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OPTIMAL STOCHASTIC CONTROL AND CARBON PRICE FORMATION
REN´
E CARMONA ∗, MAX FEHR †,AND JURI HINZ ‡
Abstract. To meet the targets of the Kyoto Protocol, European Union has established the European Emission
Trading Scheme, a mandatory market for carbon emission allowances. This regulatory framework has introduced
a market for emission allowances and created a variety of emission-related financial instruments. In this work, we
show that the economic mechanism of carbon allowance price formation can be formulated in the framework of
competitive stochastic equilibrium models, and we show that its solution reduces to an optimal stochastic control
problem. Using this mathematical set-up, we identify the main allowance price drivers and show how stochastic
control can be used to treat quantitative problems in carbon price risk management.
Key words. Stochastic control, Commodity options, Environmental risk, Carbon trading
AMS subject classifications. 93E20, 91B60, 91B62, 91B70, 91B76
∗Department of Operations Research and Financial Engineering, Princeton University, Princeton, NJ 08544.
Also with the Bendheim Center for Finance and the Applied and Computational Mathematics Program.
(rcarmona@princeton.edu).
†Institute for Operations Research, ETH Zurich, CH-8092 Zurich, Switzerland (maxfehr@ifor.math.ethz.ch).
‡National University of Singapore, Department of Mathematics, 2 Science Drive, 117543 Singapore
(mathj@nus.edu.sg), Partially supported by WBS R-703-000-020-720 / C703000 of the Risk Management Institute
at the National University of Singapore
1
2R. CARMONA AND M. FEHR AND J. HINZ
1. Introduction. The world’s changing climate and the pressing need for measures to curtail
the man-made effects on ecosystems continue to challenge policy makers. One of the issues is to
establish regulations which would provide a certain amount of flexibility, enabling agents to apply
commitments best suited to their circumstances. Presently, there are several international, national
and corporate policy frameworks aimed at pollution reduction in a flexible, cost effective manner
by the introduction of marketable emission credits. In this paper, the focus is on the European
Union Emission Trading Scheme (EU ETS), designed by the European Union as an instrument to
meet targets under the Kyoto Protocol.
In 1997 governments adopted the Kyoto Protocol, which broke new ground with its mandatory
requirements to reduce emissions of greenhouse gases (GHG). Each Annex B member, who has
ratified the Kyoto Protocol, is assigned a number of tradable credits AAU (assigned amount
units) each of which represents an allowance to emit one metric ton of carbon dioxide equivalent.
Moreover, there are a number of regional and national emission reduction projects, where agents
trade diverse emission certificates.
A remarkable example of such a project is the European initiative EU ETS, launched by the
Directive 2003/87/EC of the European Parliament and the Council of October 13, 2003. This
scheme is intended to ensure the reduction of carbon dioxide emissions from large industrial sources
within the European Union in order to contribute to the EU’s targets under the Kyoto Protocol.
The EU ETS imposes mandatory participation of nearly 10,000 [1] installations. These are power
plants and industrial users responsible for approximately 44 % of the entire EU carbon emission
in 2003 [1]. For these installations, carbon emission allowances (EUAs) are allocated yearly by the
responsible governments, according a National Allocation Plans (NAPs). Installations must cover
their emissions by allowances every single year. The precise regulation is as follows: There are two
compliance periods 2005–2007 and 2008-2012. Within each period, allowances are valid regardless
of the year in which they are allocated.
5
10
15
20
25
30
01/05/05 01/07/05 01/09/05 01/11/05 01/01/06 01/03/06 01/05/06 01/07/06
Euro/ t CO2
Fig. 1.1.EUA spot price, listed at the European Energy Exchange EEX. The price drop occurred right after
carbon emission data became public, showing that the overall market position was long.
Compliance is to be met yearly. Each operator that does not surrender sufficient allowances by
the 30th of April of each year to cover emissions for the preceeding year, is liable for the payment
of a penalty for each ton of excess emissions. The size of the potential payment is considerable:
Within the first period, the penalty was 40 EUROs, whereas for the second period, agents must
pay 100 EURO per excess ton of carbon dioxide. Moreover, the penalty payment does not release
OPTIMAL STOCHASTIC CONTROL AND CARBON PRICE FORMATION 3
from the obligation to surrender the missing allowances in the following calendar year.
Several exchanges in Europe are now committed to the trading of EUAs. The products listed there
are spot (see Figure 1.1) and forward contracts with physical delivery. The stochastic nature of this
price process is obvious. Consequently, conventional deterministic models are hardly applicable to
model allowance price formation in EU ETS. In line with this insight, [1] observe that the carbon
market is to a large extent trading on changes in stochastic fundamentals, such as relative coal/gas
prices and temperature/precipitation.
The present work proposes a dynamic stochastic equilibrium model that aims to explain allowance
price formation with respect to these fundamentals. The fundamental contribution of the paper is in
developing a model for equilibrium price formation of emission allowances when the abatement cost
is stochastic. We show that the equilibrium price process related to the solution of an appropriate
optimal control problem.
We now review the recent literature on equilibrium modeling of the emissions markets, and we
introduce the main mathematical assumptions of our approach.
