On Carbon Emission Credits Options Pricing

International Journal of Innovation Engineering and Science Research
Open Access
Volume 2 Issue 2 March 2018 8|P a g e
ABSTRACT
On Carbon Emission Credits Options Pricing
Richard OnyinoSimwa
Division of Actuarial Science& Financial Mathematics,
School of Mathematics, University of Nairobi
The effect of adverse climate change is of major concern worldwide and several approaches are being developed
to mitigate against anticipated economic and social disaster. Carbon emissions has been identified as a major
contributor to the adverse climate change and following the Kyoto protocol , European countries have , through a
caucus, effected a market to reward or fine members depending on their compliance position. The commodity for
the market is the carbon emission credits. Stochastic models for pricing of options on these credits are
considered in this paper. In particular, we determine the price basing on the Normal Inverse Gaussian and the
Brownian Motion models. Maximum Likelihood Estimation is applied to determine model parameter estimates in
each case. It is shown that the Normal Inverse Gaussian model has a better fit to the data but gives higher prices
for a given strike price , compared to the Brownian Motion model.
Key Words: Carbon Emission Credit, Brownian Motion, Kyoto Protocol Compliance, Normal Inverse Gaussian
Distribution, Fourier Transform, Risk-Neutral Option Pricing.
I. INTRODUCTION
Greenhouse gas (GHG) emissions come from the burning of fossil fuels for energy (e.g. for electricity and
transport). When oil, gas or coal burns, carbon contained within it combines with oxygen in the air to create
carbon dioxide. Globally, almost 80% of GHG emissions come from human sources. Global GHG emissions grew
by approximately 42% between 1990 and 2011, with the bulk growth occurring in emerging markets and
developing countries.
The release of GHGs and their increasing concentration in the atmosphere are already having an impact on the
environment, human health and the economy. These impacts are expected to become more severe, unless
concerted efforts to reduce emissions are undertaken. Environmental impacts include the following. Annual
temperatures are expected to rise, increased coastal flooding due to increased temperatures, heat waves that
could result to forest fires and Shrinking water supplies. Human health impacts include high temperatures that
may increase the risk of deaths from dehydration and heat stroke, an increase in water-, food-, vector- and
rodent-borne diseases, cancer diseases that develop due to air pollution. Economic impacts include agriculture,
forestry, tourism and recreation being affected by changing weather patterns and damage to infrastructure
caused by extreme weather events.
In order to address climate change globally, the Kyoto Protocol was introduced. The Kyoto Protocol is an
international agreement linked to the United Nations Framework Convention on Climate Change, which
commits its parties by setting international binding emission reduction targets.
The Kyoto Protocol was adopted in Kyoto, Japan, on 11 December 1997 and entered into force on 16th
February, 2005. Participating countries that have ratified the Kyoto Protocol have committed to cut GHG
emissions. The Kyoto Protocol sets binding emission reduction targets for participants. The goal of Kyoto was
to see participants collectively reducing emissions of greenhouse gases by 5.2% below the emission levels of
1990 by 2012.

