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MANAGEMENT SCIENCE
Vol. 57, No. 8, August 2011, pp. 1453–1468
issn 0025-1909 eissn 1526-5501 115708 1453 doi 10.1287/mnsc.1110.1358
© 2011 INFORMS
Risk-Neutral 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
The existence of mandatory emission trading schemes in Europe and the United States, and the increased
liquidity of trading on futures contracts on CO2emissions 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 risk-neutral reduced-form 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; cap-and-trade 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, cap-and-
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, market-based 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
cost-efficient and effective tool.
By its very nature, the regulatory framework of a
mandatory cap-and-trade 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
emission-related 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 CO2equivalent, the
short end of the forward curve being clearly the most
actively traded.
In this work, we propose reduced-form models for
the risk-neutral 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 cap-and-trade 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
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Carmona and Hinz: Risk-Neutral Models for Emission Allowance Prices and Option Valuation
1454 Management Science 57(8), pp. 1453–1468, © 2011 INFORMS
interrelations are investigated in Uhrig-Homburg 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
so-called 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 cap-and-trade 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 risk-neutral 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 risk-neutral 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 one-period setup to more realistic mul-
tiperiods models incorporating important features of
real-world 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
Dec-08 Cal 2001000 22000 23055 51025 5006
Dec-08 Call 1501000 26000 23055 51025 3057
Dec-08 Call 4501000 27000 23055 51025 3027
Dec-08 Call 1001000 28000 23055 51025 2099
Dec-08 Call 1251000 29000 23055 51025 2074
Dec-08 Call 5251000 30000 23055 51025 2051
Dec-08 Call 2501000 40000 23055 51025 1004
Dec-08 Call 7001000 50000 23055 51025 0045
Dec-08 Put 110001000 14000 23055 51025 0064
Dec-08 Put 2001000 15000 23055 51025 0086
Dec-08 Put 2001000 15000 23055 51025 0086
Dec-08 Put 4001000 16000 23055 51025 1013
Dec-08 Put 1001000 17000 23055 51025 1043
Dec-08 Put 110001000 18000 23055 51025 1078
Dec-08 Put 5001000 20000 23055 51025 2060
Dec-08 Put 2001000 21000 23055 51025 3007
Dec-08 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 CO2equivalent, 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. Risk-Neutral 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
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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 risk-neutral 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 real-world probability. The
description of statistical issues is not a goal of risk-
neutral models.
Absence of arbitrage is central to the risk-neutral
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
risk-neutral approaches:
• the reduced-form risk-neutral approach, which
focuses on (a) and (b);
• the detailed risk-neutral 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 risk-neutral evolution
4At5t∈601 T 7 of a futures contract on an allowance with
compliance date Tfor a penalty ∈4015is being
modeled. In this framework, the allowance price pro-
cess 4At5t∈601 T 7 is realized a filtered probability space
4ì1 F11 4Ft5t∈601 T 7 5, equipped with a distinct mea-
sure ∼, which is interpreted as the spot martingale
measure.
The reduced-form risk-neutral approach focuses on
the following problem:
Model the noncompliance event N∈FT,
which defines the -martingale 4At5t∈601 T 7
with terminal value AT=1N.
(1)
The noncompliance event Nis 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 risk-neutral approach, the non-
compliance event Nis 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 At,
the market exercises exactly those abatement mea-
sures whose costs are less than or equal to the value
of At. 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 4As5s∈601 T 7 as
ZT
0
cs4As5 ds1
where cs4a545 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∈6015in
the market scenario ∈ì. In this context, the abate-
ment volume function ct2 601 7 ×ì→6015,t∈
601 T 7 must be specified exogenously. In practice, the
abatement volume function can be be estimated from
market data: Given a risk-neutral fuel price model,
ctcan be described by an appropriate B601 7 ⊗Ft-
measurable functions for each t∈601 T 7. In this con-
text, the noncompliance event is given by
∈ì3 ET45 −ZT
0
cs4As545ds ≥01
where an exogenously specified FT—measurable
allowance demand ET—is used for the number
of excess pollution units in the business-as-usual
scenario (i.e., given zero penalty). Thus, the detailed
risk-neutral approach leads to a more complex math-
ematical problem:
Determine 4ct5t∈601 T 7 from market data
and model allowance demand ETto
obtain a -martingale 4At5t∈601 T 7 with the
terminal value AT=18ET−RT
0ct4At5 dt≥09.
