motions,

Q1tQ

2t are the logarithmic jump amplitudes, Pt is a discontinuous, one dimensional

Poisson process with constant arrival rate ω. Then the above 2-factor model can be

written in the matrix form as:

d

X1t

X2t

=

( κ1α1

κ2α2

+

−κ1 0

0 κ2

X1

t

X2t

)dt

+

σ1 0

0 σ2

dW1

dW2

+

Q1tdPt

Q2tdPt

. (4.66)

We assume again that the state vector process [X1t X2

t ]′ is under the risk-neutral

measure. Notice that (4.53) is affine where we have:

K0 =

κ1α1

κ2α2

, K1 =

−κ1 0

0 κ2

,

76

H0 =

σ21 0

0 σ22

, H(1)1 = H

(2)1 =

0 0

0 0

,

l0 = ω, l1 =

0

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)X2t + C(s1, s2, t, T, θ)

)(4.67)

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

−ω(ϕ(A(·) B(·))− 1) (4.68)


A(s1, s2, T, T, θ) = is1

B(s1, s2, T, T, θ) = is2

C(s1, s2, T, T, θ) = 0. (4.69)

4.7.1 Simultaneously Correlated Jump Amplitudes

Again, define the 2-dimensional jump transform ϕ as:

ϕ(c1, c2) = ω−1(ωaϕa(c1) + ωbϕb(c2) + ωcϕc(c1, c2)). (4.70)

77

where ϕa, ϕb, ϕc, ω, ωa, ωb, ωc are the same as defined in (4.57). In Model 3B, we

assume that there are only simultaneous jumps between X1t and X2

t , so that ωa =

ωb = 0 and ω = ωc.

Now we suppose that the jump amplitude of X1t is a normal distributed variable

Z1 with mean µ1 and variance σJ1, the jump amplitude of X2t is another normal

distributed variable Z2 with conditional mean µ2 +ρZ1 and variance σJ2, so that the

jump transform is given as the following (see Appendix B.3):

ϕ(A(·), B(·)) = exp[B(·)µ2 +

1

2σ2

J2B(·)2 + (A(·) + ρB(·))µ1 +1

2σ2

J1(A(·) + ρB(·))2].

(4.71)

Solving the system of equations (4.68) with the boundary conditions (4.69), we

get

A(·) = is1e−κ1(T−t)

B(·) = is2e−κ2(T−t)

C(·) = is1α1(1− e−κ1(T−t))− σ21s

21

4κ1

(1− e−2κ1(T−t))

+is2α2(1− e−κ2(T−t))− σ22s

22

4κ2

(1− e−2κ2(T−t))

+ω

∫(1− eB(·)µ2+ 1

2σ2

J2B(·)2+(A(·)+ρB(·))µ1+ 12σ2

J1(A(·)+ρB(·))2)dt.

(4.72)

If we suppose that t = 0 and T = 1, then the integral in C(·) can be computed

numerically. Thus, the joint conditional PDF of X1t+1, X

2t+1 given X1

t , X2t is:

f(X1t+1, X

2t+1|X1

t , X2t ) =

1

(2π)2

∫R2

e−i(s1X1t+1+s2X2

t+1)eΦ(X1t+1,X2

t+1|X1t ,X2

t )ds1ds2

=1

(2π)2

∫R2


t+1+AX1t +BX2

t +Cds1ds2. (4.73)

78

Notice that this 2-dimensional integral can not be separated as we have done in Model

3A. We have to compute it by taking the 2-dimensional inverse Fourier transform.

Consequently, the maximum likelihood estimators are obtained by:

θ = argmaxθ

n−1∑t=1

(ln f(X1t+1, X

2t+1|X1

t , X2t ). (4.74)



1 0.0003 0.0000 6.0057κ2 0.5035 0.0389 12.9269α2 3.4186 0.0232 147.4615σ2

2 0.0381 0.0043 8.8948ω 0.3541 0.0379 9.3337µ1 -0.0002 0.0017 -0.0898σ2

J1 0.0000 0.0000 0.0000µ2 0.3022 0.0501 6.0303σ2

J2 0.6812 0.0455 14.9744ρ -0.6297 17.0780 -0.0369


Table 4.10: Daily NG/EP Parameter Estimates for Model 3B

Notice: Table 4.10 gives parameter estimates for Model 3B (4.66) fitted to corre-

lated AECO daily natural gas spot prices and Alberta Power Pool daily electricity

spot prices (739 observations). The second column gives the parameter estimates.

The third column gives the standard error of the estimates. The fourth column is

the t-ratio of the second column to the third column.

