Content uploaded by Rene A. Carmona
Author content
All content in this area was uploaded by Rene A. Carmona on May 27, 2015
Content may be subject to copyright.
MANAGEMENT SCIENCE
Vol. 57, No. 8, August 2011, pp. 1453–1468
issn 00251909 eissn 15265501 11 5708 1453
doi 10.1287/mnsc.1110.1358
© 2011 INFORMS
RiskNeutral Models for Emission Allowance
Prices and Option Valuation
René Carmona
Bendheim Center for Finance, Program in Applied and Computational Mathematics, and Department of Operations Research and
Financial Engineering, Princeton University, Princeton, New Jersey 08544, rcarmona@princeton.edu
Juri Hinz
Department of Mathematics, National University of Singapore, Singapore 119076, Singapore, mathj@nus.edu.sg
T
he existence of mandatory emission trading schemes in Europe and the United States, and the increased
liquidity of trading on futures contracts on CO
2
emissions allowances, led naturally to the next step in
the development of these markets: These futures contracts are now used as underliers for a vibrant derivative
market. In this paper, we give a rigorous analysis of a simple riskneutral reducedform model for allowance
futures prices, demonstrate its calibration to historical data, and show how to price European call options
written on these contracts.
Key words: emission derivatives; emissions markets; capandtrade schemes; environmental ﬁnance
History: Received April 25, 2009; accepted March 7, 2011, by Haitao Li, guest editor, ﬁnance. Published online
in Articles in Advance June 20, 2011.
1. Introduction
Global warming and environmental problems con
tinue to challenge policy makers. In part because of
the success of the U.S. Acid Rain Program, capand
trade systems are now considered to be one of the
most promising market mechanism to reduce green
house gas emissions on an international scale. The
core principle of such a mechanism is based on the
allocation of fully tradable credits among emission
sources and a penalty to be paid per unit of pollu
tant that is not offset by a credit at the end of a pre
determined period. The introduction of trading puts
a price on emissions and helps identify and exer
cise the cheapest emission abatement measures. For
this reason, marketbased mechanisms for emission
reduction are supposed to yield pollution control at
the lowest cost for the society. Notwithstanding the
rigorous equilibrium analyses from Carmona et al.
(2009, 2010) that conﬁrm that social optimality does
not necessarily mean that the scheme is cheap for con
sumers, emission trading should be considered as a
costefﬁcient and effective tool.
By its very nature, the regulatory framework of a
mandatory capandtrade system involves its partici
pants in a risky business, necessarily creating the need
for appropriate risk management. Trading of certiﬁ
cates from a mandatory scheme is typically accom
panied by an active secondary market where diverse
emissionrelated ﬁnancial derivatives (e.g., futures)
are traded. Options have been traded since 2006, and
as we explain below, on any given day, the volume
of European call and put options traded on forward
European Union Allowance (EUA) contracts ranges
between 15 and 25 million tons of CO
2
equivalent, the
short end of the forward curve being clearly the most
actively traded.
In this work, we propose reducedform models for
the riskneutral dynamics of allowance prices, pro
viding a quantitative framework for pricing emission
derivatives.
Despite the large number of pieces in the popular
press and numerous speculative articles in magazines,
the scientiﬁc literature on capandtrade systems is
rather limited, especially if we restrict ourselves to
quantitative analysis of models including stochas
tic factors. For the sake of completeness, we brieﬂy
review the literature relevant to our contribution.
The economic theory of allowance trading can be
traced back to Dales (1968) and Montgomery (1972),
authors who proposed a market model for the pub
lic good environment described by tradable permits.
Dynamic allowance trading is addressed in Cronshaw
and Kruse (1996), Tietenberg (1985), Rubin (1996),
Leiby and Rubin (2001), Schennach (2000), Stevens
and Rose (2002), Maeda (2004), and in the literature
cited therein. Empirical evidence from existing markets is
discussed in Daskalakis et al. (2007) and Paolella and
Taschini (2008). This last paper suggests economic
implications and hints at several ways to model
spot and futures allowance prices, whose detailed
1453
INFORMS holds copyright to this article and distributed this copy as a courtesy to the author(s).
Additional information, including rights and permission policies, is available at http://journals.informs.org/.
Carmona and Hinz: RiskNeutral Models for Emission Allowance Prices and Option Valuation
1454 Management Science 57(8), pp. 1453–1468, © 2011 INFORMS
interrelations are investigated in UhrigHomburg and
Wagner (2008, 2009). There, the demand for derivative
instruments in emission markets is also addressed.
In Benz and Trueck (2009), characteristic proper
ties for ﬁnancial time series are observed for prices
of emission allowances from the mandatory Euro
pean Union Emission Trading Scheme (EU ETS) and
several Markov and AR/GARCH–like models are
proposed. Paolella and Taschini (2008) also consider
tail behavior and the heteroscedasticity of the returns.
Dynamic price equilibrium and optimal market design are
investigated in Carmona et al. (2009), which provides
a mathematical analysis of market equilibriums and
uses optimal stochastic control to show social opti
mality. Based on this approach, Carmona et al. (2010)
discuss price formation for goods whose production
is affected by emission regulations. In this setting,
an equilibrium analysis conﬁrms the existence of the
socalled windfall proﬁts (see Sijm et al. 2006) and
provides quantitative tools to analyze alternative mar
ket designs, which are applied in Carmona et al.
(2008) to optimize a capandtrade mechanism for a
proposed Japanese emission trading scheme. Seifert
et al. (2008) and Wagner (2006) also deal with risk
neutral allowance price formation within the EU ETS.
Using equilibrium properties, the price evolution is
treated in terms of marginal abatement costs and opti
mal stochastic control. Cetin and Verschuere (2010)
use hidden Markov models and a ﬁltering approach
to capture the impact of news releases. Option pric
ing within the EU ETS was considered only recently.
Chesney and Taschini (2008) rely on endogenous
emission permit price dynamics within an equilib
rium setting to value European options on emission
allowances.
The present paper is organized as follows. After
an introductory discussion of the various approaches
to riskneutral modeling in §2, we present a general
approach to modeling of an emission market with one
compliance period. More realistic multiperiod mod
els are treated in §6. The mathematical treatment of
§3 is based on the analysis of diffusion martingales
ending with only two possible values. We identify
explicit classes of such martingales, and we show
how simple deterministic time changes can provide
families of versatile riskneutral models for allowance
prices. In §4, we demonstrate how to calibrate one of
these models to historical allowance price data. We
develop a historical calibration procedure, because the
option market has not yet matured to a point we can
trust more standard calibration procedures based on
option price data. The second part of the paper gen
eralizes the oneperiod setup to more realistic mul
tiperiods models incorporating important features of
realworld markets, and §6 provides the necessary
Table 1 ECX EUA Option Quotes (in Euros) on January 4, 2008
Option Option Allowance Implied Settlement
maturity type Volume Strike price vol. (%) price
Dec08 Cal 2001000 22000 23055 51025 5006
Dec08 Call 1501000 26000 23055 51025 3057
Dec08 Call 4501000 27000 23055 51025 3027
Dec08 Call 1001000 28000 23055 51025 2099
Dec08 Call 1251000 29000 23055 51025 2074
Dec08 Call 5251000 30000 23055 51025 2051
Dec08 Call 2501000 40000 23055 51025 1004
Dec08 Call 7001000 50000 23055 51025 0045
Dec08 Put 110001000 14000 23055 51025 0064
Dec08 Put 2001000 15000 23055 51025 0086
Dec08 Put 2001000 15000 23055 51025 0086
Dec08 Put 4001000 16000 23055 51025 1013
Dec08 Put 1001000 17000 23055 51025 1043
Dec08 Put 110001000 18000 23055 51025 1078
Dec08 Put 5001000 20000 23055 51025 2060
Dec08 Put 2001000 21000 23055 51025 3007
Dec08 Put 2001000 22000 23055 51025 3057
Note. ECX, European Climate Exchange.
changes needed to extend the pricing formula to this
more general setup.
As a motivation for our derivations of option pric
ing formulas, we close this introduction with a short
discussion of the idiosyncrasies of the EUA option
markets. The facts reported below were a determin
ing factor in our decision to write the present paper.
European call and put options are actively traded on
EUA futures contracts. Since 2006, trades of options
maturing in December of each year (prior to 2012)
have produced a term structure of option prices. On
any given day, the volume varies from 5 to 25 mil
lion tons of CO
2
equivalent, the short end of the curve
being the most active with a good number of ﬁnancial
institutions involved, whereas the long end depends
mostly on a few energy companies. It is not clear
how these options are priced, and a persistent rumor
claims that traders plainly use Black’s formula. The
data reproduced in Table 1 is an extract of quotes
published on January 4, 2008. Obviously, the implied
volatility is perfectly ﬂat, and the absence of skew
or smile is consistent with the rumor. Whether or
not traders are using Black or Black–Scholes formulas
to price options on EAUs and futures contracts, and
given what we learned from equilibrium models, we
think that it is important to derive option price formu
las based on underlying martingales with binary ter
minal value, because Black–Scholes formula is based
on an underlying price martingale converging to zero!