Richard Onyino Simwa. “International Journal of Innovation Engineering and Science Research”
Volume 2 Issue 2 March 2018 9|P a g e
Binding emissions reduction commitment for participants meant that the space to pollute was limited, and
what is scarce and essential commands a price. Greenhouse gas emissions- most prevalently carbon dioxide-
became a new commodity. Kyoto Protocol now began to internalize what was now recognized as an unpriced
externality. Since carbon dioxide is the principle greenhouse gas, people speak simply of trading in carbon.
This leads us to the second, the flexible mechanisms of the Kyoto Protocol, based on the trade of
emissions permits. Kyoto Protocol countries bound to targets have to meet them largely through domestic action-
that is, to reduce their emissions onshore. But they can meet part of their targets through the “market-based
mechanisms” that ideally encourage GHG abatement to start where it is most cost-effective- for example, in the
developing world. Quite simply, it does not matter where emissions are reduced, as long as they are removed
from the planet’s atmosphere. The Kyoto mechanisms are, InternationalEmissions Trading (IET), Clean
Development Mechanism (CDM) and Joint Implementation (JI).
Parties with commitments under the Kyoto Protocol that have accepted targets for limiting or reducing
emissions are called Annex I parties. These countries are set a legally binding cut for GHG emission to 5.2%
below their 1990 level. This reduction is to be attained in sum over all Annex I members over a five year
compliance period which is from 2008 to 2012. The concrete implementation is as follows: Each Annex I member
is assigned a certain Carbon dioxide gas amount, which equals to (5 years)*(country’s emission in 1990)*(1-
0.0052). This credit is measured in the so- called Assigned Amount Units (AAUs), corresponding to one ton of
carbon dioxide. Each member faces penalties if its entire emission within the compliance period 2008-2012
exceeds member’s total number of AAUs.
The IET mechanism allows annex I members that have emission units to spare to sell the excess to
annex I members that are over their targets. Therefore, the carbon allowances traded in IET mechanism are
called AAUs.
The CDMmechanism allows an Annex I member to implement an emission-reduction project in
developing countries. Such projects can earn saleable certified emission reduction (CER) credits, each
equivalent to one tonne of Carbon dioxide gas, which can be counted towards meeting Kyoto targets.
II. LITERATURE REVIEW
Carmona and Hinz(2011) examined the spot EUA returns that exhibit a volatility clustering feature and the
carbon market system that is impacted by the announcements of CO2 emissions policies. They proposed a
regime-switching jump diffusion model (RSJM) with a hidden Markov chain to capture not only a volatility
clustering feature, but also the dynamics of the spot EUA returns that are influenced by change in the CO2
emissions policies, and thereby altering jump arrivals. They concluded that RSJM is the best model to fit the
price behavior of the carbon markets and to price its related derivatives.
Chevallier, J. and Sevi B. (2014) argue that jumps need to be explicitly taken into account when modeling
spot and future carbon price series. He says that there is evidence that suggests that carbon futures are a pure
jump process without a continuous component and a relatively high activity index. The occurrence of jumps on
the carbon market may be related to information disclosure about allocation, or changes in the perimeter of the
scheme. He concludes that derivatives models such as the models by Cont and Tankov (2004), and the
CGMY model (Carr et al. (2002)) would be plausible candidates, since these models accommodate a pure-
jump stochastic process with activity indices above unity, as found in the carbon futures data.
Seifert et al(2007) explain that European call and put options are actively traded on EUA future contracts.
Since 2006, trades of options maturing in December of each year (prior to 2012) have produced a term
structure of option prices. It is noted that, whether or not traders are using Black-Scholes to price options on
EUAs and future contracts, it is important to have option price formulas based on underlying martingale with
binary terminal value.
Bolviken and Benth(2000) argued that traders of carbon emissions need to have a valid carbon dioxide
spot price model so that it can be possible to value potential derivatives and so that carbon emitting companies
can be able to better assess their production costs and support emissions-related investment decisions. They
therefore presented a tractable stochastic equilibrium model reflecting stylized features of the emissions trading
scheme and analyzed the resulting carbon spot price dynamics. Their main findings were that carbon prices do
not have to follow any seasonal patterns, discounted prices and should possess the martingale property, and
an adequate carbon price process should exhibit a time- and price- dependant volatility structure.

Volume 2 Issue 2 March 2018 10|P a g
e
Carmona and Hinz (2011) argue that martingales finishing at two-valued random variables can be
considered as basic building blocks which form the risk-neutral futures price dynamics. They therefore suggest
a model for two-valued martingales, flexible in terms of time- and space changing volatility and capable to
match the observed historical or implied volatility of the underlying future.
However, Bolviken and Benth (2000), argue that the family of Normal Inverse Gaussian (NIG) distribution
is able to portray stochastic phenomena that have heavy tails or are strongly skewed. In addition to that, NIG
distributions are not confined to the positive half axis. Therefore, with the NIG distribution the financial analyst
has at its disposal a model that can be adapted to many different shapes while the distribution of sums of
independent random variables are still trivial to compute.
III. METHODOLOGY
3.1 .1 Levy Process
Definition 1.1 A cadlag real valued stochastic process such that is called a Levy
Process if it has stationary independent increments and is stochastically continuous.
Brownian motion
The Brownian motion is a Levy process that has a drift. The standard Browniam motion follows a normal
distribution with mean μ and variance σ2
.
3.1.2 Normal Inverse Gaussian process (NIG)
The Normal inverse Gaussian process (NIG) is a Levy process that has normal inverse Gaussian
distributed increments. Specifically, has a distribution with parameters
and .
The distribution has a probability density function
Where
is the modified Bessel function of the third kind, while the characteristic function is given by
The mean, variance, skewness and kurtosis of NIG distribution are
Mean=
Variance=
0{ ( )}tX t  (0) 0X 
0{ ( )}tX t 
( )X t ( , , , )NIG t t    
0,| | , 0       
( , , , )NIG     
2
2
2 21
( ; , , , ) 2
2
( ( ) )
exp{ ( )}
( )
NIG x
K x
f x
x
   