(2)
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Although the detailed risk-neutral 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 reduced-form risk-neutral approach yields satisfac-
tory results, at least in the area of pricing plain-vanilla
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 4At5t∈601 T 7 solv-
ing (2) requires a delicate discussion. The martin-
gale Et=ƐQ4ETFt5defined 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 4Es5s∈6t1T 7 are independent of the present
information Ftat any time t, then a solution to
(2) should be expected in the functional form At2=
4t1 Gt5with an appropriate deterministic function
2 601 T 7 ×34t1 g 5 7→ 4t1 g 5 ∈and a state process
4Gt5t∈601 T 7 given by
Gt=Et−Zt
0
cs4As5 ds1 t ∈601 T 70
This insight helps guess a solution in the standard
diffusion framework, when there exists a process
4Wt1Ft5t∈601 T 7 of Brownian motion with respect to
∼, in the simplest case dEt=dWtwith pre-
specified ∈4015and continuous, nondecreasing,
and deterministic abatement function c2 4015→.
Under these conditions, Itô’s formula, applied to the
martingale At=4t1 Gt5, leads, to a nonlinear partial
differential equation for on 601 T 5 ×:
¡t4t1 g 5 −¡g4t1 g 5c44t1 g 55 +1
2¡2
gg 4t1 g 52=01(3)
subject to the boundary condition
4T 1 g 5 =160154g5 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 4Gt5t∈601 T 7 as a solution of
the stochastic differential equation
dGt=dEt−c44t 1 Gt55 dt1 G0=E01(5)
from which we get a solution to (2) from At=4t1 Gt5.
Once 4At5t∈601 T 7 is determined, one applies standard
integration to value European options. Although
closed-form expressions are rare, option prices can
be calculated numerically. The only case that yields
quasi-explicit 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 risk-neutral 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 Tto
two years and assume the diffusion coefficient is
4 and the penalty is 100, and suppose that the
abatement function c=ctfor 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=A0=25. Determine call prices C04 5
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 C0405=0,
whereas the longest maturity call =Tmust have a
price C04T 5 =A04 −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 risk-neutral model. Figure 1
illustrates the exact curve 4C0455∈601T 7 for the param-
eters as above. Because the end points are model-
independent, merely intermediate-maturity prices
exhibit model-dependent properties. Here we observe
one remarkable issue: the so-called inverse S-shape.
In this work, we show that these features are shared
by significantly simpler option pricing schemes (see
Figure 4) based on the reduced-form approach. For this
reason, we believe that the reduced-form approach
can provide a reasonable pricing mechanism for
emission-related 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. Reduced-Form 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.
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In the one-period setting, credits are allocated at
the beginning of the period to enable allowance trad-
ing until time Tand 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 one-period
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 ATat compliance
date Tis 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 4ì1 F11 4Ft5t∈601 T 75on which
we fix an equivalent probability measure ∼,
which we call the spot martingale measure.