2. RiskNeutral Modeling of
Emission Markets
To position our contribution within the existing lit
erature, we brieﬂy review the different methodolo
gies of quantitative ﬁnancial modeling. Whereas the
INFORMS holds copyright to this article and distributed this copy as a courtesy to the author(s).
Additional information, including rights and permission policies, is available at http://journals.informs.org/.
Carmona and Hinz: RiskNeutral Models for Emission Allowance Prices and Option Valuation
Management Science 57(8), pp. 1453–1468, © 2011 INFORMS 1455
econometric approach aims to give a description of
statistical aspects in price movements, possibly ignor
ing the underlying economic underpinnings, the equi
librium approach focuses on the mechanics of price
formation: Given incentives, strategies, uncertainty,
and risk aversions, the market is described by the
cumulative effect of individual actions. Investigations
are geared toward understanding market features
appearing in steady realistic state, i.e., market equi
librium. Finally, the riskneutral approach adopted in
this paper abstracts from the mechanism driving the
market to the equilibrium state and focuses on its
basic properties. Starting from the absence of arbi
trage, the asset price evolution is introduced directly.
This approach arose from the idea that although price
movement is stochastic in nature, derivative valua
tion does not refer to the realworld probability. The
description of statistical issues is not a goal of risk
neutral models.
Absence of arbitrage is central to the riskneutral
approach. Moreover, we want to the important
insights gained from equilibrium theory (Carmona
et al. 2009, 2010) which we summarize in the follow
ing three points:
(a) There is no arbitrage from trading allowances.
(b) There are merely two ﬁnal outcomes for the
price of an allowance. Either the terminal allowance
price drops to zero or it approaches the penalty
level . Indeed, the price must vanish at maturity if
there is excess of allowances, whereas in the case of
shortage, the price will raise, reaching the level of the
penalty. It is reasonable to suppose that in reality, the
demand for allowances will coincidence with the sup
ply with zero probability. So the occurrence of such
an event will be disregarded.
(c) Allowance trading instantaneously triggers all
abatement measures whose costs are below the
allowance price because if an agent owns a tech
nology with lower reduction costs than the present
allowance price, then it is optimal to immediately use
it to reduce his or her own pollution and proﬁt from
selling allowances.
At this point, we distinguish between two types of
riskneutral approaches:
• the reducedform riskneutral approach, which
focuses on (a) and (b);
• the detailed riskneutral approach, which aims at
all three properties (a), (b), and (c).
Let us explain, at least at a formal level, the main
differences between these two classes of models. For
the sake of concreteness, we focus on a continuous
time framework in which the riskneutral evolution
4A
t
5
t∈601 T 7
of a futures contract on an allowance with
compliance date T for a penalty ∈ 401 5 is being
modeled. In this framework, the allowance price pro
cess 4 A
t
5
t∈601 T 7
is realized a ﬁltered probability space
4ì1 F1 1 4F
t
5
t∈601 T 7
5, equipped with a distinct mea
sure ∼, which is interpreted as the spot martingale
measure.
The reducedform riskneutral approach focuses on
the following problem:
Model the noncompliance event N ∈ F
T
,
which deﬁnes the martingale 4A
t
5
t∈601 T 7
with terminal value A
T
=1
N
.
(1)
The noncompliance event N is the only object that
needs to be described exogenously. However, to
obtain a useful model, several requirements, ranging
from computational tractability in derivatives valua
tion to diverse aspects of calibration, must be fulﬁlled.
Within the detailed riskneutral approach, the non
compliance event N is obtained endogenously, in
terms of other quantities that in turn must be spec
iﬁed exogenously. This is where the issue (c) comes
into play. Under natural equilibrium assumptions (see
Carmona et al. 2009), the abatement activity in the
market is driven by the allowance price in the follow
ing way: At any time t, given the allowance price A
t
,
the market exercises exactly those abatement mea
sures whose costs are less than or equal to the value
of A
t
. This is also known as the equilibrium allowance
price equals to the marginal abatement costs in envi
ronmental economics. Hence in equilibrium, the total
abatement in the market can be described in terms of
allowance prices 4A
s
5
s∈601 T 7
as
Z
T
0
c
s
4A
s
5 ds1
where c
s
4a545 stands for the total intensity of the
abatement measures at time s ∈601 T 7 available in the
market at price less than or equal to a ∈ 601 5 in
the market scenario ∈ ì. In this context, the abate
ment volume function c
t
2 601 7 × ì → 601 5, t ∈
601 T 7 must be speciﬁed exogenously. In practice, the
abatement volume function can be be estimated from
market data: Given a riskneutral fuel price model,
c
t
can be described by an appropriate B6 01 7 ⊗ F
t

measurable functions for each t ∈ 601 T 7. In this con
text, the noncompliance event is given by
∈ì3 E
T
45 −
Z
T
0
c
s
4A
s
545ds ≥0
1
where an exogenously speciﬁed F
T
—measurable
allowance demand E
T
—is used for the number
of excess pollution units in the businessasusual
scenario (i.e., given zero penalty). Thus, the detailed
riskneutral approach leads to a more complex math
ematical problem:
Determine 4c
t
5
t∈601 T 7
from market data
and model allowance demand E
T
to
obtain a martingale 4A
t
5
t∈601 T 7
with the
terminal value A
T
=1
8E
T
−
R
T
0
c
t
4A
t
5 dt≥09
.
(2)
INFORMS holds copyright to this article and distributed this copy as a courtesy to the author(s).
Additional information, including rights and permission policies, is available at http://journals.informs.org/.
Carmona and Hinz: RiskNeutral Models for Emission Allowance Prices and Option Valuation
1456 Management Science 57(8), pp. 1453–1468, © 2011 INFORMS
Although the detailed riskneutral approach seems
appealing from a methodological perspective, it is
not obvious whether its higher complexity is justi
ﬁed from the viewpoint of derivative valuation. The
present authors believe and show in this work that
the reducedform riskneutral approach yields satisfac
tory results, at least in the area of pricing plainvanilla
European options written on allowance futures. As
illustration we investigate the solution of a particular
problem of the type (2) and compare its results to a
class of solutions to (1) suggested in the present work.
The existence and uniqueness of 4A
t
5
t∈601 T 7
solv
ing (2) requires a delicate discussion. The martin
gale E
t
= Ɛ
Q
4E
T
F
t
5 deﬁned for t ∈ 601 T 7 plays an
essential role. Indeed, the analysis of the discrete
time framework shows that if the future incre
ments of 4E
s
5
s∈6t1T 7
are independent of the present
information F
t
at any time t, then a solution to
(2) should be expected in the functional form A
t
2=
4t1 G
t
5 with an appropriate deterministic function
2 6 01 T 7 × 34t1 g5 7→4t1 g5 ∈ and a state process
4G
t
5
t∈601 T 7
given by
G
t
=E
t
−
Z
t
0
c
s
4A
s
5 ds1 t ∈ 601 T 70
This insight helps guess a solution in the standard
diffusion framework, when there exists a process
4W
t
1 F
t
5
t∈601 T 7
of Brownian motion with respect to
∼, in the simplest case dE
t
= dW
t
with pre
speciﬁed ∈ 401 5 and continuous, nondecreasing,
and deterministic abatement function c2 401 5 → .
Under these conditions, Itô’s formula, applied to the
martingale A
t
= 4t1 G
t
5, leads, to a nonlinear partial
differential equation for on 601 T 5 ×:
¡
t
4t1 g5 −¡
g
4t1 g5c44t1 g55 +
1
2
¡
2
gg
4t1 g5
2
=01 (3)
subject to the boundary condition
4T 1 g5 = 1
601 5
4g5 for all g ∈1 (4)
justiﬁed by the digital nature of the terminal
allowance price. Having obtained in this way, one
constructs the state process 4G
t
5
t∈601 T 7
as a solution of
the stochastic differential equation
dG
t
=dE
t
−c44t1 G
t
55 dt1 G
0
=E
0
1 (5)
from which we get a solution to (2) from A
t
=4t1 G
t
5.
Once 4A
t
5
t∈601 T 7
is determined, one applies standard
integration to value European options. Although
closedform expressions are rare, option prices can
be calculated numerically. The only case that yields
quasiexplicit expressions (involving only numerical
integrations) is that of linear abatement functions (see
Seifert et al. 2008, Wagner 2006).
Let us elaborate on this case, to give the reader a
feeling of allowance option pricing in the framework
of detailed riskneutral modeling.