  
    
  
 
   
 
1
0
1 1
( ) exp{ ( )}
2 2
n
Z
K Z u u du
u



  
2 2 2 2
( ) exp( ( ( ) )) .i u
NIG u iu e 
          
2 2


 


2
2 2 3
( )

 

e
Skewness=
Kurtosis=
3.1.3 Symmetric Normal Inverse Gaussian distribution
The Symmetric NIG Levy process has symmetric NIG marginals. The NIG distribution is symmetric when the
skewness parameter . In this case, the density of a symmetric NIG is
It follows from the equations of the mean, variance and kurtosis, that is mean, is variance and is
kurtosis. We will denote the distribution of symmetric NIG by
The characteristic function for symmetric NIG is
(i)
By an inspection of the characteristic function we can see that the characteristic generator of symmetric Normal
inverse Gaussian distribution is given by
(ii)
where,
3.2 BASIC MODELLING OF THE COMPLIANCE EVENT ( see Carmona and Hinz(2011) )
In the one-period setting, credits are allocated at the beginning of the period to enable allowance trading until
time and to encourage agents to exercise efficient abatement strategies. At the compliance date , market
participants cover their emissions by redeeming allowances or pay a penalty per unit of pollution not offset by
credits. In this one-period model, unused allowances expire and are worthless because we do not allow for
banking into the next period. At compliance date the allowance price is a random variable taking only the
values 0 and . More precisely, if the market remains under the target pollution level, then the price approaches
zero. Otherwise, the allowance price tends to the penalty level .
All the relevant asset price evolutions are assumed to be given by adapted stochastic processes on a filtered
probability space on which we fix an equivalent probability measure Q ~P which we call
the spot martingale measure.
1
2 2 2
3
( )

   
2 2
2 2 2
4
3 1
 
  
 
 
  
0 
2
1
2
1
( )
1
NIG
x
K
f x e
x




 

        
 
  
 



3
3


( ,0, , )SNIG   
2
1 1
( )
u
iu
u e e



 
      
 

2
1 1
( ) e



 
 
   
 

 
T T

T TS


  [0, ]
, , , t t T
F F 
 

e
We denote by the price process of a future contract with maturity date T written on the allowance
price. Given the digital nature of the terminal allowance price, the central object of our study is the event
of non-compliance which settles the -dichotomy of the terminal future prices by .
Furthermore, a standard no-arbitrage argument shows that the future prices needs to be a
martingale for the spot martingale measure ,Q. Hence, the problem of allowance price modeling reduces to the
appropriate choice of the martingale
We choose our starting point to be the non-compliance event which we describe as the event where a
hypothetic positive-valued random variable exceeds the boundary 1, say . If one denotes by
the total pollution within the period which must be balanced against the total number of
credits issued by the regulator, then the event of non-compliance should be given by which
suggests that should be viewed as the normalized total emission . However in our modeling, we
merely describe the non-compliance event. Strictly speaking, so any random variable with
,
would do as well. On this account, we do not claim that represents the total normalized emission . So
the allowance Spot price is given by the martingale
(1)
.where At = St in this case. simplifying the notation, we consider the normalized futures price process
The random variable is chosen from a suitable parameterized family of random variables.
The random variable modeled by geometric Brownian motion (gbm) is given by
where is a standard Brownian motion and is the volatility parameter. Since is a martingale
with respect to the underlying Brownian motion.
which is given by,
(2)
and it solves the stochastic differential equation
(3)
[0, ]( )t t TS 
TS
TN F {0, } 1T NS 
[0, ]( )t t TS 
(1 | ), [0, ]Q
t N tA F t T  
TN F
T { 1}TN   
TE [0, ]T (0, )  
{ }TN E  
T /TE 
T
{ 1} { / 1}T TE    
T /TE 
{ 1}(1 | ), [0, ]T
Q
t tA E F t T   
{ 1}: (1 | ), [0, ]T
t
t t
S
a E F t T
    