79

Figure 4.18: Daily NG/EP Histogram Plot and Density Plot for Model 3B

Figure 4.19: Peak-finding Results for Model 3B fitting on Daily NG/EP

Note: Figure 4.18 is a comparison of the histogram and the density for Model 3B

(4.66) fitted to correlated AECO daily NG and APP daily electricity spot price. We

can see that the histogram plot is similar to the density plot. Figure 4.19 reports

corresponding Peak-finding results. All the estimates are at the peaks of the curves,

which implies that we have found the maximized likelihood points.

Chapter 5

Model Tests

As pointed by P. Wilmott (see [17]), the challenge of jump-diffusion models is the

“difficulty in parameter estimation”. The jump-diffusion model in its simplest form

needs an estimate of probability of a jump and its size. This can be made more

complicated by having a distribution for the jump size. In Chapter 4, we have

calibrated some 1-factor and 2-factor jump-diffusion models with different jump am-

plitude distributions. A natural question then appears: what model, particularly,

what jump amplitude distribution, best fits the empirical data? In this chapter, we

attempt to discover this question by imposing Monte Carol simulation and criterion

tests. Finally, we also investigate how the length of sample size affects the parameter

estimation.

5.1 Monte Carol Simulation

Simulated Price and Real Price

Using the parameter estimates we have obtained, we can simulate a series of prices

to compare with the real prices. Our simulation method is that given a set of

parameters, we simulate 50 paths of samples and set up upper and lower boundaries

for these samples. Then we randomly select one path of samples and examine whether

it is consistent with the real prices. However, it will be too restrictive to expect the

simulated prices and real prices to be exactly the same each other. Instead, we will

80

81

examine whether the real prices fall between the upper and lower boundaries. In

figure 5.1 - 5.10, we plot one typical path of simulated prices, the real prices, the

95% and 5% quantile plots. We can see that most of real prices are in the [5% 95%]

simulation boundaries, suggesing that these models provide a fit on natural gas and

electricity prices. However, it is hard to tell which model is the best to underlying

prices.

82

Figure 5.1: APP Hourly EP Simulated Prices and Real Prices for Model 1A

Figure 5.2: AECO Daily NG Simulated Prices and Real Prices for Model 1A

83

Figure 5.3: APP Hourly EP Simulated Prices and Real Prices for Model 1B

Figure 5.4: AECO Daily NG Simulated Prices and Real Prices for Model 1B

84

Figure 5.5: Daily NG/EP Simulated Prices and Real Prices for Model 2A

Figure 5.6: Daily NG/On-peak EP Simulated Prices and Real Prices for Model 2A

85

Figure 5.7: Daily NG/EP Simulated Prices and Real Prices for Model 2B

Figure 5.8: Daily NG/On-peak EP Simulated Prices and Real Prices for Model 2B

86

Figure 5.9: daily NG and daily EP Simulated Prices and Real Prices for Model 3A

Figure 5.10: daily NG and daily EP Simulated Prices and Real Prices for Model 3B

87

Mean Bias

Motivated by the simulation results obtained above, we want to test the mean bias

of the simulated price series for some models1. The best model should have the

lowest mean bias and standard deviation. We display our test results in Table 5.1 -

5.2. Notice that all these models performed quite well, since both the mean bias and

standard errors are small. We may compare the biases and standard deviations of a

single parameter estimate among the models. However, it is still hard to tell which

model performs best as a whole.

M-1A M-1B M-1CMean Bias Std. Mean Bias Std. Mean Bias Std.

κ 0.1101 -0.0059 0.0024 0.1094 -0.0056 0.0022 0.1099 -0.0058 0.0023α -0.2386 -0.0006 0.0213 -0.2513 -0.0008 0.0302 -0.2410 0.00270 0.0289σ2 0.0152 -0.0015 0.0008 0.0198 -0.0021 0.0011 0.0152 -0.0015 0.0009ω 1.1285 0.0147 0.0339 0.7715 0.0030 0.024 1.1466 -0.0007 0.0361ψ 0.5108 0.0163 0.0052 0.5316 -0.0010 0.0071 0.5237 0.0001 0.0070γ 0.3594 -0.0196 0.0072 0.241 -0.0119 0.0051 0.3584 -0.0189 0.0067

Table 5.1: Comparison of Simulation Results Fitting on Hourly EP

5.2 Criterion Test

The Monte Carlo simulation we performed above just provides a qualitative analysis

of goodness-of-fit for the jump-diffusion models calibrated in Chapter 4. Goodness-

of-fit or model comparison in general, is a complex issue that is constrained by the

limit of available market data and the shortage of effective test methods. Perhaps,

1Although we do not utilize this test on all models and all data sets, the results we get arehopefully suitable for all conditions.