Figure 1 Price of a European Call Option as a Function of Its Maturity
0.0 0.5 1.0 1.5 2.0
15
10
5
Option’s price
Option’s maturity
Example 1. Set the time to compliance date T to
two years and assume the diffusion coefﬁcient is
4 and the penalty is 100, and suppose that the
abatement function c = c
t
for t ∈ 601 T 7 is linear with
c2 a 7→ 0002 ·a. At time t = 0, we consider a family of
European calls with the same strike price of K = 25
but different maturity times running through 601 T 7.
Suppose that the initial allowance price equals to the
strike price a = A
0
= 25. Determine call prices C
0
45
at t = 0 for different maturity times ∈ 601 T 7. Inde
pendently of the model, the price of the call in front
of expiry date = 0 must be equal to zero C
0
405 = 0,
whereas the longest maturity call = T must have a
price C
0
4T 5 =A
0
4 −K5/ =25 ·0075 =18075. Because
of the digital terminal value of the underlying, such a
call is equivalent to 0075 allowances. Call prices must
increase with call’s maturity from 0 to 18075. This
must be true within any riskneutral model. Figure 1
illustrates the exact curve 4C
0
455
∈601 T 7
for the param
eters as above. Because the end points are model
independent, merely intermediatematurity prices
exhibit modeldependent properties. Here we observe
one remarkable issue: the socalled inverse Sshape.
In this work, we show that these features are shared
by signiﬁcantly simpler option pricing schemes (see
Figure 4) based on the reducedform approach. For this
reason, we believe that the reducedform approach
can provide a reasonable pricing mechanism for
emissionrelated ﬁnancial products. However, we also
agree that further development of detailed risk
neutral, econometric, and equilibrium modeling is
needed to help understand allowance price evolu
tion. Certainly, such models could be better suited to
address the impact of information asymmetry, jumps
in the information ﬂow, regulatory uncertainty, and
market idiosyncracies.
3. ReducedForm Model for a Single
Compliance Period
In this section, we introduce a simple model for an
abstract emission market. We ﬁrst restrict ourselves
to a single compliance period, say 601 T 7. The more
realistic case of multiperiod models is treated in §6.
INFORMS holds copyright to this article and distributed this copy as a courtesy to the author(s).
Additional information, including rights and permission policies, is available at http://journals.informs.org/.
Carmona and Hinz: RiskNeutral Models for Emission Allowance Prices and Option Valuation
Management Science 57(8), pp. 1453–1468, © 2011 INFORMS 1457
In the oneperiod setting, credits are allocated at
the beginning of the period to enable allowance trad
ing until time T and to encourage agents to exer
cise efﬁcient abatement strategies. At the compliance
date T , market participants cover their emissions by
redeeming allowances or pay a penalty per unit
of pollution not offset by credits. In this oneperiod
model, unused allowances expire and are worthless
because we do not allow for banking into the next
period. Under natural assumptions, equilibrium anal
ysis shows that the allowance price A
T
at compliance
date T is a random variable taking only the values
0 and (see Carmona et al. 2009, 2010). More pre
cisely, 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 ﬁl
tered probability space 4 ì1 F1 1 4F
t
5
t∈601 T 7
5 on which
we ﬁx an equivalent probability measure ∼ ,
which we call the spot martingale measure.
We denote by 4A
t
5
t∈601 T 7
the price process of a
future contract with maturity date T written on the
allowance price. Given the digital nature of the termi
nal allowance price A
T
, the central object of our study
is the event N ⊂ F
T
of noncompliance, which set
tles the 801 9dichotomy of the terminal futures price
by A
T
= 1
N
. Furthermore, a standard noarbitrage
argument shows that the futures price 4A
t
5
t∈601 T 7
needs
to be a martingale for the spot martingale mea
sure . Hence, the problem of allowance price mod
eling reduces to the appropriate choice of a model for
the martingale
A
t
=Ɛ
41
N
F
t
51 t ∈601 T 70
There are many candidates for such a process, but no
obvious choice seems to be versatile enough for the
practical requirements described below. An important
requirement is the need to match the observed volatil
ity structure. For a practitioner trying to calibrate at
time ∈ 601 T 7 a model for the martingale 4A
s
5
s∈61 T 7
that ﬁnishes at 0 or , the minimum requirements
are to match the price observed at time as well
as the observed price ﬂuctuation intensity up to this
time . Further model requirements include the exis
tence of closedform formulas, or at least fast valua
tion schemes for European options; a small number of
parameters providing sufﬁcient model ﬂexibility; and
reliable and fast parameter identiﬁcation from data.
The goal of this paper is to present and analyze sim
ple models satisfying these requirements.
In accordance with our earlier discussion of the
two reducedform approaches, we choose our starting
point to be the noncompliance event N ∈ F
T
, which
we describe as the event where a hypothetic positive
valued random variable â
T
exceeds the boundary 1,
say N = 8â
T
≥ 19. If one denotes by E
T
the total pol
lution within the period 601 T 7 that must be balanced
against the total number ∈ 401 5 of credits issued
by the regulator, then the event of noncompliance
should be given by N =8E
T
≥9, which suggests that
â
T
should be viewed as the normalized total emission
E
T
/. However, in our modeling, we merely describe
the noncompliance event. Strictly speaking, so any
random variable â
T
with
8â
T
≥19 =8E
T
/ ≥ 19
would do as well. On this account, we do not claim
that â
T
represents the total normalized emission E
T
/.
So the allowance spot price is given by the martingale
A
t
=Ɛ
1
8â
T
≥19
F
t
1 t ∈601 T 71
where the random variable â
T
is chosen from a
suitable parameterized family of random variables
8â
T
2 ∈ ä9. For reasons of model tractability, we sup
pose that the ﬁltered probability space supports a pro
cess 4W
t
5
t∈601 T 7
of Brownian motion with respect to the
spot martingale measure , and we investigate para
metric families which give allowance prices
A
t
=Ɛ
1
8â
T
≥19
F
t
1 t ∈601 T 7
with a Markovian stochastic evolution of the form
dA
t
=v
4t1 A
t
5 dW
t
1
where the diffusion term v
captures the basic prop
erties of historical price observations. In particular,
we will match exactly the observed initial allowance
price and the initial instantaneous price ﬂuctuation
intensity.
Remark 1. We propose a consistent pricing scheme
for emissionrelated ﬁnancial instruments within the
framework of diffusion processes. Although this rules
out discontinuity in allowance prices, we believe
that this approach is reasonable. It has been argued
that because of jumps in the information ﬂow, sud
den allowance price changes must be included (see,
for example, Cetin and Verschuere 2010). However,
based on our experience in the energy sector, pos
sible allowance price jumps are not likely to play
a signiﬁcant role in mature emission markets. An
increasing number of consultancies and market ana
lysts are carefully watching the European emission
market. Several agencies are providing news and peri
odical publications. Moreover, because energy gener
ation and consumption are publicity observable, one
should not expect signiﬁcant allowance price jumps
in a mature emissions market. On this account, a risk
neutral model based on continuous allowance price
evolution is reasonable.
INFORMS holds copyright to this article and distributed this copy as a courtesy to the author(s).
Additional information, including rights and permission policies, is available at http://journals.informs.org/.
Carmona and Hinz: RiskNeutral Models for Emission Allowance Prices and Option Valuation
1458 Management Science 57(8), pp. 1453–1468, © 2011 INFORMS
To simplify the notation, we consider the normal
ized futures price process
a
t
2=
1
A
t
=Ɛ
1
8â
T
≥19
F
t
1 t ∈601 T 71
and we describe it under special assumptions on â
T
.
Our goal is to identify classes of martingales 8a
t
9
t∈601 T 7
taking values in the interval 401 15 and satisfying
lim
t%T
a
t
∈801 19
=10 (6)
We ﬁrst identify a parametric family of such martin
gales by working backward from a simple model for
the random variable â
T
, motivated by intuitive under
standing of the ﬁnal cumulative level of emissions.
3.1. Basic Modeling of the Compliance Event
We use the standard notation N41
2
5 for the normal
distribution with mean and variance
2
, and we
write ê for the cumulative distribution function of
the standard normal distribution.
Proposition 1. Suppose that
â
T
=â
0
e
R
T
0
s
dW
s
−41/25
R
T
0
2
s
ds
1 â
0
∈401 5 (7)
for some continuous and squareintegrable deterministic
function 4 01 T 5 3t ,→
t
. Then the martingale
a
t
=Ɛ
1
8â
T
≥19
F
t
1 t ∈601 T 7 (8)
is given by
a
t
=ê
ê
−1
4a
0
5
q
R
T
0
2
s
ds +
R
t
0
s
dW
s
q
R
T
t
2
s
ds
!
1 (9)
and it solves the stochastic differential equation
da
t
=ê
0
4ê
−1
4a
t
55
√
z
t
dW
t
1 (10)
where the positivevalued function 401 T 5 3 t ,→ z
t
is
given by
z
t
=
2
t
R
T
t
2
u
du
1 t ∈401 T 50 (11)
Remark 2. Notice that even though the distribu
tion of â
T
depends only upon
R
T
0
2
s
ds, the noncompli
ance event N depends upon the entire function 8
s
9
s
.