We denote by 4At5t∈601 T 7 the price process of a
future contract with maturity date Twritten on the
allowance price. Given the digital nature of the termi-
nal allowance price AT, the central object of our study
is the event N⊂FTof noncompliance, which set-
tles the 801 9-dichotomy of the terminal futures price
by AT=1N. Furthermore, a standard no-arbitrage
argument shows that the futures price 4At5t∈601T 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
At=Ɛ41NFt51 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 4As5s∈6 1 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 closed-form 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 reduced-form approaches, we choose our starting
point to be the noncompliance event N∈FT, which
we describe as the event where a hypothetic positive-
valued random variable âTexceeds the boundary 1,
say N=8âT≥19. If one denotes by ETthe total pol-
lution within the period 601 T 7 that must be balanced
against the total number ∈4015of credits issued
by the regulator, then the event of noncompliance
should be given by N=8ET≥9, which suggests that
âTshould be viewed as the normalized total emission
ET/. However, in our modeling, we merely describe
the noncompliance event. Strictly speaking, so any
random variable âTwith
8âT≥19=8ET/ ≥19
would do as well. On this account, we do not claim
that âTrepresents the total normalized emission ET/.
So the allowance spot price is given by the martingale
At=Ɛ18âT≥19Ft1 t ∈601 T 71
where the random variable âTis chosen from a
suitable parameterized family of random variables
8â
T2 ∈ä9. For reasons of model tractability, we sup-
pose that the filtered probability space supports a pro-
cess 4Wt5t∈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=Ɛ18â
T≥19Ft1 t ∈601 T 7
with a Markovian stochastic evolution of the form
dA
t=v4t1 A
t5 dWt1
where the diffusion term vcaptures 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 emission-related 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.
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To simplify the notation, we consider the normal-
ized futures price process
at2=1
At=Ɛ18âT≥19Ft1 t ∈601 T 71
and we describe it under special assumptions on âT.
Our goal is to identify classes of martingales 8at9t∈601 T 7
taking values in the interval 40115and satisfying
lim
t%Tat∈80119=10(6)
We first identify a parametric family of such martin-
gales by working backward from a simple model for
the random variable âT, 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 N 41 25for the normal
distribution with mean and variance 2, and we
write êfor the cumulative distribution function of
the standard normal distribution.
Proposition 1. Suppose that
âT=â0eRT
0sdWs−41/25RT
02
sds1 â0∈4015(7)
for some continuous and square-integrable deterministic
function 401 T 5 3t ,→t. Then the martingale
at=Ɛ18âT≥19Ft1 t ∈601 T 7 (8)
is given by
at=ê ê−14a05qRT
02
sds +Rt
0sdWs
qRT
t2
sds !1(9)
and it solves the stochastic differential equation
dat=ê04ê−14at55√ztdWt1(10)
where the positive-valued function 401 T 5 3t ,→ztis
given by
zt=2
t
RT
t2
udu1 t ∈401 T 50 (11)
Remark 2. Notice that even though the distribu-
tion of âTdepends only upon RT
02
sds, the noncompli-
ance event Ndepends upon the entire function 8s9s.
Proof. A direct calculation shows
at=Ɛ18âT≥19Ft=8âT≥1Ft9
=8âteRT
tsdWs−41/25RT
t2
sds ≥1Ft9
=ê ln ât−41/25RT
t2
sds
qRT
t2
sds !
=ê ln4â05−41/25RT
02
sds
qRT
02
sds qRT
02
sds
qRT
t2
sds +Rt
0sdWs
qRT
t2
sds!1
and taking into account the initial condition
a0=ê ln â0−41/25RT
02
sds
qRT
02
sds !1
we obtain the desired expression (9). To show (10), we
start with at=ê4t5,t∈601 T 7, where
t=01 T +Rt
0sdWs
qRT
t2
sds
for t∈601 T 71
with 01 T =ln â0−1
2ZT
0
2
sds (12)
and 0=ê−14a05with deterministic a0∈40115. Com-
puting its Itô differential, we get
dt=ZT
t
2
sds−1/2
tdWt
+1
2x0+Zt
0
sdWsZT
t
2
sds−3/2
2
tdt
=ZT
t
2
sds−1/2
tdWt+1
2tZT
t
2
sds−1
2
tdt
=√ztdWt+1
2zttdt
d67t=ztdt0
Next, Itô’s formula gives the differential of the nor-
malized allowance prices as
dat=ê04t5dt+1
2ê004t5d67t
=ê04t5√ztdWt+1
2zttdt+1
2ê004t5ztdt
=ê04ê−14at55√ztdWt
because xê04x 5 +ê004x5 ≡0.