To the best of our knowledge, the first equilibrium model for environmental trading schemes
was proposed by [6]. In his seminal paper, Montgomery considers a static perfect market of
several agents obliged to cover their entire emissions through allowances. To this end, the agents
are allocated a certain initial amount of permits, reflecting the overall emission target. Agents
face different reduction costs and, based on allowance prices, optimize their emissions and trade
allowances. It turns out that, optimal emission rates equate marginal control costs and allowance
prices. [6] proves the existence of a market equilibrium, where the emission target is reached
at minimal costs. Numerous subsequent studies have addressed environmental trading schemes
within a static framework of non-bankable permits. This line of research is not considered in the
present work as our focus is on the dynamic aspects of emission trading. Publications [14], [3]
and [10] provide a foundation for dynamic permit trading in a deterministic environment. They
derive market equilibria, similar to [6], but consider different multi-period settings where banking
and borrowing of allowances are allowed. The terminology banking stands for a mechanism where
credits remain valid for compliance in all future periods, while borrowing means that credits from
future periods can be used for compliance. The work [14] considers one dynamic compliance period
with discrete time-steps, where the abatement costs curve is static and reflects only short term
abatement. All allowances are allocated at the beginning of the compliance period, while the total
emissions have to be covered by allowances only at the end of the period. It is concluded that
before the permits are used up, their price must rise at the rate of interest, while also in this
dynamic framework, emission levels are chosen such that marginal control costs and allowance
price coincide. In [3] a permit system with banking is examined, where permits are allocated to
firms in each of Tcompliance periods. Further [10] extends the work of [14] and [3] by providing
a more general treatment of permit trading in continuous time through the use of optimal-control
theory. Considering both, borrowing and banking with restrictions on borrowing as a special case,
this formulation allows the extension of the results from [14] and [3]. Further generalization is given
in [13], where also explicit constraints on permit trading are examined. In the area of stochastic
modeling, to the best of our knowledge, only [5] takes into account the uncertainty aspects in an
equilibrium with banking. In this two-step stochastic equilibrium model, it is assumed that both,
emissions and abatement costs are stochastic, although obeying a specific structure. The study
4R. CARMONA AND M. FEHR AND J. HINZ
[5] discusses how uncertainty, technological progress and different types of market participants
affect allowance prices. Beyond equilibrium modeling, the issues of uncertainty are addressed in
several papers. We single out [11] which discusses the optimal decisions of the representative
agent under uncertainty. The approach of [11] is based on [10] and presents a model where both
allowance demand and abatement costs are stochastic although in a very restrictive way. However,
equilibrium formation is not considered. Finally, we mention the recent works [12] and [16] which
deal with the allowance price formation within EU ETS. Here, rather than modeling equilibrium
allowance prices, the EUA’s evolution is introduced as the marginal abatement costs when the
market follows an overall minimal-cost abatement policy. In [2] valuation of options on emission
allowances is addressed, and [7] treats econometric aspects of emission allowance prices.
Our contribution to equilibrium modeling of emission markets is two-fold. On one hand, we
address the need for a model which takes into account the particularities of the EU ETS. Indeed,
due to the existence of a penalty payment, the fundamental connection with electricity production,
the stochastic nature of demand for power and fuel prices, the abatement costs, etc, there is no
obvious way to extend the existing models reviewed above to fit the EU ETS. On the other hand,
the theoretical value of our model will go beyond EU ETS since it encompasses most of the stylized
facts in a generic cap-and-trade system, and yields a straightforward extension of [14] (see [15],
Chapter 2) to a stochastic framework. In particular, we expect that our work will be relevant for
the cap-and-trade schemes to be implemented in Japan, Canada, Australia and the US.
2. Emissions Markets Modeling.
2.1. Allowance Prices and Abatement Costs. In a cap and trade system, the allowance
price is determined by the existing abatement strategies, their flexibility and costs. Moreover, it
is important to distinguish abatement measures according to the time they require to return
a profit. In this regard, one can conceive a continuum of measures ranging from short-term
measures (no initial investment, savings returned within days), to long-term measures (high and
irreversible investments, savings returned over decades). Examples of long-term measures are
optimization/substitution of highly polluting production units, installation of scrubbers, investment
in CDM and JI projects. On the other end of the time scale, typical short term abatement measures
yield emission savings by replacing fuels or skipping / re-scheduling the production.
Because of their differences in time scales, we assume in this work, that decisions of middle and
long term investments are made in a different manner than those of short term abatements. Long
term investments are on a much larger time scale than a single compliance period. For instance
according to [8] the time scale for new electricity generating capacity is 20-30 years. Thus, the
influence of such projects can be modeled as an endogenous stochastic process which stands for the
residual demand on short term reduction, when the uncertainty about the amount of realized long
and middle term measures and their allowance supply is taken into account. Our prime focus is on
short-term measures, referred to as abatement measures in the following. The study of Dresdner
Kleinwort Wasserman Research (see [4], p. 57) finds that in the case of EU ETS, the main short
term abatement potential stems from the electricity sector, where the production can be switched
from hard coal to gas. Clearly marginal abatement costs need to be modeled as stochastic, but
before turning to the modeling of this stochasticity, we discuss marginal abatement cost curves for
fixed fuel prices.
In the existing literature, most of the research is done under the assumption that marginal
abatement costs increase with abatement volumes, while for mathematical reasons they restrict
marginal abatement curves to be continuous. The recent study [4] reveals however that for fixed fuel
prices, marginal abatement curves are stepwise constant and increasing. Each step is associated
OPTIMAL STOCHASTIC CONTROL AND CARBON PRICE FORMATION 5
with a certain abatement measure, its reduction capacity and marginal costs. Notice that the
marginal abatement cost curves in [4] could look continuous, as in most plots the steps are linearly
interpolated. In this work, we follow the same strategy in the sense that we suppose that a finite
number of abatement measures (here abatement measures are on the level of single plants) are
available and assume that there is a limit on the maximally possible emission reduction of each
measure per time unit. Furthermore, at each time step, marginal reduction costs of each abatement
measure are supposed to be constant, though dependent on instantaneous fuel prices. Consequently,
the marginal abatement curves in our model are non-decreasing and piecewise constant, with a
finite number of steps. Note, however that since in our model marginal abatement costs are
stochastic, abatement curves and merit order change randomly over time.