Proof. A direct calculation shows
a
t
= Ɛ
1
8â
T
≥19
F
t
=8â
T
≥1 F
t
9
= 8â
t
e
R
T
t
s
dW
s
−41/25
R
T
t
2
s
ds
≥1 F
t
9
= ê
ln â
t
−41/25
R
T
t
2
s
ds
q
R
T
t
2
s
ds
!
= ê
ln4â
0
5 −41/25
R
T
0
2
s
ds
q
R
T
0
2
s
ds
q
R
T
0
2
s
ds
q
R
T
t
2
s
ds
+
R
t
0
s
dW
s
q
R
T
t
2
s
ds
!
1
and taking into account the initial condition
a
0
=ê
ln â
0
−41/25
R
T
0
2
s
ds
q
R
T
0
2
s
ds
!
1
we obtain the desired expression (9). To show (10), we
start with a
t
=ê4
t
5, t ∈601 T 7, where
t
=
01 T
+
R
t
0
s
dW
s
q
R
T
t
2
s
ds
for t ∈601 T 71
with
01 T
=ln â
0
−
1
2
Z
T
0
2
s
ds (12)
and
0
= ê
−1
4a
0
5 with deterministic a
0
∈ 401 15. Com
puting its Itô differential, we get
d
t
=
Z
T
t
2
s
ds
−1/2
t
dW
t
+
1
2
x
0
+
Z
t
0
s
dW
s
Z
T
t
2
s
ds
−3/2
2
t
dt
=
Z
T
t
2
s
ds
−1/2
t
dW
t
+
1
2
t
Z
T
t
2
s
ds
−1
2
t
dt
=
√
z
t
dW
t
+
1
2
z
t
t
dt
d67
t
= z
t
dt0
Next, Itô’s formula gives the differential of the nor
malized allowance prices as
da
t
= ê
0
4
t
5d
t
+
1
2
ê
00
4
t
5d67
t
= ê
0
4
t
5
√
z
t
dW
t
+
1
2
z
t
t
dt
+
1
2
ê
00
4
t
5z
t
dt
= ê
0
4ê
−1
4a
t
55
√
z
t
dW
t
because xê
0
4x5 +ê
00
4x5 ≡ 0.
We notice for later use that if t < ,
is given
explicitly as a function of
t
by
=e
41/25
R
t
z
s
ds
t
+
Z
t
e
41/25
R
s
z
u
du
√
z
s
dW
s
0 (13)
3.2. Construction via Time Change
The stochastic differential Equation (10) can be inter
preted in the following way. Because of the factor
√
z
t
in front of dW
t
, a
t
can be viewed as the timechange
of a martingale 8Y
t
9
t∈601 5
given by the strong solution
of the stochastic differential equation
dY
t
=ê
0
4ê
−1
4Y
t
55 dW
t
1 (14)
for t ∈ 601 5, with Y
0
∈ 401 15. This solution stays in
the open interval 401 15 and converges to the bound
aries 0 or 1 with certainty when t approaches :
lim
t%
Y
t
∈801 19
=10 (15)
INFORMS holds copyright to this article and distributed this copy as a courtesy to the author(s).
Additional information, including rights and permission policies, is available at http://journals.informs.org/.
Carmona and Hinz: RiskNeutral Models for Emission Allowance Prices and Option Valuation
Management Science 57(8), pp. 1453–1468, © 2011 INFORMS 1459
This construction is in fact a special case of a gen
eral program where the martingale 8a
t
9
t∈601 T 7
satisfy
ing (6) is constructed in two steps: First determine a
401 15valued martingale 8Y
t
9
t∈601 5
satisfying (15) and
then search for a time change bringing the halfaxis
601 5 onto the bounded interval 601 T 5. With this in
mind, it appears natural to consider the solutions of
the stochastic differential equation
dY
t
=ä4Y
t
5 dW
t
1 Y
0
∈401 151 t ≥01 (16)
where ä is a nonnegative continuous function on
601 17 satisfying ä405 = ä415 = 0. We can then use
Feller’s classiﬁcation (see, for example, Karatzas and
Shreve 1991, Karlin and Taylor 1981) to check that
such a diffusion is conservative, does not reach the
boundaries 0 and 1 in ﬁnite time, and satisﬁes (15).
This is the case if v40+5 = v41−5 = , where v4x5 is
deﬁned by
v4x5 =2
Z
x
005
4x −y5
dy
ä4y5
1 x ∈401 150
Straightforward computations show that the solution
of the stochastic differential Equation (14) does indeed
satisfy these conditions; hence it does not hit 0 and 1
in ﬁnite time with probability one.
Explicit families of such martingales can easily be
constructed. Case in point, a twoparameter family
of examples can be constructed from the analysis of
Carmona et al. (2001), which we learned from Mike
Terhanchi (who extended the argument of Carmona
et al. 2001 to Lévy processes in Tehranchi 2009). If
we set
X
t
=e
−W
t
+ct
X
0
−
Z
t
0
e
W
s
−cs
4a ds +dB
s
5
1 X
0
∈
for c > 0 and a ∈ , where 8W
t
9
t∈601 5
and 8B
t
9
t∈601 5
are independent Wiener processes, then 8X
t
9
t∈601 5
satisﬁes
dX
t
=
c +
1
2
X
t
−a
dt −X
t
dW
t
−dB
t
1 t ∈601 5
and
lim
t→
X
t
= − on the set
Z
0
e
W
s
−cs
4a ds +dB
s
5 > X
0
1
lim
t→
X
t
= + on the set
Z
0
e
W
s
−cs
4a ds +dB
s
5 < X
0
0
Now if we deﬁne the function G by G4x5 =
R
x
−
g4y5 dy
for all x ∈, where the function g is
g4y5 =C
e
2a tan
−1
y
41 +y
2
5
c+1/2
1 y ∈ 1
with the constant C > 0 chosen so that
R
+
−
g4y5 dy =1,
then it is easy to check that
1
2
g
0
4y541 +y
2
5 +g4y5
c +
1
2
y −a
=01
which in turn implies that Y
t
=G4X
t
5 is a martingale.
Clearly, this martingale satisﬁes the limits (15). More
over, a simple application of Itô’s formula shows that
8Y
t
9
t
is a solution of the stochastic differential Equa
tion (16) with ä4y5 = g4G
−1
4y55
p
1 +G
−1
4y5
2
.
It is now plain to see that the basic model of
Proposition 1 is a particular case of this construction.
Indeed, if 4Y
t
=ê4X
t
55
t∈6016
for
X
t
=e
t/2
x
0
+
Z
t
0
e
−s/2
dW
s
1 x
0
∈1 for all t ∈601 50
and 4z
s
5
s∈601T 5
is a positivevalued, continuous func
tion, then the 401 15valued process
a
t
=Y
R
t
0
z
s
ds
1 t ∈601 T 51
satisﬁes
da
t
=ê
0
4ê
−1
4a
t
55
√
z
t
d
˜
W
t
1 t ∈601 T 5
for the process 4
˜
W
t
5
t∈601T 5
deﬁned by
d
˜
W
t
=
1
√
z
t
dM
t
1
M
t
=W
R
t
0
z
s
ds
t∈601 T 5
1
which follows a Brownian motion with respect to the
time changed ﬁltration
˜
F
t
=F
R
t
0
z
s
ds
1 t ∈601 T 50
Moreover,
lim
t→T
Z
t
0
z
s
ds = + implies that
lim
t%T
a
t
∈801 19
=10
4. Model Parametrization and
Calibration
We now show how to calibrate the basic model
introduced in §3.1. As explained above, historical
option prices are not appropriate for calibration in
the present state of the market, and we limit our
selves to historical calibration of the model. Note that
according to Proposiotion 1, the choice of the function
401 T 5 3t ,→
s
affects only the timechange 4z
t
5
t∈401 T 5
.
Moreover, Proposition 1 shows that when modeling
the random variable â
T
by (7), we must assume that
the function 401 T 5 3 t ,→
s
is not constant. Indeed,
a constant volatility
s
≡ ¯ ∈ 401 5 for all s ∈ 601 T 7
would give, independently on the choice of ¯, the
same process
a
t
=ê
ê
−1
4a
0
5
√
T +W
t
√
T −t
(17)
INFORMS holds copyright to this article and distributed this copy as a courtesy to the author(s).
Additional information, including rights and permission policies, is available at http://journals.informs.org/.
Carmona and Hinz: RiskNeutral Models for Emission Allowance Prices and Option Valuation
1460 Management Science 57(8), pp. 1453–1468, © 2011 INFORMS
with dynamics
da
t
=ê
0
4ê
−1
4a
t
55
1
√
T −t
dW
t
0 (18)
Thus, with a constant and deterministic ¯ it is impos
sible to match both observed allowance prices and
their observed (instantaneous) ﬂuctuation intensity.