We notice for later use that if t < ,is given
explicitly as a function of tby
=e41/25R
tzsdst+Z
t
e41/25R
szudu√zsdWs0(13)
3.2. Construction via Time Change
The stochastic differential Equation (10) can be inter-
preted in the following way. Because of the factor √zt
in front of dWt,atcan be viewed as the time-change
of a martingale 8Yt9t∈6015given by the strong solution
of the stochastic differential equation
dYt=ê04ê−14Yt55 dWt1(14)
for t∈6015, with Y0∈40115. This solution stays in
the open interval 40115and converges to the bound-
aries 0 or 1 with certainty when tapproaches :
lim
t% Yt∈80119=10(15)
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This construction is in fact a special case of a gen-
eral program where the martingale 8at9t∈601T 7 satisfy-
ing (6) is constructed in two steps: First determine a
40115-valued martingale 8Yt9t∈6015satisfying (15) and
then search for a time change bringing the half-axis
6015onto the bounded interval 601 T 5. With this in
mind, it appears natural to consider the solutions of
the stochastic differential equation
dYt=ä4Yt5 d Wt1 Y0∈401151 t ≥01(16)
where äis a nonnegative continuous function on
60117satisfying 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 =2Zx
005
4x −y5 dy
ä4y51 x ∈401150
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 two-parameter 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
Xt=e−Wt+ct X0−Zt
0
eWs−cs 4a ds +dBs51 X0∈
for c > 0 and a∈, where 8Wt9t∈6015and 8Bt9t∈6015
are independent Wiener processes, then 8Xt9t∈6015
satisfies
dXt=c+1
2Xt−adt −XtdWt−dBt1 t ∈6015
and
lim
t→ Xt= − on the setZ
0
eWs−cs 4a ds +dBs5 > X01
lim
t→ Xt= + on the setZ
0
eWs−cs 4a ds +dBs5 < X00
Now if we define the function Gby G4x5 =Rx
− g4y 5 dy
for all x∈, where the function gis
g4y5 =Ce2atan−1y
41+y25c+1/21 y ∈1
with the constant C > 0 chosen so that R+
− g4y 5 dy =1,
then it is easy to check that
1
2g04y541+y25+g4y5c+1
2y−a=01
which in turn implies that Yt=G4Xt5is a martingale.
Clearly, this martingale satisfies the limits (15). More-
over, a simple application of Itô’s formula shows that
8Yt9tis a solution of the stochastic differential Equa-
tion (16) with ä4y5 =g4G−14y55p1+G−14y52.
It is now plain to see that the basic model of
Proposition 1 is a particular case of this construction.
Indeed, if 4Yt=ê4Xt55t∈6016for
Xt=et/2x0+Zt
0
e−s/2dWs1 x0∈1for all t∈60150
and 4zs5s∈601T 5 is a positive-valued, continuous func-
tion, then the 40115-valued process
at=YRt
0zsds1 t ∈601 T 51
satisfies
dat=ê04ê−14at55√ztd˜
Wt1 t ∈601 T 5
for the process 4˜
Wt5t∈601T 5 defined by
d˜
Wt=1
√zt
dMt1Mt=WRt
0zsdst∈601 T 5 1
which follows a Brownian motion with respect to the
time changed filtration
˜
Ft=FRt
0zsds1 t ∈601 T 50
Moreover,
lim
t→TZt
0
zsds =+ implies that lim
t%Tat∈80119=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 ,→saffects only the time-change 4zt5t∈401 T 5 .