T
T
2
0 0
1
,
2
0
T T
s s sdW ds
T e
    
tW  0,gbm  T
1 2
0 0 0
2
( )
T t
s s s
t T
st
a ds dW
a
ds
 

 
  
  
 
 
 

' 1
( ( ))t t t tdS S Z dW 
  

e
where the positive valued function is given by
(4)
The martingale is a random variable taking only the values 0 and and satisfies
(6)
As an alternative model for we introduce the geometric Levy process
where is a normal inverse Gaussian Levy process and , where are i.i.d.
If we specify the parameters at time t=1, the first two moments (expectation and variance) are
,
This means that the log returns with time increment equal to 1 have expectation (or volatility)
when they are modeled as independent normally inverse Gaussian distributed variables. We should
note that when using the NIG process for option pricing, the location parameter of the distribution has no effect
on the option value, so for convenience we will take
Since we have an infinitely divisible characteristic function, we can define the NIG process ,
which starts at zero and has independent and stationary increments each with a distribution and
the entire process has a law.
Note: for each , where is an inverse Gaussian subordinator, , which is
independent of the standard Brownian motion with parameters and
We have the following connection between the parameters in the geometric Brownian motion and normal inverse
Gaussian models
,
The Levy-Kintchine representation of is
,
(0, ) tT t Z 
2
2
,t
t T
ut
Z
du




[0, ]t T

{lim {0,1}} 1t
t T
P a

 
T
0 exp( ( ) )T nig tt Y   
tY 1
T
T tt
Y Y
  1,...,t N
1 2 2
[ ]E Y

 

  
2
1 3/22 2
[ ]Var Y

 


1[ ]nig E Y 
1[ ]Var Y
0nig 
 1 , 0tY Y t 
( , , )NIG   
( , , )NIG t  
( ( ))t T tY Y 0t  ( )T t ( ; , )IG l a b
1a  2 2
b    
2 2gbm nig

 
 
 
  
2
2
3/22 2
.


 


tY
~
0 {0}
z ( , z),
t
t R
Y t N dt d    1[ ]E Y 

e
where is a Poisson random measure and is the
compensated Poisson random measure associated to . The levy measure of is
.
It follows from the Levy-Khintchine representation that the normal inverse Gaussian Levy process is a pure jump
process.
In the symmetric case (i.e., when ), we have and . is a martingale with respect
to its own filtration when which is equivalent to .
Therefore
(5)
(6)
The characteristic triplet of is , where c is used to define the Brownian component. In this
case we will assume that .
We denote by the distribution of whose characteristic function is of the form
, (7)
The function is called the characteristic generator. It is unique up to scaling and if chosen such
that yields that and are the mean and variance of respectively.
UnderQ , remains a symmetric NIG Levy process with characteristic triplet , where
r is the risk free rate.
Now it is easy to see that Q1 is a natural equivalent martingale measure. Indeed, since is a Levy process
with P-characteristic triplets and since the distribution of is , Q1 is chosen
so that
( , z)N dt d
~
( , z) ( , z) ( z)N dt d N dt d dt n d  
tY tY
1( z) exp( z) ( | z |) z
| z |
n d K d

 


0  gbm nig  2 


 tY
1[ ] 0,E Y   0 
0 exp( )t tY  
    
 
1
0
1 | 1|
e 1|
T
t
Q
t t T t
Y
t
a E F Q F
Q F
 
   
  
tY ( , , ( (z))c n d
0c 
2
( , , )S    1Y
1
2
2
( )
2
iu
Y u e u 
 
 
  
 
( ):[0, ]u 
'(0) 1,    2
 1Y
tY
~
2
, ,   
 
 
 