Indeed, the entire process is completely determined
by the value of a
0
. This suggests that we introduce
extra degrees of freedom in (18). In this paper, we
choose to work with the model
da
t
=ê
0
4ê
−1
4a
t
55
p
4T −t5
−
dW
t
1 (19)
parameterized by ∈ and ∈ 401 5. This leads to
a parametric family of functions 4
s
5
s∈601 T 7
, which we
denote by
4
s
41 55
s∈401 T 5
1 ≥11 > 01 (20)
and we show how to calibrate the parameterized fam
ily (20) to historical data.
As seen from (7), the function 401 T 5 3s ,→
s
enters
the dynamics of 4a
t
5
t∈601 T 7
indirectly through the time
change function 401 T 5 3 t ,→ z
t
deﬁned in (11). The
correspondence between the functions and z is elu
cidated in the following lemma.
Lemma 1. (a) Given any squareintegrable continu
ous and positive function 401 T 5 3 s ,→
s
, the function
401 T 5 3 t ,→z
t
deﬁned by
z
t
=
2
t
R
T
t
2
u
du
1 t ∈ 401 T 51 (21)
is positive and continuous and satisﬁes
lim
t%T
Z
t
0
z
u
du =+0 (22)
(b) Conversely, if the positive and continuous function
401 T 5 3 t ,→ z
t
satisﬁes (22) then the function 401 T 5 3
s ,→
s
deﬁned by
t
=
q
z
t
e
−
R
t
0
z
u
du
1 t ∈401 T 51
is positive and continuous and satisﬁes (21).
Proof. (a) Let us write (21) as z
t
t
=
2
t
for t ∈
401 T 5, where
t
=
Z
T
t
2
u
du for all t ∈601 T 70
Then, ˙
t
=−
2
t
for t ∈401 T 5 and satisﬁes the differ
ential equation z
t
t
=− ˙
t
for t ∈ 401 T 5. Its solution is
given by
t
=
0
e
−
R
t
0
z
u
du
1 t ∈601 T 50
From the terminal condition
T
=
R
T
T
2
u
du = 0, we
get (22).
(b) Let us now suppose that 4z
t
5
t∈401 T 5
is positive
and continuous and satisﬁes (22), and let us deﬁne
the positive and continuous function 4
t
5
t∈601 T 5
by
t
=e
−
R
t
0
z
u
du
1 t ∈601 T 50 (23)
Clearly, it satisﬁes ˙
t
= −z
t
t
for t ∈ 401 T 5, and
because the divergence of the integral implies that
T
=0, we have
t
=−
Z
T
t
˙
u
du1 t ∈ 401 T 5 and −z
t
=−
˙
t
R
T
t
˙
u
du
0
(24)
Setting
2
t
=− ˙
t
for t ∈ 401 T 5, (21) is satisﬁed. More
over, this function is positive, continuous in the open
interval, and integrable because 1 =405 =−
R
T
0
˙
u
du,
which follows from (23) and (24). Consequently, the
function deﬁned by
t
2=
p
2
t
for t ∈ 401 T 5 is square
integrable, continuous and positive and is related to
4z
t
5
t∈401 T 5
by (21), as required.
We return to the expression (7) for â
T
, using now
the targeted family (20) to determine the stochas
tic differential Equation (19). In light of the previous
lemma, the function
4z
t
41 5 = 4T −t5
−
5
t∈401 T 5
(25)
must satisfy (22), implying the following restrictions
on the parameters and :
> 0 and ≥10 (26)
However, we will let vary freely over for calibra
tion purposes, interpreting the ﬁtted values in light of
these conditions.
Remark 3. If we use the parametric family
z
t
41 5 = 4T − t5
−
, then the actual time change is
given by the integral
Z
t
0
z
s
ds =
4log4T 5 −ln4T −t55 if =11
1 −
6T
1−
−4T −t5
1−
7 otherwise0
Notice that is a multiplicative parameter in the
sense that z
t
41 5 = z
t
41 15. Also, the emission
volatility 4
t
41 55
t∈401 T 5
associated to the parameteri
zation 4 z
t
41 55
t∈601 T 7
is given by
t
41 5
2
=z
t
41 5e
−
R
t
0
z
u
415 du
(27)
=
4T −t5
−
e
−4/41−556T
1−
−4T −t 5
1−
7
if 6= 11
4T −t5
−1
T
−
if = 10
(28)
INFORMS holds copyright to this article and distributed this copy as a courtesy to the author(s).
Additional information, including rights and permission policies, is available at http://journals.informs.org/.
Carmona and Hinz: RiskNeutral Models for Emission Allowance Prices and Option Valuation
Management Science 57(8), pp. 1453–1468, © 2011 INFORMS 1461
4.1. Historical Calibration
Consider historical observations of the futures prices
4A
t
5
t∈601T 7
, recorded at times t
1
< t
2
<1 ··· 1 < t
n
0 result
ing in a data set
1
1 0 0 0 1
n
, where
t
i
=ê
−1
4a
t
i
5 =ê
−1
1
A
t
i
1 i =11 ··· 1n (29)
The objective measure governing the statistics of
the observations can be recovered from the spot mar
tingale measure via its Radon–Nikodym density
d
d
=e
R
T
0
H
t
dW
t
−41/25
R
T
0
H
2
t
dt
0
For the sake of simplicity, we follow the timehonored
approach assuming that the market price of risk pro
cess 4H
t
5
t∈601 T 7
is constant and deterministic, H
t
≡ h
for t ∈ 601 T 7, for some ﬁxed h ∈ . According to
Girsanov’s theorem, the process 8
˜
W
t
9
t∈601 T 5
deﬁned as
˜
W
t
=W
t
−ht for t ∈ 601 T 5 is a Brownian motion with
respect to the objective measure ; under this mea
sure ,
t
satisﬁes
d
t
=
1
2
z
t
t
+h
√
z
t
dt +
√
z
t
d
˜
W
t
and the analog of (13) reads
= e
41/25
R
t
z
s
ds
t
+h
Z
t
e
41/25
R
s
z
u
du
√
z
s
ds
+
Z
t
e
41/25
R
s
z
u
du
√
z
s
dW
s
(30)
for 0 ≤ t ≤ ≤ T . Consequently, for each i =
11 0 0 0 1 n, the conditional distribution of
t
i
given
t
i−1
is Gaussian with mean
i
and variance
2
i
given by
i
4h115 =e
41/25
R
t
i
t
i−1
z
s
ds
t
i−1
+h
Z
t
i
t
i−1
e
1/2
R
t
i
s
z
u
du
√
z
s
ds1 (31)
2
i
4h1 1 5 =
Z
t
i
t
i−1
z
s
e
R
t
i
s
z
u
du
ds1 (32)
Figure 2 History of Futures Prices on EUA with Maturity 2012
35
30
25
20
15
10
Time in months
Price
01/07 04/07 06/07 08/07 10/07 12/07 02/08 05/08 07/08 09/08 11/08 01/09 04/09 06/09 08/09 10/09 12/09
provided we ﬁx t
0
and
0
by convention. So for a
given realization 8
t
i
9
n
i=1
∈
n
, the loglikelihood is
L
1
10001
n
4h115
=
n
X
i=1
−
4
t
i
−
i
4h1155
2
2
2
i
415
−log
q
2
2
i
415
(33)
for all h1 1 ∈ . Although there is no closedform
estimate for this parameter, the maximum of the like
lihood function can be determined numerically. Let us
illustrate this procedure.
Consider historical daily prices of a futures contract
written on the EUA with maturity in December 2012,
which are depicted in Figure 2. Based on these data,
the function (33) is calculated numerically, using the
following integral approximations:
i
4h1 1 5 ∼ e
41/254t
i
−t
i−1
5z
t
i−1
t
i−1
+h4t
i
−t
i−1
5
p
z
t
i−1
e
41/254t
i
−t
i−1
5z
t
i−1
1 (34)
2
i
4h1 1 5 ∼4t
i
−t
i−1
5z
t
i−1
e
4t
i
−t
i−1
5z
t
i−1
0 (35)
Starting with initial parameter h 2= 0, =005, =005,
a numerical method for unconstrained maximization
returned the maximizer
∗
=003321
∗
=001611 h
∗
=−000780 (36)
To conﬁrm the validity of our procedure, we deter
mine the residuals
w
i
=
t
i
−
i
4h1 1 5
p
2
i
4h1 1 5
1 i =11 0 0 0 1 n0 (37)
Under the model assumptions, this series must be a
realization of independent standard normal random
INFORMS holds copyright to this article and distributed this copy as a courtesy to the author(s).
Additional information, including rights and permission policies, is available at http://journals.informs.org/.