Moreover, Proposition 1 shows that when modeling
the random variable âTby (7), we must assume that
the function 401 T 5 3t ,→sis not constant. Indeed,
a constant volatility
s≡ ¯∈4015for all s∈601 T 7
would give, independently on the choice of ¯, the
same process
at=êê−14a05√T+Wt
√T−t(17)
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with dynamics
dat=ê04ê−14at55 1
√T−tdWt0(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 a0. This suggests that we introduce
extra degrees of freedom in (18). In this paper, we
choose to work with the model
dat=ê04ê−14at55p4T −t5−dWt1(19)
parameterized by ∈and ∈4015. This leads to
a parametric family of functions 4s5s∈601 T 7 , which we
denote by
4s41 55s∈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 ,→senters
the dynamics of 4at5t∈601 T 7 indirectly through the time-
change function 401 T 5 3t ,→ztdefined in (11). The
correspondence between the functions and zis elu-
cidated in the following lemma.
Lemma 1. (a) Given any square-integrable continu-
ous and positive function 401 T 5 3s ,→s, the function
401 T 5 3t ,→ztdefined by
zt=2
t
RT
t2
udu1 t ∈401 T 51 (21)
is positive and continuous and satisfies
lim
t%TZt
0
zudu =+0(22)
(b) Conversely, if the positive and continuous function
401 T 5 3t ,→ztsatisfies (22) then the function 401 T 5 3
s ,→sdefined by
t=qzte−Rt
0zudu1 t ∈401 T 51
is positive and continuous and satisfies (21).
Proof. (a) Let us write (21) as ztt=2
tfor t∈
401 T 5, where
t=ZT
t
2
udu for all t∈601 T 70
Then, ˙t=−2
tfor t∈401 T 5 and satisfies the differ-
ential equation ztt= − ˙tfor t∈401T 5. Its solution is
given by
t=0e−Rt
0zudu1 t ∈601 T 50
From the terminal condition T=RT
T2
udu =0, we
get (22).
(b) Let us now suppose that 4zt5t∈401 T 5 is positive
and continuous and satisfies (22), and let us define
the positive and continuous function 4t5t∈601 T 5 by
t=e−Rt
0zudu1 t ∈601 T 50 (23)
Clearly, it satisfies ˙t= −zttfor t∈401 T 5, and
because the divergence of the integral implies that
T=0, we have
t=− ZT
t˙udu1 t ∈401 T 5 and −zt= − ˙t
RT
t˙udu0
(24)
Setting 2
t= − ˙tfor t∈401 T 5, (21) is satisfied. More-
over, this function is positive, continuous in the open
interval, and integrable because 1 =405=−RT
0˙udu,
which follows from (23) and (24). Consequently, the
function defined by t2=p2
tfor t∈401 T 5 is square
integrable, continuous and positive and is related to
4zt5t∈401 T 5 by (21), as required.