To provide the reader with further insight, we illustrate the price of fuel switching from coal to gas
in electricity generation. This illustration is of great practical importance since it represents one
of the major short term abatement measures within EU ETS.
Consider an agent iswitching from a hard coal plant to a cleaner CCGT plant. CCGT is the
0
50
100
150
200
250
300
01/05/05 01/07/05 01/09/05 01/11/05 01/01/06 01/03/06
Euro/tCO2
EUA Intraday
Switch Coal-Gas
0
5
10
15
20
25
30
01/04 01/05 01/06 01/07 01/08 01/09 01/10
Fig. 2.1.The price for EUA versus fuel switching price calculated from gas/coal spot prices.
acronym for Combined Cycle Gas Turbine, a relatively novel technology where the waste heat
from the gas turbine is used to run a steam turbine in order to enhance the efficiency of electricity
generation. Using CCGT, the agent’s technology possesses specific emissions for gas
ei
g= 0.202 tCO2
MWhtherm
·1
0.52
MWhtherm
MWhel
= 0.388 tCO2
MWhel
(2.1)
and coal
ei
c= 0.341 tCO2
MWhtherm
·1
0.38
MWhtherm
MWhel
= 0.897 tCO2
MWhel
(2.2)
measured in tons of emitted carbon for the generation of one MWh of electricity. Here, tCO2and
MWhtherm , MWhel denote a ton of carbon dioxide, and a Mega Watt of thermic and electrical
power respectively. The CO2emission factors are default values provided by the Intergovernmental
Panel on Climate Change (IPCC). The switch of production technology at time tyields per MWh
of electricity a reduction of ei
c−ei
g= 0.509 ton of carbon dioxide. At the same time, this fuel
switch causes costs of hi
gGi
t−hi
cCi
tEURO per MWh, where Gi
t,Ci
tare gas and coal spot prices
6R. CARMONA AND M. FEHR AND J. HINZ
for the agent iat time t(expressed in EURO per MWhtherm and in EURO per ton respectively).
The coefficients
hi
g=1
0.52
MWhtherm
MWhel
= 1.92 MWhtherm
MWhel
hi
c=1
6.961
tcoal
MWhtherm
1
0.38
MWhtherm
MWhel
= 0.378 tcoal
MWhel
are specific heat rates, expressing how much fuel is consumed for the generation of one MWh
of electricity. Here we have assumed that the amount of coal is measured in tons, whereas the
amount of gas is expressed in Mega Watt hours of thermal power since gas prices we use are given in
EURO/MWhtherm . The calculation of hi
gis based on the reference value of 6000kcal/kg reported
in McCloskey’s NWE Steam Coal Marker. With these quantities, we have the fuel switching price
Ei
t=hi
gGi
t−hi
cCi
t
ei
g−ei
c
for all t= 0,...,T −1 (2.3)
measured in EURO per ton of carbon dioxide. Based on a given time series for coal and gas spot
prices, formula (2.3) yields the corresponding fuel switching price process appearing in Figure 1.1.
Remark. In the case of Figure 1.1, gas prices are too high to trigger a notable number of fuel
switches from CCGT technology. For the actual abatement from other measures, we refer the
interested reader to the comprehensive study [4].
Remark. In the commodity business, companies exposed to risks from fluctuations in the
prices of commodities used as production inputs, hedge themselves with an appropriate portfolio
of futures contracts. Hence correlations between the prices of the diverse commodities become
essential. In particular, the European energy business is concerned about the correlation between
EUA and fuel prices. However as Figure 2.1 shows, their inter-dependence is not obvious. A study
based on our model could shed light on this important problem.
2.2. Mathematical Model. In this paper, we consider a stylized cap-and-trade scheme
where market participants trade allowances and apply abatement policies to comply with the
emissions targets. We suppose that each agent acts rationally: to increase her own wealth, she
applies abatement and trading based on past and current information. Our goal is to identify and
study an equilibrium elucidating the EUA price formation mechanism. However, in order to deal
with a tractable model we have to make simplifications which we articulate below.
Since trading and production decisions occur at discrete times, we choose to work in a discrete-time
setting. However, notice that the quantitative study in Section 4 is based on parameter estimation
for discretely sampled diffusion processes. A further simplification is to suppose that the penalty
consists only of a financial penalty which is paid once at the end of the compliance period. Current
regulation stipulates annual compliance, and allowances are valid during the entire period, even
though companies are issued allowances on February 28, two months prior to the date when they
are required to cover their emissions from the previous year. Consequently for compliance in one
given year, companies may surrender allowances which are issued for the following year, a form of
borrowing. Considering that the yearly allocations are sufficiently high, the penalty must only be
paid at the end of the compliance period, if ever.