Carmona and Hinz: RiskNeutral Models for Emission Allowance Prices and Option Valuation
1462 Management Science 57(8), pp. 1453–1468, © 2011 INFORMS
Figure 3 Statistical Analysis of Series 4w
i
5
n
i=1
25 321
0
1
2
3
0–1
–1
–2–3
–3
2015
Lag Theoretical quantiles
Time in months
01/07
02/07
04/07
05/07
06/07
07/07
08/07
09/07
10/07
11/07
12/07
01/08
02/08
04/08
05/08
06/08
07/08
08/08
09/08
10/08
Sample quantiles
0
1
2
3
–1
–3
ACF
W
1050
0.0
0.2
0.4
0.6
0.8
1.0
variables. Thus, standard statistical analysis of these
residuals can be applied to verify the the quality of
the model ﬁt. In Figure 3 we show this series, its
empirical autocorrelation function (ACF), and its QQ
plot. The empirical properties of the residuals are not
completely in line with model assumptions, but the
reader should keep in mind that riskneutral models
are not designed to capture all the statistical particu
larities of the underlying ﬁnancial time series. In fact,
the target of riskneutral modeling is to describe the
price evolution with respect to the risk premium cor
rected pricing measure.
Remark 4. The original rationale for the choice of
our basic model was based on equilibrium considera
tions and the assumption that the terminal allowance
price was binary. However, the real market EU ETS
(second phase), whose data are used for historical cal
ibration, operates under uncertainty. One of the major
price determinant here is the unknown impact of the
international credits, the socalled Certiﬁed Emission
Reductions, or CERs. Most likely, market participants
believe that a signiﬁcant number of cheap interna
tional credits will be used to fulﬁll the compliance
within the EU ETS if needed, and that noncompliance
because of a shortage of certiﬁcates will not occur
at compliance time. Under such condition, the dis
tribution of the terminal allowance price should not
be binary any more. Namely, in the case of national
allowance shortage, it would reach a level deter
mined by supply and demand for international cred
its, which is likely to fall below the EU ETS penalty
of 100 euros. It is interesting to see that historical data
seem to reﬂect this concern, suggesting a value for the
parameter below 1 which would yield a martingale
with a nondigital terminal value because the integral
giving the time change does not diverge when < 1!
Remark 5. The above maximum likelihood calibra
tion from historical data used strongly the explicit
form (8) of the normalized allowance price and the
Gaussian property of the diffusion 4
t
5
t∈601 T 7
given
by the explicit form of the solution given by (13).
For general models of onedimensional diffusion pro
cesses with volatility given by a more function ä
more general than the speciﬁc ä =ê
0
ê
−1
, the max
imum likelihood estimates of , and h can be
computed using AïtSahalia (2002) approach.
4.2. Option Pricing
Now we turn our attention to the valuation of
European call options written on allowance futures
price 4A
t
5
t∈601 T 7
. The payoff of a European call with
maturity ∈ 601 T 7 and strike price K ≥ 0 is given
by 4A
−K5
+
. Under the assumption that the savings
account 8B
t
9
t∈601 T 7
is given by B
t
= e
R
t
0
r
s
ds
for t ∈ 601 T 7
for some deterministic short rate 8r
s
9
s∈601 T 7
, this price
can be computed in the model proposed in this paper.
INFORMS holds copyright to this article and distributed this copy as a courtesy to the author(s).
Additional information, including rights and permission policies, is available at http://journals.informs.org/.
Carmona and Hinz: RiskNeutral Models for Emission Allowance Prices and Option Valuation
Management Science 57(8), pp. 1453–1468, © 2011 INFORMS 1463
Proposition 2. In a oneperiod 601 T 7 compliance
model, with riskneutral parameters ∈ and > 0, the
price of a European call with maturity ∈ 601 T 7 and strike
K ≥0 written on an allowance futures maturing at the end
T of the compliance period is given at time t ∈ 601 7 by
C
t
=e
−
R
t
r
s
ds
Z
4ê4x5 −K5
+
N 4
t1
1
2
t1
5 4dx51 (38)
with
t1
and
2
t1
given by formulas (39) and (40) below.
This result is obtained by a straightforward calcu
lation because the conditional distribution of
given
t
is Gaussian with mean
t1
41 5
=
t
T −t
T −
/2
if = 11
t
exp
241 −5
64T −t5
1−
−4T −5
1−
7
if 6= 1
(39)
and variance
2
t1
41 5
=
T −t
T −
−1 if =11
exp
1 −
64T −t5
1−
−4T −5
1−
7
−1
if 6= 10
(40)
Let us illustrate the role of the parameter on
option prices. In the following example, we ﬁx =1
and assume that the penalty is = 100. Further
more, we suppose that at the initial time t = 0 four
years prior to the compliance date T = 4, the price of
a futures contract, written on allowance price at T ,
is A
0
= 25. For constant and deterministic continu
ously compounded interest rate r = 0005, we con
sider European calls written on the forward price with
strike price of K = 25 and varying maturity date ∈
601 T 7. The option price is calculated from (38) at time
t = 0. In Figure 4, we also illustrate the dependence of
the option price upon the parameter (recall Figure 1
for a plot for ﬁxed ). Comparing three cases = 005,
=008, and =101, Figure 4 shows that the call price
is increasing in . Less surprisingly, the dependence
on shows that longermaturity calls (with the same
strike) are more valuable than their shortmaturity
counterparts.
Let us stress that although there are no closedform
formulas for call prices, their numerical evaluations
can be performed very efﬁciently.
Figure 4 Plots of the Prices C
0
4 5 at Time t = 0 as Functions of Option
Maturity
01234
2
4
6
8
10
12
14
Time in years
Call price
Notes. The graphs marked by , Î, and È stand for = 005, = 008,
and = 101, respectively. The values of the other parameters are given in
the text.
5. Implied Calibration
Because emission futures prices are supposed to fol
low a bounded martingale in 6 01 1007, the option price
must vanish for strike prices exceeding the penalty
level 100. At this point, our valuation technique
(38)–(40) differs from the traditional Black 76 formula
frequently applied to price options on futures. How
ever, it turns out that the difference is rather minor,
for parameter values relevant to the current situation
of the EU ETS and for low strike prices. As an illus
tration, we plotted in Figure 5 the call option price
with maturity
o
= 3044 depending on the strike K,
calculated at t = 0 for the underlying futures price
of A
0
4
f
5 = 17054, supposing that the futures contract
matures at
f
= 3046. The calculation is done for two
cases, the Black 76 model with volatility = 0045 and
our pricing formula (38)–(40) for the maximum likeli
hood parameters
∗
≈003321
∗
≈00161 (41)
from (36). In all three cases, we have supposed that
the time to compliance date is T = 5 and set the
short rate at r = 0007. Figure 5 shows that for low
strikes K in the range 601 207, Black 76 call prices
are very similar to those given by our model with
maximumlikelihood parameters. Although for strike
Figure 5 Option Prices (Expressed in Euro) Against Their Strikes
0
0
5
Option prices
10
15
20 40 60 80
Black 76 price with sigma = 0.45
Theoretical price with = 0.332, = 0.161
k
100
INFORMS holds copyright to this article and distributed this copy as a courtesy to the author(s).
Additional information, including rights and permission policies, is available at http://journals.informs.org/.
Carmona and Hinz: RiskNeutral Models for Emission Allowance Prices and Option Valuation
1464 Management Science 57(8), pp. 1453–1468, © 2011 INFORMS
Figure 6 Evidence that European Calls Quoted on ECX Are Most
Likely Priced Using Black 76 Formula
02468
B76
10
8
6
4
2
0
Market
Market prices vs Black 76
at quoted volatility
prices in the range 6201 807, the difference is consider
able, it becomes negligible again at strikes above 80.
This approximation may explain why the traditional
Black 76 formula is considered to be useful for EUA
option pricing. As mentioned at the beginning, the
European Climate Exchange (ECX) provides beyond
call option quotas also the implied volatilities associ
ated with these quotas via Black 76 formula. In most
cases, the implied volatility remains in the interval
600401 00657. Based on a representative history of daily
call option prices quoted at the ECX during the period
January 4, 2007–September 30, 2009, we plot in Fig
ure 6 the theoretical Black 76, where we have adopted
the quoted volatility and set the short rate at r =0005,
against market price of the option. Although we could
not reach a complete coincidences due to unknown
historical interest rates entering these option prices,
this ﬁgure strongly suggests that the traders polled
by ECX priced EUA options using Black 76 formula.
In Figure 7 we used a ﬁxed approximative volatil
ity value of 0045, instead of the implied volatilities
quoted by the ECX, and the same short rate of r =
0005. Clearly, these data are more scattered, but the
concentration of price points across a straight line
shows that the option pricing practice of the ECX can
be approximatively described by the Black 76 formula
with a constant volatility of 0045.