~
2
1 ln / 2r   
tY
( , , ( z))c n d 1Y 2
( , , )S   
 
~
2
1 ln / 2r   

e
By the uniqueness of , Q1 is unique.
Under Q1, is a symmetric NIG Levy process with marginal distribution from the family
Proposition: Denote by the P-distribution function of the standardized variable
Then
We will approximate the standardized symmetric NIG distribution by the standard normal.
(8)
By (ii) and we obtain
.
Equation (8) becomes
, (9)
Remark: If the excess emissions have a normal distribution (Normal inverse Gaussian) then the price process
automatically becomes a normal distribution (Normal inverse Gaussian).
3.3 Estimation of Model Parameters
Parameter estimates that are such that the model lead to best fit to the carbon price data are determined . By
using the NIG model, it means that we have four parameters that need to be estimated. While for the normal
distribution, only two parameters need to be estimated. Given data we obtain these estimates for the
parameters.
To obtain estimates of the parameters , maximum likelihood estimation method is applied. Thus the likelihood
function is maximized with respect to the parameters. Hence, we seek the parameter values that maximizes the
likelihood function
We obtain the MLE estimate by maximizing the log likelihood function.The log likelihood function given a random
sample of size n is given by
~
1( )
tY
~
( , , )tS t t  
TF    / :TY T T 
 ( ) .T TF y P Y T y T   
~ ~
1 1( )T T TF y Q Y T y T Q Y T y T      
        
   
    1
2
0
1 | 1|
ln ( ln ( / 2))
T
Q
t t T ta E F Q F
r t
t
 

 
   
    
  
 
2 2
,    
2 2
2 2 2 2
ln 1 1 1
2
 
     

  
             
2 2 2 2
0ln ( 1)
t
r t
a
t
   

     
  
 
 
1
( ) ( ; )
n
i
i
L f x 

 

e
The log likelihood function of the normal distribution, given a sample of size n is given by
The maximization of the log-likelihood function is done numerically using an optimization algorithm. For further
details about this, see (Myung, 2003). However, I have chosen to use the R-package.
3.4 Goodness of fit
There are various approaches for measuring the goodness-of-fit of a given model. These include the following.
3.4.1 QQ-plots
We use the QQ-plot graphical technique for determining which distribution best fits the data set.
3.4.2 Anderson-Darling test statistic:
Where, (no. of )/n, is the empirical cumulative distribution function and is the cumulative
distribution function.
A smaller value of AD means that the empirical distribution and fitted distribution are closer.
3.5 Risk-neutral Option Pricing
We assume that the price of a risk-free asset satisfies the ordinary differential equation
, where is the interest rate.
By first fundamental theorem of asset pricing, if a risk-neutral probability measure exists, then there is no
arbitrage. This risk-neutral probability is a martingale measure that is equivalent to the original probability
measure P and such that the underlying asset price is a local martingale.
A European call option is the right but not obligation to buy a contingent claim at the time of maturity T to a fix
strike price K. Thus the payoff function is given by
.
The arbitrage-free value of the option at time can be defined as
3.6 Option Pricing Using Fast Fourier Transform (FFT)
1/2
1
1 1 1
1
ln( ) ln( ) ( ) ( ) ( ( ) )
2
n n n
i i i
i i i
L n n n x x K x      
  
           
( , )N  
/2 2
2
1
1
(2 ) exp ( )
2
n
n n
i
i
L x  

 

 
   
 

 
( ) ( )
max
( ) 1 ( )
n
x R
F x F x
AD
F x F x



( )nF x  ix x ( )F x
( )B t
( ) ( )dB t rB t dt 0r 


 max ( ) ,0S T K
t T
 ( )
max ( ) ,0r T t
tC e E S T K 
   


e
The European call option price will be based on the asset price process, , with maturity time and strike
price . Write and . denote the option price and the risk –neutral
probability density function of price .
The characteristic function of the density is given by
(11)
The option value which is related to the risk-neutral density is given by
(12)
Here is not square integrable because when so that , we have . To
obtain a square integrable function, we consider the modified price given by
(13)
for a suitable . The value affects the speed of convergence.
The Fourier transform of is defined by
. (14)
We first develop an analytical expression for in terms of characteristic function, , and then obtain call
prices numerically using the inverse transform
(15)
(16)
Call values are determined by substituting (16) into (15) and performing the required integration. Since the FFT
evaluates the integrand at , the use of is required. Therefore it is necessary to make an
appropriate choice of the coefficient . Positive values of assist in the integrability of the modified call value
over the negative log strike axis, but aggravate the same condition for the positive log strike direction, and hence
tS T
K log( )k K ( ) log( ( ))s T S T ( )TC k Tf
Ts
Tf
( ) ( ) .ius
T Tu e f s ds