Once a frictionless transfer of allowances over time is agreed upon, we can assume that EUA is a
purely financial asset, and the difference between spot and forward prices is captured by interest
OPTIMAL STOCHASTIC CONTROL AND CARBON PRICE FORMATION 7
rate effects only. That is, it suffices to consider either EUA spot or the EUA forward price. To
avoid discounting, we decided to model the forward price for EUA with maturity at the final
compliance date. Remember that a forward is an agreement to exchange an asset at a future date,
at the price which is fixed when the parties enter the contract. Because of this, all incentives are
described by the maximization of the individual terminal wealths, expected from the perspective
of the historical objective measure. This is a rather strong assumption, and the advantages of
the ensuing simplification are far-reaching: due to social optimality (which applies only to the
non-risk-averse case), the model boils down to a low dimensional optimal control problem and
is amenable to numerical treatment. In some sense, we obtain a similarity to auction theory,
where the celebrated revenue equivalence theorem plays an essential role and enjoys applications,
although it applies only to non-risk-averse agents.
We start from the realization that carbon price development reflects the private economic interests
of installations, concerned with emission regulations in a cap-and-trade framework. Thus, the main
aspect in our modeling is to face the individual strategy optimization of single market participants
exposed to carbon price risk. We consider personal incentives of stylized agents who possess the
flexibility of short term emission reduction, which is exercised whenever emission allowance prices
indicate that this is reasonable. Here, the decisions may range from a simple replacement of the
output by a supplementary material to a partial shut down of production. For instance, in the
cement industry, customers are offered cinders remaining from the iron production, whereas in the
energy business, electricity producers occasionally cancel their long-term supply contracts.
As already mentioned, the study [4] indicates that the major carbon emission reduction resource
is fuel switch (in the simplest case, from coal to gas) in heating and electricity generation. On this
account, commodity price models (in particular, fuel price models) form an intrinsic part of carbon
price description. Consequently, we attempt to find how the emission allowances price evolution
relates to the fuel price development.
We consider N∈Nmarket participants producing electricity from fossil fuel and trading carbon
allowances at discrete times t∈ {0,1,...,T} ⊂ N. The entire time horizon corresponds to one
compliance period, that is, at maturity T, all agents have to cover their carbon emissions by
allowances or pay penalties. We agree to describe all prices and strategies by adapted stochastic
processes on a filtered probability space (Ω,F, P, (Ft)T
t=0) . So the decisions of all market participants
are based on the information flow given by the filtration (Ft)T
t=0 . In particular, each agent observes
the allowance price and the short term abatement prices of all market participants. From a practical
perspective, it is not obvious whether modeling of private information is crucial for EU ETS, since
the collection and distribution of news, reports, and analysis of the carbon market is becoming a
business line for many consultancies. Another model simplification is that each agent i= 1,...,N
observes her own short term abatement price (Ei
t)T−1
t=0 which is supposed to follow an adapted
stochastic process (note that fuel switch price is not necessarily positive).
We write A= (At)T
t=0 for the forward price with delivery date Tof one carbon allowance
certificate, and assume that it is a positive–valued stochastic process. Note that due to the
definition of a forward price, ATequals the spot price for carbon allowances at the final time
T. A forward trading strategy for agent iis a process θi= (θi
t)T−1
t=0 , where θi
tfor t= 0,...,T −1
stands for the number of forward contracts held by agent i= 1,...,N at time t. Note that holding
position θi
tfrom tto t+ 1 yields
θi
t(AT−At)−θi
t(AT−At+1) = θi
t(At+1 −At)
8R. CARMONA AND M. FEHR AND J. HINZ
at T. That is, at compliance date T, the wealth of strategy (θi
t)T−1
t=0 is given by
T−1
X
t=0
θi
t(At+1 −At).(2.4)
Moreover, each agent/producer imust face the cost associated with the difference between emitted
carbon and allowances allocated at the beginning of the period. We model this quantity by an
FT–measurable random variable Γi. We allow for both negative and positive realizations of Γi,
occurring if the credit exceeds or drops below the realized emission. In order to comply at the
end of the period, each agent adjusts the number of credits depending on the actually realized
allowance demand. This action is described by the number of allowances θi
Twhich the agent i
purchases at Tgiving the final emission balance Γi−θi
T. In the case Γi−θi
T≥0 emissions top
allowances, a penalty of π∈(0,∞) EURO is to be paid for each ton which is not covered. Thus,
the final profit from trading allowances equals
T−1
X
t=0
θi
t(At+1 −At)−θi
TAT−π(Γi−θi
T)+.(2.5)
According to our previous discussion, we should include the effect of long-term abatement measures
by adjusting the emission expectation with the estimated reduction from the proposed long-term
measures, but we shall refrain from doing that in the present contribution.
We suppose that each producer ipossesses a technology which at any time t= 0,...,T−1 allows
a reduction ξi
tof at most λi∈[0,∞) ton of carbon emitted within the period [t, t + 1] by fuel
switching. It is straight forward to extend this work to several abatement measures per agent.
Then each agent would face a stepwise constant increasing marginal abatement cost curve in each
state and at each time point. At time t, the decision is based on the public information Ft. The
fuel switching policy ξi= (ξi
t)T−1
t=0 yields expenses which are modeled by a cash payment of the
amount
T−1
X
t=0
ξi
t
Ei
t
pt(T)=
T−1
X
t=0
ξi
tEi
t(2.6)
at maturity T. Discounting is taken care of by relating the fuel switch spot price Etto the price
pt(T) of the zero bond maturing at Tby
Ei
t:= Ei
t/pt(T)t= 0,...,T.