Figure 7 Black 76 Formula with Volatility 0045 Explains Market
Call Prices
02468
10
8
6
4
2
0
B76
Market
Market prices vs Black 76
at volatility of 0.45
Figure 8 Market Prices vs. (38) and (39) with
?
?
from (42)
0
0
2
2
4
4
Market
6
6
Theoretical call price
8
8
10
10
Market prices
vs model price
To compare our technique to the market option
prices, we decided to ﬁt our model to actual option
prices. To do so, we suggest determining those param
eters and that minimize the sum of squared
deviations between historical market prices and their
theoretical values, based on our model (38)–(40). Hav
ing implemented the function describing the sum of
squared deviations depending on model parameters
and , we applied a numerical procedure based on the
Nelder–Mead method to determine the minimizer
?
≈003181
?
≈001440 (42)
These values are very close to the maximum likeli
hood estimates in (41). Coincidences of historical and
implied calibration are usually interpreted as model
validation. However, the deviations of market call
prices from their theoretical values plotted in Figure 8
are still strong. This will change when the option
market matures, the market participants realize the
differences between allowance price evolutions and
model assumptions underlying Black 76 formula, and
the ECX ﬁnally changes its way of reporting option
prices.
6. Multi–Compliance Periods Markets
So far, we focused on a generic capandtrade scheme
modeled after the ﬁrst phase of the EU ETS, namely,
limited to one compliance period and without bank
ing in the sense that unused allowances become
worthless at the end of the period. This is a strong
simpliﬁcation because as already mentioned above,
realworld markets are operating in a multiperiod
framework. Furthermore, subsequent periods are con
nected by market speciﬁc regulations. In what fol
lows, we consider an abstract but generic model of
such a market and focus on most natural rules for the
period interconnection.
Presently, three regulatory mechanisms connect
successive compliance periods in a capandtrade
scheme. Their rules go under the names of borrowing,
banking, and withdrawal.
INFORMS holds copyright to this article and distributed this copy as a courtesy to the author(s).
Additional information, including rights and permission policies, is available at http://journals.informs.org/.
Carmona and Hinz: RiskNeutral Models for Emission Allowance Prices and Option Valuation
Management Science 57(8), pp. 1453–1468, © 2011 INFORMS 1465
• Borrowing allows for the transfer of a (limited)
number of allowances from the next period into the
present one.
• Banking allows for the transfer of a (limited)
number of (unused) allowances from the present
period into the next.
• Withdrawal penalizes ﬁrms that fail to comply in
two ways: by penalty payment for each unit of pollu
tant not covered by credits and by withdrawal of the
missing allowances from their allocation for the next
period.
From the nature of the existing markets and the
designs touted for possible implementation, it seems
that policy makers tend to favor unlimited banking
and forbid borrowing. Furthermore, the withdrawal
rule is most likely to be included. Banking and with
drawal seem to be reasonable rules to reach an emis
sion target within a ﬁxed number of periods because
each success (respectively, failure) in the previous
period results in stronger (respectively, weaker) abate
ment in the subsequent periods.
6.1. Market Model
For the remainder of this section, we consider a two
period market model without borrowing but with
withdrawal and unlimited banking. We denote the
two periods by 601 T 7 and 6T 1 T
0
7 and consider a
stochastic basis 4ì1 F1 1 4F
t
5
t∈601 T
0
7
5 with a distinct
measure ∼, which we view as the spot martingale
measure. Further, we introduce processes 4A
t
5
t∈601 T 7
,
4A
0
t
5
t∈601 T
0
7
for the futures contracts with maturities at
compliance dates T , T
0
written on allowance prices
from the ﬁrst and the second period, respectively.
To exclude arbitrage, we suppose that the prices
4A
t
5
t∈601 T 7
and 4A
0
t
5
t∈601 T
0
7
are martingales with respect
to the spot martingale measure . Noncompliance in
the ﬁrst and second periods occurs on events N ∈F
T
and N
0
∈F
T
0
, respectively. As before, we assume that
the savings account 4B
t
5
t∈601 T
0
7
is given by
B
t
=e
R
t
0
r
s
ds
1 t ∈601 T
0
7 (43)
for some deterministic short rate 4r
s
5
s∈601 T
0
7
. The results
of the previous section imply that in the case ì\N of
the ﬁrstperiod compliance, the allowance price drops
A
T
1
ì\N
=A
0
T
1
ì\N
1 (44)
where ∈401 5 stands for discount factor describing
the interest rate effect
=B
T
B
−1
T
0
=e
−
R
T
0
T
r
s
ds
0
The relation (44) is justiﬁed by considering spot
prices. The random variable A
0
T
is nothing but the
spot price at time T of the secondperiod allowance.
Because futures and spot price agree at maturity, A
T
must be the spot price of the ﬁrstperiod allowance
at T . In the case of compliance in the ﬁrst period, the
unused allowances can be banked; hence we have the
equality in (44).
In the case of noncompliance at the end of the ﬁrst
period, the withdrawal regulation implies that
A
T
1
N
=A
0
T
1
N
+1
N
0 (45)
Namely, the noncompliance in one pollutant unit at
time T costs a penalty in addition to one allowance
from the next period, which must be withdrawn at
the spot price A
0
T
.
Combining the results (44) and (45), we ﬁnd out
that the difference is
A
t
−A
0
t
=Ɛ
4A
T
−A
0
T
F
t
5 =Ɛ
41
N
F
t
51 t ∈601 T 7
and must be modeled as 801 9valued martingale. We
suggest using the same methodology as in the one
period model
A
t
−A
0
t
=ê4
1
t
51 t ∈ 601 T 71 (46)
where the Gaussian process 4
1
t
5
t∈601 T 7
is introduced as
previously in (12), with 4
415
s
5
s∈601 T 7
in parameterized
form (28) and driven by a process 4W
1
t
1 F
t
5
t∈601 T
0
7
of
Brownian motion.
To model the secondperiod allowance futures
price, a continuation of the capandtrade system
must be speciﬁed. If there is no agreement on long
term regulatory framework (as is the case for most of
the existing emission markets), the process 4A
0
t
5
t∈601 T 7
should be speciﬁed exogenously. The simplest choice
would be a geometric Brownian motion with constant
volatility. Another idea to handle the uncertain con
tinuation is to suppose that the capandtrade sys
tem will be terminated after the second period. In
this case,
A
0
t
=Ɛ
41
N
0
F
t
51 t ∈ 601 T
0
7
can also be modeled as in the oneperiod model
A
0
t
=ê4
2
t
51 t ∈601 T
0
70 (47)
Again, 4
2
t
5
t∈601 T 7
is introduced as in (12), with a pro
cess 8
425
s
9
s∈601 T 7
chosen in parameterized form (28) and
driven by another Brownian motion 4W
2
t
1 F
t
5
t∈601 T
0
7
.
6.2. Option Pricing
As an application of our twoperiod model, we con
sider pricing of European calls. Consider European
call option with strike price K ≥ 0 and maturity
∈601 T 7 written on futures price of allowance from
the ﬁrst period. This contract yields a payoff
C
=4A
−K5
+
at time ∈6 01 T 7.
INFORMS holds copyright to this article and distributed this copy as a courtesy to the author(s).
Additional information, including rights and permission policies, is available at http://journals.informs.org/.
Carmona and Hinz: RiskNeutral Models for Emission Allowance Prices and Option Valuation
1466 Management Science 57(8), pp. 1453–1468, © 2011 INFORMS
Under the assumptions of the previous section, we
start with the computation of the price C
0
C
0
=e
−
R
0
r
s
ds
Ɛ
44A
−K5
+
5
of this option at time t =0. Using the decomposition
4A
−K5
+
=4A
−A
0
+A
0
−K5
+
1
we utilize our modeling of 801 9valued martingales
(46) and (47) to express the terminal payoff as
4A
−K5
+
=4ê4
1
5 +ê 4
2
5 −K5
+
with expectation
C
0
= e
−
R
0
r
s
ds
Ɛ
44A
−K5
+
5
= e
−
R
0
r
s
ds
Ɛ
44ê4
1
5 +ê 4
2
5 −K5
+
5
= e
−
R
0
r
s
ds
Z
2
4ê4x
1
5 +ê 4x
2
5 −K5
+
·N 4
1
54dx
1
1 dx
2
51 (48)
where N 4
1
5 =F
1
1
2
stands for joint normal distri
bution of
1
and
2
.