 
Tf
( ) ( ) ( )rT s k
T Tk
C k e e e f s ds


 
( )TC k k   0K  (0)TC S
( )Tc k
( ) ( ),k
T Tc k e C k

0  
( )Tc k
( ) ( )ivk
T Tv e c k dk


 
( )T v T
0
exp( )
( ) ( )
2
exp( )
( )
ivk
T T
ivk
T
k
C k e v dv
k
e v dv

















 
 (1 )
( 1 ) ( 1 )
2 2
( ) ( )
( )
( )
1
( ( 1) )
(2 1)
ivk k rT s k
T Tk
s
rT s k k ivk
T
iv s iv s
rT
T
rT
T
v e e e e e f s
e f s e e e dkds
e e
e f s ds
iv iv
e v i
v i v

 
 

 
 
  
 



  
 
   




 
 
 
  
   
 

   
 
 

0v  exp( )k
 

e
for it to be square-integrable as well, a sufficient condition is provided by being finite provided that
is finite. From the definition of the characteristic function, this requires that
(17)
Carr and Madan suggest that, one may determine an upper bound on from the analytical expression for the
characteristic function and the condition (17). One quarter of this upper bound serves as a good choice for ,
that is .
At equation (15) becomes,
Where,
Therefore,
The characteristic function of the log of , which follows a NIG distribution, is given by
The equation to get the option price of an NIG model is therefore,
We now get the expression for the characteristic function of the normal distribution and use it to get the option
price
The characteristic function of normal distribution is given by
The Brownian motion is a type of Levy process. Therefore, under , is a Levy process with characteristic
triplet and the distribution of becomes , where
(0)
  1T i  
1
[ ]TE S
 


0.75 
0v 
exp( )
( ) (0)T T
k
C k





2
( ( 1) )
(0)
rT
T
T
e i 

 

 


2
exp( ) ( ( 1) )
( ) *
rT
T
T
k e i
C k
  
 

  

 
TS
2 2 2 2
( ) exp ln ( )T u g T r iu            
   
2 2 2 2
( ( 1) ) exp ln ( ) ( 1)T i g T r                 
   
 
2 2 2 2
2
exp ln ( ) ( 1)exp( )
( ) *T
g T rk
C k
     
 
           
 
2 21
2
( )
i u u
T u e
 



 N
tS
~
, ,c  
 
 
1
N
S
~
, ,s    
 
 
2~
ln
2
r

 
 
   
 

e
The standard Brownian motion is
Therefore,
The option valuation for the normal distribution is given by
Remark: We will use AM92 Actuarial tables to get the values of and .
IV. Application, Results and Conclusions
Data for analysis was sourced from [11], which is in the public domain. Codes for the main analysis appear in the
Appendix. The results from the analysis are as follows.
4.1 Stylized features
4.1.1 Skewness and kurtosis
The skewness of log returns of price of carbon is -0.1212164. Since this value is near zero it is safe to say that
our data is indeed symmetric. The value of kurtosis is -1.36154. This justifies the use of NIG levy process to
model our data. The skewness and kurtosis of Gaussian distributions are 0 and 3 respectively.
4.2 Parameter Estimation
Asymmetric Normal Inverse Gaussian distribution:
Table 1: Maximum Likelihood Estimates for Parameters:
NIG 1.562073917 0.003563 -1.650997 12.358663
Normal -0.0952422 0.44958767
 2
0
0,
t
N ds
2 2
0
1
2
( )
T
t u dt
T u e


 
2 2
0
1
( ( 1) ) exp ln ( ) ( 1)
2
T
T ti h T r dt    
  
          