The fuel switch helps to meet compliance, since instead of the allowance demand Γimerely Γi−
PT−1
t=0 ξi
tton of carbon dioxide are to be covered at the end of the period. Thus, we correct Γi
in the equation (2.6) by Γi−PT−1
t=0 ξi
twhich, combined with (2.5) expresses the profit/loss of the
producer iby
IA,i(θi, ξ i) =
T−1
X
t=0
θi
t(At+1 −At)−θi
TAT−π(Γi−
T−1
X
t=0
ξi
t−θi
T)+−
T−1
X
t=0
ξi
tEi
t.(2.7)
3. Mathematical Analysis of the Model. In order to guarantee the existence of expected
values of (2.7) and later on (3.6), we suppose that
Γi,Ei
tare integrable for i= 1,...,N,t= 0,...,T −1 . (3.1)
OPTIMAL STOCHASTIC CONTROL AND CARBON PRICE FORMATION 9
We shall use the Banach spaces L1=L1(FT) and L∞=L∞(FT) of P–equivalent classes of
integrable and essentially bounded FT–measurable random variables respectively. Further, we
introduce the following spaces of adapted processes
L1:= {(Ξt)T−1
t=0 : Ξt∈L1(Ft), t = 0,...,T −1}
L∞:= {(ξt)T−1
t=0 :ξt∈L∞(Ft), t = 0,...,T −1}
Ui:= {(ξi
t)T−1
t=0 : adapted [0, λi]-valued process}
U:= ×N
i=1Ui
Given that each market participant maximizes her own profit by trading allowances and applying
abatement measures, given an allowance price process A= (At)T
t=0 , we formulate the individual
optimization problem for agent ias
sup
(θi,ξi)∈(L1×L1)×U i
E{IA,i(θi, ξi)}.(3.2)
With these notations, we define the equilibrium as
Definition 3.1. Given a fuel switch price process (Ei
t)T−1
t=0 ∈ LN
1for each agent i= 1,...,N,
an adapted process A∗= (A∗
t)T
t=0 is said to be an equilibrium carbon price process, if there exists
for each agent i= 1,...,N a trading and an abatement strategies (θi∗, ξ i∗)∈(L1×L1)× U ifor
which financial positions are in zero net supply
N
X
i=1
θ∗i
t= 0 at any time t= 0,...,T.(3.3)
and individual agents cannot do better in the sense that:
E{IA∗,i(θi∗, ξi∗)} ≥ E{IA∗,i(θi, ξi)}for all (θi, ξi)∈(L1×L1)× U i,(3.4)
for i= 1,...,N.
Remark. It should be emphasized that zero net supply (3.3) is stated at t= 0,...,T −1 for
different reasons than at t=T. Indeed, for t= 0,...,T −1, (θ∗i
t)N
i=1 are forward positions,
whereas at maturity, (θ∗i
T)N
i=1 stand for the change in the initial physical allocation of the agents
i= 1,...,N. Let us elaborate on the economic meaning of this remark. In the above definition,
we use a standard equilibrium notion, based on the intuition that at low prices most agents will
have a tendency to purchase allowances, while at high prices, they will most likely produce and
sell the goods. Hence, the equilibrium price is characterized by vanishing excess demand. Note
that according to this concept, (θi∗, ξi∗) is indeed a maximizer of (θi, ξi)7→ E{IA∗,i (θi∗, ξi∗)}on
the entire set (L1×L1)× U iwithout any restrictions from zero net supply.
The most important property of the above notion of equilibrium is that it enjoys the property of
social optimality. As we are about to show, an equilibrium in the above sense automatically results
in the solution of a certain global optimization problem, where the total pollution is reduced at
minimal overall cost. Beyond the economic interpretations of social-optimality, the importance of
the global optimization is that its solutions help to show the existence of an equilibrium, and to
calculate the corresponding carbon prices.
Suppose we are given the fuel switch prices (Ei
t)T−1
t=0 ∈ L1for i= 1,...,N. For any given switching
policy ξ= (ξ1
t,...,ξN
t)T−1
t=0 ∈ U of the agents i= 1,...,N, we denote the final overall switching
10 R. CARMONA AND M. FEHR AND J. HINZ
costs by
F(ξ) =
N
X
i=1
T−1
X
t=0
ξi
tEi
t.
Further, we write
Π(ξ) =
N
X
i=1
T−1
X
t=0
ξi
t,(3.5)
for the total savings in emissions when the abatement policy ξis used, and we denote by
Γ =
N
X
i=1
Γi
the overall allowance demand. Finally, we define the total costs from fuel switching and penalty
payments as
G(ξ) = −F(ξ)−π(Γ −Π(ξ))+, ξ ∈ U,(3.6)
and we introduce the global optimization problem
ξ∗= arg sup
ξ∈U
E{G(ξ)},(3.7)
where a switching policy ξ∗∈ U for all agents is to be determined, which minimizes the social
costs of non-compliance. We now prove that the existence of such an optimal policy ξ∗is ensured
by standard functional analytic arguments.
Proposition 3.2. With the above notations and under the above assumptions, there exists a
solution ξ∗∈ U to the global optimal control problem (3.7).
Remark. In economic terms, this proposition states that the financial trading of allowances
decouples from physical abatement. Indeed, the proof actually shows that there is no contribution
from financial trading to individual wealth because the equilibrium allowance price process is
a martingale. In other words, the individual wealth is effectively increased only by physical
abatement, and by the adjustment of the final physical position.