We return to the expression (7) for âT, using now
the targeted family (20) to determine the stochas-
tic differential Equation (19). In light of the previous
lemma, the function
4zt41 5 =4T −t5−5t∈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
zt41 5 =4T −t5−, then the actual time change is
given by the integral
Zt
0
zsds =
4log4T 5 −ln4T −t55 if =11
1−6T 1−−4T −t51−7otherwise0
Notice that is a multiplicative parameter in the
sense that zt41 5 =zt41 15. Also, the emission
volatility 4t41 55t∈401 T 5 associated to the parameteri-
zation 4zt41 55t∈601 T 7 is given by
t41 52
=zt41 5e−Rt
0zu415 du (27)
=
4T −t5−e−4/41−556 T 1−−4T −t51−7if 6=11
4T −t5−1T−if =10
(28)
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4.1. Historical Calibration
Consider historical observations of the futures prices
4At5t∈601T 7, recorded at times t1< t2<1 ···1 < tn0result-
ing in a data set 110001n, where
ti=ê−14ati5=ê−11
Ati1 i =11···1 n (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=eRT
0HtdWt−41/25RT
0H2
tdt0
For the sake of simplicity, we follow the time-honored
approach assuming that the market price of risk pro-
cess 4Ht5t∈601 T 7 is constant and deterministic, Ht≡h
for t∈601 T 7, for some fixed h∈. According to
Girsanov’s theorem, the process 8˜
Wt9t∈601 T 5 defined as
˜
Wt=Wt−ht for t∈601 T 5 is a Brownian motion with
respect to the objective measure ; under this mea-
sure ,tsatisfies
dt=1
2ztt+h√ztdt +√ztd˜
Wt
and the analog of (13) reads
=e41/25R
tzsdst+hZ
t
e41/25R
szudu√zsds
+Z
t
e41/25R
szudu√zsdWs(30)
for 0 ≤t≤≤T. Consequently, for each i=
110001n, the conditional distribution of tigiven ti−1
is Gaussian with mean iand variance 2
igiven by
i4h115=e41/25Rti
ti−1zsdsti−1+hZti
ti−1
e1/2Rti
szudu√zsds1 (31)
2
i4h1 1 5 =Zti
ti−1
zseRti
szudu 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 t0and 0by convention. So for a
given realization 8ti9n
i=1∈n, the log-likelihood is
L110001n4h115
=
n
X
i=1−4ti−i4h11552
22
i415 −log q22
i415(33)
for all h1 1 ∈. Although there is no closed-form
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:
i4h1 1 5 ∼e41/254ti−ti−15zti−1ti−1
+h4ti−ti−15pzti−1e41/254ti−ti−15zti−11(34)
2
i4h1 1 5 ∼4ti−ti−15zti−1e4ti−ti−15zti−10(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
wi=ti−i4h1 1 5
p2
i4h1 1 5 1 i =110001n0 (37)
Under the model assumptions, this series must be a
realization of independent standard normal random
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Figure 3 Statistical Analysis of Series 4wi5n
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 risk-neutral models
are not designed to capture all the statistical particu-
larities of the underlying financial time series. In fact,
the target of risk-neutral 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 so-called 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 4t5t∈601T 7 given
by the explicit form of the solution given by (13).
For general models of one-dimensional diffusion pro-
cesses with volatility given by a more function ä
more general than the specific ä=ê0ê−1, the max-
imum likelihood estimates of ,and hcan be
computed using Aït-Sahalia (2002) approach.
4.2. Option Pricing
Now we turn our attention to the valuation of
European call options written on allowance futures
price 4At5t∈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 8Bt9t∈601 T 7 is given by Bt=eRt
0rsds for t∈601 T 7
for some deterministic short rate 8rs9s∈601 T 7 , this price
can be computed in the model proposed in this paper.
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Proposition 2. In a one-period 601 T 7 compliance
model, with risk-neutral parameters ∈and > 0, the
price of a European call with maturity ∈601 T 7 and strike
K≥0written on an allowance futures maturing at the end
Tof the compliance period is given at time t∈601 7 by
Ct=e−R
trsds Z
4ê4x5 −K5+N 4t1 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
tis Gaussian with mean
t1 41 5
=
tT−t
T−/2
if =11
texp
241−564T −t51−−4T − 51−7
if 6=1
(39)
and variance
2
t1 41 5
=
T−t
T−
−1 if =11
exp
1−64T −t51−−4T − 51−7−1
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 A0=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 longer-maturity calls (with the same
strike) are more valuable than their short-maturity
counterparts.
Let us stress that although there are no closed-form
formulas for call prices, their numerical evaluations
can be performed very efficiently.
Figure 4 Plots of the Prices C04 5 at Time t=0as Functions of Option
Maturity
01234
2
4
6
8
10
12
14
Time in years
Call price
Notes. The graphs marked by ,Î, and Èstand 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 6011007, 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 A04f5=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 Kin the range 601207, Black 76 call prices
are very similar to those given by our model with
maximum-likelihood 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
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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 6201807, 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
60040100657. 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 cap-and-trade 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,
real-world 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 cap-and-trade
scheme. Their rules go under the names of borrowing,
banking, and withdrawal.