88

M-2A M-2B M-2CMean Bias Std. Mean Bias Std. Mean Bias Std.

κ1 0.0349 -0.007 0.0121 0.0319 -0.0041 0.0078 0.0324 -0.0045 0.0094α1 0.9731 0.001 0.0722 0.9811 -0.006 0.0922 0.9801 -0.0329 0.0774σ2

1 0.0593 -0.0012 0.0013 0.0585 -0.0004 0.0017 0.0591 -0.001 0.0015κ2 0.6944 -0.1937 0.0362 0.682 -0.188 0.0333 0.6983 -0.1974 0.0343α2 3.3788 0.0000 0.0197 3.3813 0.0106 0.0254 3.3727 -0.0046 0.0302σ2

2 0.2382 -0.0605 0.0168 0.2476 -0.0585 0.0142 0.2359 -0.0597 0.0171ω 0.4344 0.0683 0.0861 0.3937 -0.0351 0.0529 0.6121 -0.0344 0.0794ψ 0.0005 0.232 0.0741 0.7549 -0.0031 0.0429 0.7345 0.0143 0.0424γ 0.5686 -0.1439 0.0708 0.3993 -0.0975 0.0377 0.5714 -0.1457 0.0655ρ -0.0292 -0.0017 0.0581 -0.023 -0.0088 0.0541 -0.0366 0.0056 0.0582

Table 5.2: Comparison of Simulation Results Fitting on Daily NG/EP

the most efficient way is using the calibrated models to do option pricing or hedging.

Then we can compare their performances to market quotes or compare their hedging

efficiency. As a result, the following criterion tests can only be used as a reference.

We should be very careful to interpret the testing results.

Akaike Information Criterion

Given a sample of data, the Akaike information criterion (AIC) quantifies the rel-

ative goodness-of-fit of various statistical models by using a rigorous framework of

information analysis. AIC produces a measure which balances between the complex-

ity of the model and the goodness of its fit to the sample data. The basic formula is

defined as:

AIC = 2k − 2 ln |L| (5.1)

where k is the number of parameters, and L is the value of likelihood function.

Although a model with many parameters will fit the data very well, it has less use.

Therefore, the utility of a model with too many parameters is limited in practice.

89

This is the problem of over-fitting. The AIC methodology attempts to find the

minimal model that correctly explains the data, so we say a preferred model is the

one that has the lowest AIC value.

Another variation of AIC is AICc, which is better than AIC when the sample

size is small. The formula for AICc is:

AICc = −2 ln |L|+ 2k(n/(n− k − 1)) (5.2)

where n is the sample size. Since AICC converges to AIC as n gets large, it is

suggested that we should always use AICC in any case.

Schwarz Criterion

An alternative criterion is the Schwarz criterion (SC) 2. It imposes a penalty for

including too many terms in a model. The general formula is given by:

SC = −2 ln |L|+ k ln(n) (5.3)

Again, L here is the value of likelihood function, and n is the sample size. The

preferred model is the one with the lowest value of the criterion.

Table 5.3 is a summary of the criteria tests for Model 1A (M-1A), Model 1B

(M-1B) and Model 1C (M-1C). Notice that the two kinds of exponential distributed

jump amplitudes (M-1A vs. M-1C) have almost the same values in terms of the

Likelihood, AICC and RC, suggesting that the two models are inter-exchangeable.

We also observe that M-1B has the lowest AICC and SC values fitting hourly elec-

tricity prices, indicating that it has the best goodness-of-fit for modeling hourly EP

2SC criterion is also called Bayesian information criterion (BIC).

90

prices. This is perhaps because of the fact that gamma distributed jump amplitudes

are more consistent with the big jumps (spikes) as observed from the evolution of

hourly electricity prices. On the contrary, the AICC and RC of M-1A and M-1C are

lower than that of M-1B in the case of modeling daily natural gas prices, which is

consonant with our observation too3.

Since the AIC and SC criteria can be compared among various models, we

present a comparison among all the 2-factor models in Table 5.3 fitted to the daily

natural gas and electricity prices. Compared to Model 2A-2B-2C (M-2A, M-2B,

M-2C), Model 3A (M-3A) and Model 3B (M-3B) have the comparably lowest AICC

and RC values. From this comparison we discover that, instead of considering dif-

ferent distributions of jump amplitude to get better goodness-of-fit, it is essential to

select a good model in advance. As we can see, the improvement produced from the

selection of the jump amplitude distribution is eventually limited.