Let us derive the mean
and the covariance
matrix
under the standing assumption
1
=
2
=1,
for
1
> 0,
2
> 0. We have
1
= ê
−1
A
0
−A
0
0
s
T
T −
1
+
1/2
1
R
0
4T −u5
4
1
−15/2
W
1
u
du
4T −5
1
/2
1
2
= ê
−1
A
0
0
s
T
0
T
0
−
2
+
1/2
2
R
0
4T
0
−u5
4
2
−15/2
W
2
u
du
4T
0
−5
2
/2
0
Denoting by the correlation of the two Brownian
motions 4 W
1
t
5
t∈601 T
0
7
and 4 W
2
t
5
t∈601 T
0
7
6W
1
1 W
2
7 dt = dt1 ∈6−11 171
we can apply the same argumentation to obtain the
means
1
=Ɛ4
1
5 =ê
−1
A
0
−A
0
0
s
T
T −
1
1
2
=Ɛ4
2
5 =ê
−1
A
0
0
s
T
0
T
0
−
2
3
the variances
111
=Var4
1
5 =
T
T −
1
−11
212
=Var4
2
5 =
T
0
T
0
−
2
−13
and the covariance as
11 2
=
21 1
=Cov4
1
1
2
5
=
1/2
1
1/2
2
R
0
4T −u5
4
1
−15/2
4T
0
−u5
4
2
−15/2
du
4T −5
1
/2
4T
0
−5
2
/2
0
At times t ∈6 01 7 prior to maturity, the price C
t
of the
call is obtained similarly.
Proposition 3. In a two–compliance periods model as
above, with parameters
1
1
2
> 0, and ∈ 4−11 15, the
price of the European call with strike price K ≥0 and matu
rity ∈ 601 T 7 written on ﬁrstperiod allowance futures
price is given at time t ∈601 7 by
C
t
= e
−
R
t
r
s
ds
Z
2
4ê4x
1
5 +ê 4x
2
5 −K5
+
·N 4
t1
1
t1
54dx
1
1 dx
2
5 (49)
with mean
t1
1
t1
=ê
−1
A
t
−A
0
t
s
T −t
T −
1
1 (50)
2
t1
=ê
−1
A
0
t
s
T
0
−t
T
0
−
2
1 (51)
and covariance matrix
t1
11 1
t1
=Var4
1
5 =
T −t
T −
1
−11 (52)
21 2
t1
=Var4
2
5 =
T
0
−t
T
0
−
2
−11 (53)
11 2
t1
=
21 1
t1
=
1/2
1
1/2
2
R
t
4T −u5
4
1
−15/2
4T
0
−u5
4
2
−15/2
du
4T −5
1
/2
4T
0
−5
2
/2
0
(54)
If we take a closer look at the computations in
volved in the valuation of the call price
C
t
=C
t
41 T 1 T
0
1 A
0
1 A
0
0
1 K1 r1
1
1
2
1 5
given by the formulas (49)–(54), we see that be
cause of
C
t
41 T 1 T
0
1 A
0
1 A
0
0
1 K1 r1
1
1
2
1 5
=C
0
4 −t1 T −t1 T
0
−t1 A
0
1 A
0
0
1 K1 r1
1
1
2
1 5
for all t ∈ 601 7, it sufﬁces to consider the case t =0.
The numerical evaluation of twodimensional inte
gral is easily performed by using a decomposition of
the twodimensional normal distribution. To ease the
notation, let us skip t1 to write
i
=
i
t1
,
i1 j
=
i1 j
t1
for i1 j =11 2. It holds
N 41 54dx
1
1 dx
2
5
=N 4
11 c
4x
2
51
11 11 c
54dx
1
5N 4
2
1
21 2
54dx
2
51 (55)
INFORMS holds copyright to this article and distributed this copy as a courtesy to the author(s).
Additional information, including rights and permission policies, is available at http://journals.informs.org/.
Carmona and Hinz: RiskNeutral Models for Emission Allowance Prices and Option Valuation
Management Science 57(8), pp. 1453–1468, © 2011 INFORMS 1467
where the conditional mean and the conditional vari
ance are given by
11 c
4x
2
5 =
1
+
21 1
21 2
4x
2
−
2
51
11 11 c
=
11 1
−
4
21 1
5
2
21 2
0
With factorization (55), the inner integral is calculated
explicitly in the following cases:
Z
4ê4x
1
5 +ê 4x
2
5 −K5
+
N 4
11 c
4x
2
51
11 11 c
54dx
1
5
=
0 if K −ê4x
2
5 ≥1
ê
11 c
4x
2
5
√
1 +
11 11 c
+ê4x
2
5 −K
if K −ê4x
2
5 ≤00
That is, the numerical valuation is required only in
the case 0 < K −ê4x
2
5 < , where
Z
ê
−1
4K/−ê4x
2
55
4ê4x
1
5 +ê 4x
2
5 −K5
·N 4
11 c
4x
2
51
11 11 c
54dx
1
5
needs to be calculated.
Having obtained the inner integral, the numerical
evaluation of the outer integral is straightforward.
Because the density of the normal distribution decays
Figure 9 Surface Plots of the Initial Call Price 4 1
1
5 7→ C
0
4 1
1
5 as
Function of Maturity and
1
for Correlation = 008 (Top)
and = −008 (Bottom)
1
2
3
0.6
0.8
1.0
1.2
2
4
6
8
Maturit
y
Maturity
1
2
3
Beta
Beta
0.6
0.8
1.0
1.2
Call price
Call price
0
2
4
6
8
Note. The values of the other parameters are given in the text.
sufﬁciently fast, we expect neither numerical difﬁcul
ties nor long computation times. In fact, we did not
encounter any problem implementing this formula.
For the sake of completeness, we illustrate the
dependence of the call price on
1
and maturity of
the call. To make the results comparable with the one
period example given above, we chose the following
parameters: four years to the ﬁrstperiod compliance
date T = 4, eight years to the secondperiod compli
ance date T
2
=8, initial ﬁrstperiod allowance futures
price A
0
=25, initial secondperiod allowance futures
price A
0
0
=15, strike price of the European call K =25,
interest rate r = 0005, and
2
= 002. Figure 9 depicts
the dependence of the call price on the value of
1
for
the ﬁrst period and of the call maturity .
7. Conclusion
Mandatory emission markets are being established
throughout the world. In the most mature market,
the EU ETS, beyond physical allowances, a large
volume of allowance futures is traded. Furthermore,
European options written on these futures have been
introduced and traded although no theoretical foun
dation for their pricing is available yet.
The goal of this work is to ﬁll this gap. In our
analysis, we gradually move from a oneperiod mar
ket model to a more realistic situation of twoperiod
markets (covering the present EU ETS regulations)
and show that martingales ﬁnishing at twovalued
random variables can be considered as basic build
ing blocks for riskneutral futures price dynamics.
We suggest a model for twovalued martingales, ﬂex
ible in terms of time and spacechanging volatil
ity and capable of matching the observed historical
or implied volatility of the underlying future. From
a hedging perspective, this issue could be one of
the most desirable model properties. Other practi
cal aspects like ease of calibration and simple option
valuation schemes are also fulﬁlled in our approach.
We show how parameters can be estimated from his
torical price observation and suggest efﬁcient option
valuation schemes. Although option price formulas
are not available in a closed form, a simple and fast
numerical integration can be applied.
Acknowledgments
The research of René Carmona was partially supported by
NSFFRGDMS0455982 and DMS0806591. The research of
Juri Hinz was partially supported by WBS R703000020
720/C703000 of the Risk Management Institute and by WBS
R146000107133 of the National University of Singapore.
References
AïtSahalia, Y. 2002. Maximum likelihood estimation of discretely
sampled diffusions: A closedform approximation approach.
Econometrica 70(1) 223–262.
INFORMS holds copyright to this article and distributed this copy as a courtesy to the author(s).
Additional information, including rights and permission policies, is available at http://journals.informs.org/.
Carmona and Hinz: RiskNeutral Models for Emission Allowance Prices and Option Valuation
1468 Management Science 57(8), pp. 1453–1468, © 2011 INFORMS
Benz, E., S. Trueck. 2009. Modeling the price dynamics of CO
2
emis
sion allowances. Energy Econom. 31(1) 4–15.
Carmona, P., F. Petit, M. Yor. 2001. Exponential functionals of Lévy
processes. O. E. BarndorffNielsen, T. Mikosch, S. I. Resnick,
eds. Lévy Processes: Theory and Applications. Birkhaüser Boston,
New York, 41–56.
Carmona, R., M. Fehr, J. Hinz. 2008. Properly designed emis
sions trading schemes do work! Technical report, Princeton
University, Princeton, NJ.
Carmona, R., M. Fehr, J. Hinz. 2009. Optimal stochastic control
and carbon price formation. SIAM J. Control Optim. 48(12)
2168–2190.
Carmona, R., M. Fehr, J. Hinz, A. Porchet. 2010. Market design for
emission trading schemes. SIAM Rev. 52(3) 403–452.
Cetin, U., M. Verschuere. 2010. Pricing and hedging in carbon
emissions markets. Internat. J. Theoretical Appl. Finance 12(7)
949–967.
Chesney, M., L. Taschini. 2008. The endogenous price dynamics of
the emission allowances: An application to CO
2
option pricing.
Swiss Finance Institute Research Paper 0802, Swiss Finance
Institute, Zurich.
Cronshaw, M., J.