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• 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 07and consider a
stochastic basis 4ì1 F11 4Ft5t∈601 T 075with a distinct
measure ∼, which we view as the spot martingale
measure. Further, we introduce processes 4At5t∈601T 7 ,
4A0t5t∈601 T 07for the futures contracts with maturities at
compliance dates T,T0written on allowance prices
from the first and the second period, respectively.
To exclude arbitrage, we suppose that the prices
4At5t∈601 T 7 and 4A0
t5t∈601 T 07are martingales with respect
to the spot martingale measure . Noncompliance in
the first and second periods occurs on events N∈FT
and N0∈FT0, respectively. As before, we assume that
the savings account 4Bt5t∈601 T 07is given by
Bt=eRt
0rsds1 t ∈601 T 07(43)
for some deterministic short rate 4rs5s∈601 T 07. The results
of the previous section imply that in the case ì\Nof
the first-period compliance, the allowance price drops
AT1ì\N=A0
T1ì\N1(44)
where ∈4015stands for discount factor describing
the interest rate effect
=BTB−1
T0=e−RT0
Trsds0
The relation (44) is justified by considering spot
prices. The random variable A0
Tis nothing but the
spot price at time Tof the second-period allowance.
Because futures and spot price agree at maturity, AT
must be the spot price of the first-period 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
AT1N=A0
T1N+1N0(45)
Namely, the noncompliance in one pollutant unit at
time Tcosts a penalty in addition to one allowance
from the next period, which must be withdrawn at
the spot price A0
T.
Combining the results (44) and (45), we find out
that the difference is
At−A0
t=Ɛ4AT−A0
TFt5=Ɛ41NFt51 t ∈601 T 7
and must be modeled as 801 9-valued martingale. We
suggest using the same methodology as in the one-
period model
At−A0
t=ê41
t51 t ∈601 T 71 (46)
where the Gaussian process 41
t5t∈601 T 7 is introduced as
previously in (12), with 4415
s5s∈601 T 7 in parameterized
form (28) and driven by a process 4W 1
t1Ft5t∈601 T 07of
Brownian motion.
To model the second-period allowance futures
price, a continuation of the cap-and-trade 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 4A0
t5t∈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 cap-and-trade sys-
tem will be terminated after the second period. In
this case,
A0
t=Ɛ41N0Ft51 t ∈601 T 07
can also be modeled as in the one-period model
A0
t=ê42
t51 t ∈601 T 070 (47)
Again, 42
t5t∈601 T 7 is introduced as in (12), with a pro-
cess 8425
s9s∈601 T 7 chosen in parameterized form (28) and
driven by another Brownian motion 4W 2
t1Ft5t∈601 T 07.
6.2. Option Pricing
As an application of our two-period 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 ∈601 T 7.
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Under the assumptions of the previous section, we
start with the computation of the price C0
C0=e−R
0rsdsƐ44A−K5+5
of this option at time t=0. Using the decomposition
4A−K5+=4A−A0
+A0
−K5+1
we utilize our modeling of 801 9-valued martingales
(46) and (47) to express the terminal payoff as
4A−K5+=4ê41
5+ê42
5−K5+
with expectation
C0=e−R
0rsdsƐ44A−K5+5
=e−R
0rsdsƐ44ê41
5+ê42
5−K5+5
=e−R
0rsds Z24ê4x15+ê4x25−K5+
·N 41 54dx11 dx251 (48)
where N 41 5=F1
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−1A0−A0
0
sT
T−1
+1/2
1R
04T −u541−15/2W1
udu
4T −51/21
2
=ê−1A0
0
sT0
T0−2
+1/2
2R
04T 0−u542−15/2W2
udu
4T 0−52/20
Denoting by the correlation of the two Brownian
motions 4W 1
t5t∈601 T 07and 4W 2
t5t∈601 T 07
6W 11 W 27 dt = dt1 ∈6−11171
we can apply the same argumentation to obtain the
means
1
=Ɛ41
5=ê−1A0−A0
0
sT
T−1
1
2
=Ɛ42
5=ê−1A0
0
sT0
T0−2
3
the variances
111
=Var41
5=T
T−1
−11
212
=Var42
5=T0
T0−2
−13
and the covariance as
112
=211
=Cov41
1 2
5
=1/2
11/2
2R
04T −u541−15/24T 0−u542−15/2 du
4T −51/24T 0−52/20
At times t∈601 7 prior to maturity, the price Ctof the
call is obtained similarly.