Proof. First, note that LN
1equipped with the norm
kΞk1=
T−1
X
t=0
N
X
i=1
E{|Ξi
t|}
is a Banach space with dual LN
∞, the canonical bilinear pairing being
hΞ, ξi:=
T−1
X
t=0
N
X
i=1
E{Ξi
tξi
t}Ξ∈ LN
1, ξ ∈ LN
∞.
For the weak topology σ(LN
∞,LN
1) on LN
∞(see [9]), a basis of neighborhoods of a point ξ∈ LN
∞is
given by the finite intersections of sets
Bξ(Ξ, δ) := {ξ′∈ LN
∞:|hΞ, ξ′−ξi| < δ},Ξ∈ LN
1, δ > 0.(3.8)
OPTIMAL STOCHASTIC CONTROL AND CARBON PRICE FORMATION 11
In other words, σ(LN
∞,LN
1) is the weakest topology for which all the linear functionals
LN
∞→R, ξ 7→ hΞ, ξi,Ξ∈ LN
1(3.9)
are continuous. A function f:LN
∞→Ris lower semi-continuous at ξif for each ε > 0 there exists
a neighborhood Bξof ξsuch that f(ξ′)> f(ξ)−εfor all ξ′∈Bξ. Such a function is called lower
semi-continuous, if it is lower semi-continuous at each point. We prove existence of a minimizer
ξ∗of ξ7→ E{−G(ξ)}on Uby proving that Uis compact and that ξ7→ E{−G(ξ)}is lower semi-
continuous with respect to σ(LN
∞,LN
1), and the fact that lower semi-continuous functions attain
their minima on compact sets.
Given ξ∈ LN
∞,
E{−G(ξ)}=
T−1
X
t=0
N
X
i=1
E{Ei
tξi
t}+πE{(Γ −Π(ξ))+},
and since the first term is a continuous linear functional of the type (3.9) (evaluated at ξ), it
suffices to prove lower semi-continuity of
ξ7→ E{(Γ −Π(ξ))+}.
In order to do so, we fix the point ξ. For any ξ′∈ LN
∞we have:
(Γ −Π(ξ′))+≥(Γ −Π(ξ′))1{Γ−Π(ξ)≥0}
≥(Γ −Π(ξ))1{Γ−Π(ξ)≥0}−(Π(ξ′)−Π(ξ))1{Γ−Π(ξ)≥0}
≥(Γ −Π(ξ))+−(Π(ξ′)−Π(ξ))1{Γ−Π(ξ)≥0},
thus
E{(Γ −Π(ξ′))+} ≥ E{(Γ −Π(ξ))+} − E{(Π(ξ′)−Π(ξ))1{Γ−Π(ξ)≥0}}
≥E{(Γ −Π(ξ))+} − hΞ, ξ′−ξi(3.10)
where Ξ ∈ LN
1is given by Ξi
t=E{1{Γ−Π(ξ)≥0}|Ft}for all i= 1,...,N and t= 0,...,T −1.
Hence, given ε, define the neighborhood Bξ(Ξ, ε) of ξas in (3.8) which ensures that |hΞ, ξ′−ξi| < ε
for all ξ∈Bξ(Ξ, ε) and finally with (3.10) yields the lower semi-continuity
E{(Γ −Π(ξ′))+} ≥ E{(Γ −Π(ξ))+} − εfor all ξ′∈Bξ(Ξ, ε).
Now let (ξ(n))n∈N⊂ U be a sequence approaching the infimum
lim
n→∞ E{−GT(ξ(n))}= inf
ξ∈U E{−GT(ξ)}.
By the theorem of Banach–Alaoglu, it contains a subsequence (ξ(nk))k∈Nwhich converges to ξ∗
in the weak topology. Since Uis convex and norm-closed in LN
1, Lemma 3.3 shows that
Uis σ(LN
∞,LN
1)-closed, (3.11)
hence ξ∗∈ U . Finally, the semi-continuity ensures that E{−GT(ξ∗)}= infξ∈U E{−GT(ξ)}.
Since LN
1is not the dual space of LN
∞we can not conclude that the k · k∞-closed subset Uof LN
∞
is σ(LN
∞,LN
1)-closed. However, since Uis k · k1-closed in LN
∞, we obtain the assertion.
Lemma 3.3. For the convex subset Uof L∞it holds that:
12 R. CARMONA AND M. FEHR AND J. HINZ
(i) Uis norm-closed in L1
(ii) Uis σ(LN
∞,LN
1)-closed in L∞.
Proof. (i) If (ξn)n∈Nis a sequence in Uconverging in L1to some random variable ξ, then
extracting a subsequence if necessary, one concludes that it converges almost surely showing that
the constraints defining Uare satisfied in the limit, which implies that ξ∈ U .
(ii) Since Uis a convex and a norm-closed subset of L1it follows from the Hahn-Banach Theorem
that Uis the intersection of half spaces Hξ,c ={X∈ L1|hX, ξi ≤ c}with ξ∈ L∞and c∈R.
Since L∞⊆ L1it holds for each of these half spaces Hξ ,c that ξ∈ L1. Thus we conclude that
Hξ,c ∩ L∞={X∈ L∞|hX, ξi ≤ c}is closed in (L∞, σ(L∞,L1)). Since by definition it holds
that U ⊆ L∞it follows that Uis given by the intersection of the sets Hξ,c ∩ L∞which are
σ(LN
∞,LN
1)-closed so (3.11) follows.
The following result will be crucial in our analysis of the equilibrium.