Hourly EP Daily NGLikelihood AICC SC Likelihood AICC SC

M-1A -6701.7003 -5.6134 38.9242 5667.5633 -5.2548 30.3005M-1B -6726.0418 -5.6207 38.9169 5664.3332 -5.2536 30.3017M-1C -6701.6994 -5.6134 38.9242 5667.4837 -5.2547 30.3006

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.

91

Daily Gas/Electricity Daily Gas/On-peak ElectricityLikelihood AICC SC Likelihood AICC SC

M-2A 902.5042 -1.4956 26.0214 838.5339 -1.34856 26.1685M-2B 902.4670 -1.4955 26.0215 837.6027 -1.34633 26.1707M-2C 902.5291 -1.4956 26.0214 838.7119 -1.34898 26.1685M-3A 1294.1783 -2.2165 25.3005M-3B 1314.5294 -2.2477 25.2693

Table 5.4: Criteria Statistics for 2-factor Jump-Diffusion Models

5.3 Effects of Sample Size

A common problem in calibrating energy models is the shortage of enough market

data. However, the sample size is a very important statistical factor that may affect

the results of our estimation. We therefore want to know how the length of the sample

size affects the estimation. This section gives empirical and simulation studies about

the effects of data length in the calibration. Although we do not investigate all

possible combinations of data sets and models, we believe that the results we get

from this section are hopefully suitable for all conditions.

Empirical Data

We first take a look at Figure 5.11, in which the parameter estimates are plotted

with respect to the sample sizes. The corresponding statistics are also provided in

Table 5.5. We can see that our parameter estimates are not stable when the sample

sizes are changed. However, neither the statistics nor the curves can provide us a

clear judgment about the effects of the sample sizes on parameter estimation. So we

need to take more close look at the effects of sample size.

92

Figure 5.11: Parameter Estimates vs. Sample Size for M-1C fitting on Hourly EP

True Theta Mean Bias Std.κ 0.1041 0.1817 -0.0776 0.1061α -0.2383 -0.133 -0.1053 0.1411σ2 0.0137 0.012 0.0017 0.0034ω 1.1459 1.6515 -0.5056 0.6582ψ 0.5238 0.5468 -0.0230 0.0358γ 0.3395 0.3303 0.0092 0.0194

Table 5.5: Statistics for M-1C fitting on Hourly EP

93

A comparison of parameter estimates with respect to different sample sizes is

then plotted in Figure 5.12. We observe that in general, the parameter estimates are

more and more close to the true values when the sample size increases. This result

is consistent with our expectation.

Figure 5.12: Comparison of Parameter Estimates for M-1C fitting on Hourly EP

Notice: The bold solid line represents true parameter values, the line with diamonds

represents estimation of sample size 500, the line with x-marks represents estimation

of sample size 1000, the line with circles represents estimation of sample size 2000,

the line with points represents estimation of sample size 4000. We can see that the

estimates of sample size 4000 are generally the closest estimates to the true values.

94

Simulated Data

Ideally, if we can get large empirical data sets with different sample sizes, then we

can observe the biases of parameter estimations. Due to the limit of available data,

we use a simulation method described as the following: given a set of parameters, we

simulate 50 paths of samples with a given sample size. Using each sample path, we

reestimate the parameters. Put the 50 sets of reestimated parameters together and

compute the mean and standard deviation. The corresponding results are shown in

Table 5.6 - 5.7. We can see that the standard deviation tends to increase when the

length of the sample size decrease. Theoretically, this result is consistent with our

expectation: the more data we can use in our calibration, the better accuracy we

can get.

12000 6000 3000 1500 750Mean Std. Mean Std. Mean Std. Mean Std. Mean Std.

κ 0.1106 0.0020 0.1104 0.0032 0.1107 0.0042 0.1108 0.0069 0.0094 0.0095α -0.2444 0.0322 -0.2356 0.0419 -0.2318 0.0517 -0.2235 0.0913 0.0996 0.1006σ2 0.0154 0.0008 0.0153 0.0013 0.0151 0.0019 0.0153 0.0027 0.0030 0.0030ω 1.1417 0.0321 1.1474 0.0524 1.1594 0.0707 1.1385 0.1069 0.1415 0.1430ψ 0.5260 0.0082 0.5244 0.0099 0.5229 0.0128 0.5214 0.0227 0.0272 0.0275γ 0.3595 0.0070 0.3593 0.0110 0.3540 0.0144 0.3603 0.0186 0.0360 0.0363

Table 5.6: Effects of Sample Size for M-1C Fitting on Hourly EP

95

12000 6000 3000 1500 750Mean Std. Mean Std. Mean Std. Mean Std. Mean Std.