Proposition 3. In a two–compliance periods model as
above, with parameters 11 2>0, and ∈4−1115, the
price of the European call with strike price K≥0and matu-
rity ∈601 T 7 written on first-period allowance futures
price is given at time t∈601 7 by
Ct=e−R
trsds Z24ê4x15+ê4x25−K5+
·N 4t1 1 t1 54dx11dx25(49)
with mean t1
1
t1 =ê−1At−A0
t
sT−t
T−1
1(50)
2
t1 =ê−1A0
t
sT0−t
T0−2
1(51)
and covariance matrix t1
111
t1 =Var41
5=T−t
T−1
−11(52)
212
t1 =Var42
5=T0−t
T0−2
−11(53)
112
t1 =211
t1 =1/2
11/2
2R
t4T −u541−15/24T 0−u542−15/2 du
4T −51/24T 0−52/20
(54)
If we take a closer look at the computations in-
volved in the valuation of the call price
Ct=Ct4 1 T 1 T 01 A01 A0
01 K1 r 1 11 21 5
given by the formulas (49)–(54), we see that be-
cause of
Ct4 1 T 1 T 01 A01 A0
01 K1 r 1 11 21 5
=C04 −t1 T −t 1 T 0−t1 A01 A0
01 K1 r 1 11 21 5
for all t∈601 7, it suffices to consider the case t=0.
The numerical evaluation of two-dimensional inte-
gral is easily performed by using a decomposition of
the two-dimensional normal distribution. To ease the
notation, let us skip t1 to write i=i
t1 ,i1 j =i1 j
t1
for i1 j =112. It holds
N 41 54dx11 dx25
=N 411 c 4x251 1111 c 54dx15N 421 21254dx251 (55)
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Carmona and Hinz: Risk-Neutral 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 c4x25=1+211
2124x2−251
1111 c =111−421152
2120
With factorization (55), the inner integral is calculated
explicitly in the following cases:
Z
4ê4x15+ê4x25−K5+N 411 c 4x251 1111c 54dx15
=
0 if K−ê4x25≥1
ê11c 4x25
√1+1111 c + ê4x25−K
if K−ê4x25≤00
That is, the numerical valuation is required only in
the case 0 < K −ê4x25 < , where
Z
ê−14K/ −ê4x255
4ê4x15+ê4x25−K5
·N 411 c 4x251 1111 c 54dx15
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 1 157→ C04 1 15as
Function of Maturity and 1for 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 1and 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 first-period compliance
date T=4, eight years to the second-period compli-
ance date T2=8, initial first-period allowance futures
price A0=25, initial second-period allowance futures
price A0
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 1for
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 one-period mar-
ket model to a more realistic situation of two-period
markets (covering the present EU ETS regulations)
and show that martingales finishing at two-valued
random variables can be considered as basic build-
ing blocks for risk-neutral futures price dynamics.
We suggest a model for two-valued martingales, flex-
ible in terms of time- and space-changing 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
NSF-FRG-DMS-0455982 and DMS-0806591. The research of
Juri Hinz was partially supported by WBS R-703-000-020-
720/C703000 of the Risk Management Institute and by WBS
R-146-000-107-133 of the National University of Singapore.
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