 
2 2
0
2
1
exp ln ( ) ( 1)
exp( ) 2
( ) *
T
t
T
h T r dt
k
C k
  

 
  
        
 

( )g ( )h
   

e
4.3 Q-Q Plot
The QQ-plots indicate that the empirical data fits much better to the NIG levy process than the Brownian motion.

e
4.4 Anderson-Darling (AD) test Statistic:
Table 2: AD statistic for NIG and Brownian Motion
AD
NIG 131.7
Normal 208.5
The AD-statistic value is relatively smaller for the NIG Levy process than Brownian motion.
4.5 Model Parameter estimates
Suppose the strike price, K, is 20 Euros, 25 Euros or 30 Euros, T=4 (compliance phase), , and
r=0.05. The value of the call option using NIG model and Brownian motion are:
Table 3 : NIG model Parameter estimates
K k
20 2.9957322 1.2535
25 3.2188758 1.06
30 3.4011973 0.92
Table 4 : Brownian Motion Model Parameter estimates
K k
20 2.9957322 0.00806
25 3.2188758 0.00682
30 3.4011973 0.00595
The option prices in the NIG model are higher than those of the Brownian model with respect to the given strike
prices. As the strike price increase, the value of the option decreases in both models.
V. Conclusion
The significant parameters in the NIG model are the ∝ and σ, while for the Brownian Motion model the
significant parameter is σ . The NIG model has a better fit to data compared to the Brownian Motion model.
Using the Fast Fourier transform, NIG model gives higher option prices than the Brownian motion.
References
[1.] Anderson, T. W. and Darling, D.A. (1954). A test of goodness of fit. Journal of the American Statistical
Association, 43, 26-37.
[2.] Bu, Yongqiang (2007). Option pricing using Levy Processes. Chalmers tekniska hogskola. Chapman &
Hall, New York.
[3.] Cont, R. and Tankov, P. (2004) Financial Modelling with Jump Processes. Chapman & Hall, New York.
[4.] Erik Bolviken and Fred Espen Benth(2000) Quantification of Risk in Norwegian stokes via the Normal
Inverse Gaussian distribution. Proceedings of the AFIR 2000 Colloqium,Tromso, Norway, Vol. 8798,
2000.
[5.] Hamza, Kais et al( 2015) Option pricing for Symmetric Levy Returns with Applications. Asia-Pacific
Financial Markets. 22, 1, 27-52
[6.] Jan Seifert, Marliese Uhrig-Homburg and Michael Wagner(2008). Dynamic Behaviour of Carbon Spot
Prices. Journal of Environmental Economics and Management. 24, 180-194
[7.] Julien Chevallier and Benoit Sevi (2014) On the Stochastic Properties of Carbon Futures Prices.
Environmental and Resource Economics. 58,1, 127-153
0 5  3.14 
( )TC k
( )TC k

e
[8.] Kitchen, Cliff (2009). Normal Inverse Gaussian (NIG) Process with Applications in Mathematical
Finance. Mathematical and Computational Finance . 41(2) 25-32
[9.] Rene Carmona and Juri Hinz(2011) Risk-neutral Models for Emission Allowance Prices and Option
Valuation. Management Science. 57(8) 1453-1468
[10.] Saebo, Karsten Krog (2009) Pricing Exotic Options with the Normal Inverse Gaussian Market Model
using Numerical Path Integration. Norwegian University of Science and Technology
[11.] Data Link: http://www.investing.com/commodities/carbon-emissions-historical-data
Appendix: R codes used in the Analysis
Carbon = read.csv(file.choose())
Carbon
Price <-carbon$Price
Price
Plot(price, xlab=”Fig1.1:carbon prices”)
Plot (log_returns4, type=”1”, xlab=”Figure 2.2: Log returns of carbon prices”, ylab = “Log returns “)
Log_returns4 < -diff(log(price), lag=364 )
Log_returns4
par(mfrow=c(1,2))
qqnig(log_returns4, mu=0, delta=1, alpha=1, beta=0, xlab=”Figure 1.3: NIG QQ plot”)
qqnorm(log_returns4, xlab = “ Figure 1.4 : Normal QQ plot “)
qqline (log_returns4)
y1
ad.test(y1)

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