κ1 0.0282 0.0017 0.0284 0.0029 0.0293 0.0042 0.0301 0.0058 0.0327 0.0090α2 0.9669 0.0170 0.9678 0.0237 0.9776 0.0403 0.9470 0.0531 0.9579 0.0823σ2

1 0.0589 0.0003 0.0590 0.0006 0.0589 0.0008 0.0587 0.0011 0.0588 0.0015κ2 0.6950 0.0078 0.6933 0.0115 0.6988 0.0171 0.7002 0.0211 0.6943 0.0348α2 3.3636 0.0071 3.3635 0.0097 3.3619 0.0149 3.3632 0.0204 3.3603 0.0298σ2

2 0.2353 0.0053 0.2354 0.0058 0.2370 0.0086 0.2347 0.0111 0.2309 0.0168ω 0.6191 0.0281 0.6228 0.0333 0.6138 0.0537 0.6209 0.0625 0.6301 0.1076ψ 0.7488 0.0100 0.7454 0.0129 0.7541 0.0199 0.7552 0.0282 0.7490 0.0437γ 0.5653 0.0159 0.5606 0.0208 0.5666 0.0275 0.5701 0.0333 0.5700 0.0574ρ -0.0338 0.0153 -0.0350 0.0178 -0.0317 0.0326 -0.0412 0.0452 -0.0350 0.0458

Table 5.7: Effects of Sample Size for M-2C Fitting on Daily NG/EP

Chapter 6

Conclusion

Beginning in the United States, the deregulation of natural gas was launched in the

early 1990s. This process has often gone on in parallel with the deregulation of elec-

tricity. Nowadays, the competitive natural gas markets in the United Stated and

Canada have become fairly liquid. Electricity markets, on the other hand, are still

experiencing rapid restructuring and their development has proved to be uneven.

While a great amount of research has been dedicated to financial markets, empirical

studies on natural gas and electricity markets are not well developed so far. Due

to its physical characteristics, electricity markets need real-time balancing between

instantaneous supply and demand. This peculiar feature makes the research of elec-

tricity markets a challenging undertaking. Another restriction of researching gas and

electricity is the absence of available historical data, which as we know is abundant

in financial markets.

At the beginning of this thesis, we performed some empirical tests on natural gas

and electricity prices, including the distribution of log-returns, mean reversion be-

havior, volatility structure and correlations between gas and electricity prices. From

the distribution test we can reject the normality hypothesis for the log-returns of

natural gas and electricity prices. From the mean reversion test we observed that

electricity prices have strong mean-reverting behavior, which however could be due

to the presence of spikes. The historical volatilities for the log-returns of gas and elec-

tricity prices are not constant, and the correlation between gas and electricity also

96

97

varies with time. In a word, all these empirical tests have indicated the complexity

of modeling natural gas and electricity prices. To get more realistic representations

of gas and electricity prices, we add additional effects such as mean reversion, jumps

and correlation into the geometric Brownian motion. As a result, we adopted mean-

reverting jump-diffusion models for fitting the evolution of gas or electricity prices.

The pair of 1-factor models, Model 1A and Model 1B, are formed of mean re-

version, noises and jump components, differing only in the way they form the dis-

tributions of jump amplitudes. Comparing the calibration results with previous

researchers’ work (Model 1C), we found that double gamma distributed jump am-

plitude (with the shape parameter equals to 2) is a bit superior to the exponentially

distributed one in capturing the high spikes of electricity prices. The Double expo-

nential distribution, on the other hand, is a bit better for fitting natural gas prices.

However, there is not sufficient evidence to determine which model, Model 1A, Model

1B or Model 1C, is the best in general.

Encouraged by the relatively simple 1-factor models, we examined how to model

correlated gas and electricity prices by a couple of 2-factor jump-diffusion models

that have correlations between noises. Model 2A and Model 2B have the same form

as Model 2C, except the jump amplitude distributions are different. Each of the

two models (Model 2A and Model 2B) either has a double gamma or has a double

exponentially distributed jump amplitude for the electricity prices. Compared to

Model 2C, the calibration results have shown us that neither double exponential nor

double gamma distribution of the jump amplitude could improve calibration accu-

racy quantitatively.

In the last two 2-factor jump-diffusion models, the correlation between gas and

98

electricity prices were specified to be between jumps. We studied a 2-factor model

(Model 3A), in which the jump components are independent but the arrive rate is

the same. The calibration result indicated that Model 3A is a little better than

M-2A and M-2B for fitting on gas and electricity prices. Finally, we examined an-

other 2-factor model (Model 3B) that has simultaneously correlated jumps. Model

3B turned out to be the best one among all the 2-factor models calibrated in this

thesis in terms of matching the evolution of gas and electricity prices.