Theorem 3.4. Suppose that
the FT−1-conditional distribution of Γ
posseses almost surely no point mass. (3.12)
Then, the equilibrium carbon price process is a martingale given in terms of the global-optimal
policy ξ∗∈ U from Proposition 3.2 by
A∗
t=πE{1{Γ−Π(ξ∗)≥0}| Ft}for t= 0,...,T. (3.13)
Remark. Let us highlight the connection of this theorem to results given in the literature. Due
to (3.13), the equilibrium allowance price is economically interpreted as the marginal contribution
of an additional allowance to lower the potential penalty payment, when the global-optimal policy
ξ∗∈ U is followed. Indeed, expressing the indicator function appearing in (3.13) as a derivative
we get
A∗
t=−∂
∂x E{π(Γ −Π(ξ∗)−x)+| Ft}|x=0.
This justifies rigorously the folk principle often found in the literature on emission markets
the equilibrium allowance price equals the marginal abatement costs, (3.14)
and considered as crucial in the economic analysis of these markets. In the proof given below,
the equilibrium allowance price plays the role of an exercise boundary which drives the abatement
measures to a global optimum. To understand this, we point out that in the presence of allowance
trading, the optimal decision of each market participant iis to apply abatement at full intensity if
the value of the allowance price indicates that this is reasonable {At>Ei
t} ⊆ {ξi
t=λ}. Indeed, all
saved credits can be immediately sold on the market. Otherwise, no–abatement action is optimal
{At<Ei
t} ⊆ {ξi
t= 0}since instead of saving emissions in the producers own business, the agent
is better off purchasing emissions allowances on the market. In the proof (see assertions (3.25)
and (3.26)) we show that the equilibrium allowance price A∗
tactually triggers the overall-optimal
abatements ξi∗
t(i= 1,...,N) in this way. Now we explain why this issue is actually nothing
but a version of the classical result (3.14). Being the optimal-exercise boundary (in the above
sense), A∗
tis greater or equal than the costs of the most expansive among all active abatement
OPTIMAL STOCHASTIC CONTROL AND CARBON PRICE FORMATION 13
measures. Moreover, there is no inactive abatement measure whose cost is below A∗
t. Therefore,
by an appropriate interpretation of the notion of marginality, we see that (3.14) holds in our model.
Proof. The equilibrium property of (A∗
t)T
t=0 can be shown by an explicit construction of θi∗∈
L1×L1such that the individual strategies
(θi∗, ξi∗)∈(L1×L1)× Ui, i = 1,...,N
fulfill (3.4) and (3.3). To proceed, let ξ∗∈ U be as given in Proposition 3.2, and define (θi∗)N
i=1 ∈
(L1×L1)Nby
θ∗i
t= 0 for all i= 1,...,N,t= 0,...,T −1,(3.15)
θi∗
T= Γi−
T−1
X
t=0
ξi∗
t−(Γ −Π(ξ∗))/N.
Note that due to martingale property of (A∗
t)T
t=0 , any other choice of (3.15) which respects the
zero net supply condition is also admissible. Since (3.3) is obviously fulfilled, we focus on the proof
of (3.4).
For θ∈(L1×L1)N,ξ∈ U , and carbon price processes (3.13), we can express the expectation of
(2.7) as
E{IA∗,i(θi, ξ i)}=E{−θi
TA∗
T−π(Γi−
T−1
X
t=0
ξi
t−θi
T)+} − E{
T−1
X
t=0
ξi
tEi
t}
since the process A∗∈ L∞is a martingale by definition (3.13) and (θi
t)T−1
t=0 is an element of L1.
Thus, in order to show (3.4), it suffices to prove that for each ξi∈ Ui, the supremum
m(ξi) := sup
θi
T∈L1
E{−θi
TA∗
T−π(Γi−
T−1
X
t=0
ξi
t−θi
T)+}(3.16)
is attained on L1at
θi
T(ξi) = Γi−
T−1
X
t=0
ξi
t−(Γ −Π(ξ∗))/N (3.17)
and
ξi7→ m(ξ) + E{
T−1
X
t=0
ξi
tEi
t}is maximized on Uiat ξi∗. (3.18)
First, we turn to (3.16) and (3.17) showing that the maximum is attained pointwise. In view of
(3.13), ω∈ {Γ−Π(ξ∗)<0}implies that
A∗
T(ω) = 0 and θi∗
T(ξi)(ω)>Γi(ω)−
T−1
X
t=0
ξi
t(ω).
Moreover, the maximum of
z7→ −zA∗
T(ω)−π(Γi(ω)−
T−1
X
t=0
ξi
t(ω)−z)+(3.19)
14 R. CARMONA AND M. FEHR AND J. HINZ
is attained at each point of the interval [Γi(ω)−PT−1
t=0 ξi
t(ω),∞), thus θi
T(ξi)(ω) is a maximizer.
In the other case, namely when ω∈ {Γ−Π(ξ∗)≥0}, we have
A∗
T(ω) = πand θi
T(ξi)(ω)≤Γi(ω)−
T−1
X
t=0
ξi
t(ω).
Here the maximum of (3.19) is attained at each point of the interval [0,Γi(ω)−PT−1
t=0 ξi
t(ω)], thus
θi
T(ξi)(ω) is again a maximizer. In both cases, the maximum of (3.16) is
m(ξi) = E{−(Γi−
T−1
X
t=0
ξi
t)A∗
T}.