Although they were not directly represented in this thesis, the numerical methods

developed in our calibration programs have extended our capability of calibrating

the so-called affine jump-diffusion models. If the computational cost of multi-Fourier

transform can be accepted, we do not have to require a explicit probability density

function for the maximum likelihood estimation.

In future research, there are a couple of directions we can explore. First and

the most important, we can use the models we calibrated in this thesis to do option

pricing. Then we can investigate the performances of these models by comparing the

option prices obtained from these models with market quotes. Second, we can fur-

ther explore the 2-factor jump-diffusion models using the results we have achieved.

For example, we can investigate the possibility of applying these models on a pair

of correlated futures price series. We can think about deploying correlation between

drifts. More generally, we can even put all possible correlations (between drifts,

between noises, and between jumps) together in one 2-factor model. Finally, we

can consider proposing a stochastic volatility process for modeling natural gas and

electricity prices. As we mentioned before, the jump-diffusion model is not the only

explanation of the special behavior of gas and electricity prices. Stochastic volatility

99

models may perform better for modeling gas and electricity prices if we can fully

exploit the information of the joint conditional characteristic function.

Natural gas and electricity are the leading commodities in energy industry, and

are becoming more and more important given the world-wide energy crisis. It is

definitely worthy of pushing forward more research on these peculiar but interesting

energy assets. However, we are just unable to discuss more in this thesis.

Appendix A

Transform Analysis for Affine Jump-diffusion

Models

Suppose that Xt is an affine jump-diffusion process as defined in Chapter 3. Given

an initial condition X0, the coefficient tuple θ = (K0, K1, H0, H1, l0, l1) determines a

transform Φθ : Cn × [0,∞)× [i,∞)×D 7→ C of XT conditional on Xt:

Φθ(u, t, T,Xt) = Eθ[eu·XT |Xt] (A.1)

where Eθ denotes expectation under the distribution of XT determined by θ. In [7],

Duffie et al have proved that if we suppose θ = (K0, K1, H0, H1, l0, l1) is well-behaved

at (u, T ), then the transform Φθ of Xt defined by (A.1) exists and is given by:

Φθ(u, t, T,Xt) = exp(M(u, t, T ) +N(u, t, T ) ·Xt

)(A.2)

where M(·) and N(·) satisfy the following complex-valued Riccati equations:

∂M(u, t, T )

∂t= −K0(t)N − 1

2N

′H0(t)N + l0(ϕ(N, t)− 1)

∂N(u, t, T )

∂t= −K1(t)

′N − 1

2N

′H1(t)N + l1(ϕ(N, t)− 1) (A.3)


M(u, T, T ) = 0

N(u, T, T ) = u, (A.4)

where ϕ(c) =∫

Rn exp(cz)dvt(z) is so-called “jump transform” that determines the

jump amplitude distribution.

100

Appendix B

Some Integrals

B.1 Integral 1

Suppose

φ(c) =

∫ ∞

0

ec·zλe−λzdz (c ∈ C). (B.1)

Let c = it, then we have:

φ(c) =

∫ ∞

0

eit·xλe−λxdx

= λ

∫ ∞

0

(cos(tx) + i sin(tx))e−λxdx. (B.2)

Since ∫ ∞

0

cos(tx)e−λxdx =λ

λ2 + t2∫ ∞

0

sin(tx)e−λxdx =t

λ2 + t2. (B.3)

Then we have

φ(c) = λ(λ

λ2 + t2+ i

t

λ2 + t2)

= (1− it

λ)−1

=λ

λ− c. (B.4)

101

102

B.2 Integral 2

consider the following integral:

ϕ(c) =

∫ ∞

0

eczzβ−1ezλ

Γ(β)λβdz. (B.5)

Let c = it, then we have

ϕ(c) =

∫ ∞

0

xβ−1e−(1−λit) xλ

Γ(β)λβdx. (B.6)

Let y = (1−λit)xλ

, then dy = 1−λitλdx. Hence,

ϕ(c) =

∫ ∞

0

(λy

1− λit)β−1 e−y

Γ(β)λβ

λdy

1− λit

=1

(1− λit)βΓ(β)

∫ ∞

0

yβ−1e−ydy

=1

(1− λc)β. (B.7)

103

B.3 Integral 3

Consider the following integral where Z2 is conditional on Z1:

ϕ(A(·) B(·)) =1√

2πσJ1

∫R

eA(·)Z1e− 1

2σ2J1

(Z1−µ1)2

dZ1

× 1√2πσJ2

∫R

eB(·)Z2e− 1

2σ2J2

(Z2−µ2−ρµ1)2

dZ2

(B.8)

We fist solve the second integral in (B.8) :

1√2πσJ2

∫R

eB(·)Z2e− 1

2σ2J2

(Z2−µ2−ρµ1)2

dZ2

=1√

2πσJ2

∫R

e− 1

2σ2J2

(Z22−2Z2(µ2+ρZ1+σ2

J2B(·))+(µ2+ρZ1)2

= e1

2σ2J2

((µ2+ρZ1+σ2J2B(·))2)−(µ2+ρZ1)2

= eB(·)(µ2+ρZ1)+ 12σ2

J2B(·)2 . (B.9)

Therefore,

ϕ(A(·) B(·)) =1√

2πσJ1

∫R

eA(·)Z1e− 1

2σ2J1

(Z1−µ1)2

dZ1

× 1√2πσJ2

∫R

eB(·)Z2e− 1

2σ2J2

(Z2−µ2−ρµ1)2

dZ2

=1√

2πσJ1

∫R

eA(·)Z1e− 1

2σ2J1

(Z1−µ1)2

eB(·)µ2+ 12σ2

J2B(·)2+B(·)ρZ1dZ1

= exp[B(·)µ2 +

1

2σ2

J2B(·)2 + (A(·) + ρB(·))µ1 +1

2σ2

J1(A(·) + ρB(·))2].

(B.10)

Appendix C

Parameter Estimates for Model 1C and Model 2C

C.1 Parameter Estimates for Model 1C

Model 1C has the same form as Model 1A and Model 1B, except that the jump

amplitude for logarithmic gas or electricity spot prices is specified as exponentially

distributed and the sign of the jumps is defined as a Bernoulli random variable. Since

we have discussed this issue in Chapter 4 when we calibrate Model 1A and 1B, here

we only present the calibration results of Model 1C for the purpose of comparison.

Estimate Std. T-ratioκ 0.1041 0.0047 22.1489α -0.2383 0.0312 -7.6378σ2 0.0137 0.0009 15.2222ω 1.1459 0.0386 29.6865µ 0.5238 0.0078 67.1538γ 0.3395 0.0072 47.1528


Table C.1: Alberta Power Pool Hourly EP Parameters Estimates for M-1C

Estimate Std. T-ratioκ 0.0004 0.0008 0.5000α 2.6927 4.0620 0.6629σ2 0.0003 0.0000 7.6497ω 0.6261 0.1141 5.4873µ 0.4885 0.0310 15.7581γ 0.0264 0.0023 11.4783


Table C.2: AECO Daily NG Parameters Estimates for M-1C

104

105

C.2 Parameter Estimates for Model 2C

Model 2C has the same form as Model 2A and Model 2B, except that the jump am-

plitude for logarithmic electricity spot price is specified as exponentially distributed

and the sign of the jumps is defined as a Bernoulli random variable. Since we have

discussed this issue in Chapter 4 when we calibrate Model 2A and 2B, here we only

present the calibration results of Model 2C for the purpose of comparison.


1 0.0581 0.0015 38.7333κ2 0.5009 0.0387 12.9432α2 3.3681 0.0287 117.3554σ2

2 0.1762 0.0142 12.4085ω 0.5777 0.1091 5.2951µ 0.7488 0.0404 18.5347γ 0.4257 0.0505 8.4297ρ -0.0310 0.0397 -0.7809


Table C.3: Daily NG/EP Parameter Estimates for M-2C


1 0.0581 0.0015 38.7333κ2 0.5705 0.0483 11.8116α2 3.4873 0.0289 120.6678σ2

2 0.1739 0.0164 10.6037ω 0.6872 0.1280 5.3688µ 0.7758 0.0354 21.9153γ 0.4515 0.0515 8.7670ρ -0.0468 0.0501 -0.9341


Table C.4: Daily NG/On-peak EP Parameter Estimates for M-2C

Appendix D

Solving the Riccati Equations

D.1 Riccati Equations for Model 1A

The complex-valued Riccati equations in Model 1A is given as the following:

∂A(s, t, T )

∂t= −καB(s, t, T )− 1

2σ2B(s, t, T )2

+ω


1 + γ2s2e−2κ(T−t))dt

∂B(s, t, T )

∂t= κB(s, t, T, θ) (D.1)


A(s, T, T ) = 0

B(s, T, T ) = is, (D.2)

It is easy to solve the second equation for B(s, t, T ) with respect to t, given the

initial condition B(s, T, T ) = is,

B(s, t, T ) = ise−κ(T−t). (D.3)

Then, we integrate both sides of the first equation in (D.1) with respect to t,

A(s, t, T ) =

∫−καise−κ(T−t)dt+

∫1

2σ2s2e−2κ(T−t)dt

+ω


1 + γ2s2e−2κ(T−t))dt. (D.4)

106

107

We can compute the first two integrals in (D.4) explicitly,∫−καise−κ(T−t)dt = −καis1

κe−κ(T−t) + C1 = −αise−κ(T−t) + C1 (D.5)

∫1

2σ2s2e−2κ(T−t)dt =

σ2s2

2

1

2κe−2κ(T−t) + C2 =

σ2s2

4κe−2κ(T−t) + C2 (D.6)

Under the initial condition A(s, T, T ) = 0, these constants C1 and C2 can be

chosen in the way such that each of the two integrals equals to zero. Hence,

C1 = αis

C2 = −σ2s2

2κ(D.7)

We have no explicit solution for the last integral in (D.4). However, suppose that

t = 1 and T = 1, then the last integral in (D.4) can be computed numerically.

Therefore,

A(s, t, T, θ) = iα(1− e−κ(T−t))− σ2s2

4κ(1− e−2κ(T−t))

+ω


1 + γ2s2e−2κ(T−t)dt (D.8)

108

D.2 Riccati Equations for Model 1B

Consider the following complex-valued Riccati equations:

∂A(s, t, T )

∂t= −καB(s, t, T )− 1

2σ2B(s, t, T )2

+ω[ψ(1 + λB(s, t, T )β + (1− ψ)(1− λB(s, t, T )β

(1− λ2B(s, t, T )2)β− 1

]∂B(s, t, T )

∂t= κB(s, t, T, θ) (D.9)


A(s, T, T ) = 0,

B(s, T, T ) = is. (D.10)

It is easy to solve the second equation for B(s, t, T ) with respect to t, given the

initial condition B(s, T, T ) = is,

B(s, t, T ) = ise−κ(T−t) (D.11)

Then, we integrate both sides of the first equation in (D.9) with respect to t,

A(s, t, T ) =

∫−καise−κ(T−t)dt+

∫R

1

2σ2s2e−2κ(T−t)dt

−ω∫ψ(1 + λB)β − ψ(1− λB)β + (1− λB)β(1− (1 + λB)β)

(1− λ2B2)βdt.

(D.12)

Now we compute the five integrals in (D.12) as the following:∫−καise−κ(T−t)dt = −καis1

κe−κ(T−t) + C1 = −αise−κ(T−t) + C1, (D.13)

109

∫1

2σ2s2e−2κ(T−t)dt =

σ2s2

2

1

2κe−2κ(T−t) + C2 =

σ2s2

4κe−2κ(T−t) + C2, (D.14)

ψ

∫ ( 1 + λB

1− λ2B2

)dt =

ψ

κ

∫(1 + x)β−1

xdx (x =

iλse−κ(T−t)

1− iλse−κ(T−t))

=ψ

κ

∫ ∑β−10 (β−1

m )xm

xdx

=ψ

κ

β−1∑m=1

(β−1m )

∫xm−1dx+

ψ

κ

∫1

xdx

=ψ

κ

β−1∑m=1

(β−1m )

xm

m+ψ

κln |x|+ C3, (D.15)

ψ

∫ ( 1− λB

1− λ2B2

)dt =

ψ

κ

∫(1 + x)β−1

xdx (x =

−iλse−κ(T−t)

1− iλse−κ(T−t))

=ψ

κ

β−1∑m=1

(β−1m )

xm

m+ψ

κln |x|+ C4, (D.16)

∫(1− λB)β(1− (1 + λB)β)

1 + λ2s2e−2κ(T−t)dt =

∫1− (1 + λB)β

(1 + λB)βdt

=

∫1− xβ

xβdt (x = 1 + λise−κ(T−t))

=1

κ

∫1

xβ

β−1∑o

xidx

=1

κ

β−1∑i=0

1

i− β + 1xi−β+1 + ln |x|+ C5. (D.17)

Under the initial condition A(s, T, T ) = 0, these constants Ci=1,2,3,4 can be chosen

in the way such that each of the four integrals equals to zero.

We can choose different β corresponding to different distribution of jump am-

plitudes. For example, if we set β = 1, we get the exponential distributed jump

110

amplitudes. In Model 1B we set β = 2, then we have the solution for A(·) and B(·)

as is given in (4.30).

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