If we plug the above expression for m(ξi) into (3.18) and use the martingale property of (A∗
t)T−1
t=0
we conclude that
m(ξi)−E{
T−1
X
t=0
ξi
tEi
t}=−E{ΓiA∗
T}+E{ξi
t(A∗
T− Ei
t)}
=−E{ΓiA∗
T}+E{ξi
t(A∗
t− Ei
t)}.
In order to show (3.18), it suffices to check that for each i= 1,...,N and t∈ {0,...,T −1}the
following inclusions hold almost surely
{A∗
t− Ei
t>0} ⊆ {ξ∗i
t=λi},and {A∗
t− Ei
t<0} ⊆ {ξ∗i
t= 0}.(3.20)
First, we remark that
for any ξ∈ U with ξs=ξ∗
sfor s= 0,...,t−1.
E{G(ξ)|Ft} ≤ E{G(ξ∗)|Ft}holds almost surely. (3.21)
This assertion is proved as follows. If untrue, the Ft-measurable set
M:= {E{G(ξ)|Ft}>E{G(ξ∗)|Ft}}
would be of positive measure, P{M}>0 , and could be used to construct an abatement strategy
ξ′which would outperform ξ∗. Indeed, setting
ξ′
s= 1Mξs+ 1Ω\Mξ∗
s(3.22)
for all s= 0,...,T −1 we see that since ξ∗and ξ′coincide at times 0,...,t−1 , this definition
indeed yields an adapted process ξ′∈ U . The decomposition
G(ξ′) = 1MG(ξ) + 1Ω\MG(ξ∗),
leads to a contradiction to the optimality of ξ∗. Indeed,
E{G(ξ′)}=E{E{1MG(ξ) + 1Ω\MG(ξ∗)|Ft}}
=E{1ME{G(ξ)|Ft}+ 1Ω\ME{G(ξ∗)|Ft}}
>E{1ME{G(ξ∗)|Ft}+ 1Ω\ME{G(ξ∗)|Ft}} =E{G(ξ∗)}.
To prove (3.20) we consider for each λin the countable set Q:= [0, λi]∩Qwhere Qdenotes the
set of rational numbers, the strategy ξ(λ, i)∈ U defined by
ξk
s(λ, i) = λif s=tand k=i
ξ∗k
sotherwise ,
OPTIMAL STOCHASTIC CONTROL AND CARBON PRICE FORMATION 15
In other words, ξ(λ, i) coincides with ξ∗with the exception of time t, where only for the agent
ithe fuel switch intensity is changed from ξ∗i
tto a deterministic value λ∈Q. This abatement
policy ξ(λ, i) satisfies
Π(ξ(λ, i)) = Π(ξ∗)−(ξ∗i
t−λ)
F(ξ(λ, i)) = F(ξ∗)−(ξi∗
t−λ)Ei
t
for all λ∈Q. (3.23)
Since the set Qis countable due to (3.21), there exists a set ˜
Ω with P{˜
Ω}= 1 such that
E{G(ξ∗|Ft)}(ω)−E{G(ξ(λ, i)|Ft)}(ω)
|ξ∗i
t(ω)−λ|≥0 for all ω∈˜
Ω with λ6=ξ∗i
t(ω).
Using (3.23) and (3.6), we conclude from this inequality that
0≤ − ξ∗i
t(ω)−λ
|ξ∗i
t(ω)−λ|Ei
t(ω)
−Eπ(ΓT−Π(ξ∗))+−(ΓT−Π(ξ∗) + (ξ∗i
t−λ))+
|ξ∗i
t−λ|| Ft(ω) (3.24)
holds for all ω∈˜
Ω with λ6=ξ∗i
t(ω). Let us denote the term in in (3.24) by D(ξ∗, λ)(ω)
Approaching ξ∗i
t(ω) by λ∈Q\ {ξ∗i
t(ω)}, we apply dominated convergence theorem to obtain
lim
λ↑ξi∗
t(ω)D(ξ∗, λ)(ω) = −Eπ1{Γ−Π(ξ∗)≥0}| Ft(ω) for ξ∗i
t(ω)∈]0, λi],
lim
λ↓ξi∗
t(ω)D(ξ∗, λ)(ω) = Eπ1{Γ−Π(ξ∗)>0}| Ft(ω) for ξ∗i
t(ω)∈[0, λi[.
Now (3.12) gives
Eπ1{Γ−Π(ξ∗)≥0}| Ft=Eπ1{Γ−Π(ξ∗)>0}| Ft=A∗
t
which with (3.24) implies that the following inclusions hold almost surely: Calculating left limit
λ↑ξi
t(ω), we have
{ξ∗i
t∈]0, λi]} ⊆ {A∗
t− Ei
t≥0} ⇔ {A∗
t− Ei
t<0} ⊆ {ξ∗i
t= 0}(3.25)
For the right limit λ↓ξi
t(ω), we obtain
{ξ∗i
t∈[0, λi[} ⊆ {A∗
t− Ei
t≤0} ⇔ {A∗
t− Ei
t>0} ⊆ {ξ∗i
t=λi}.(3.26)
The assertions (3.25) and (3.26) give (3.20).
4. Quantitative Analysis. This section is devoted to the numerical analysis of some of the
quantitative aspects of the carbon market equilibrium model introduced in this paper. Recall that
it only deals with one compliance period. First, we emphasize the main differences between our
generic model and the actual EU ETS implementation.
Streamlined EU ETS As explained earlier, we took care of the discounting effects by working
with forward prices. In this section, we go even further and we suppose that the interest rate is
equal to zero.
1) Working with one compliance period in isolation does not fully reflect the situation of EU ETS in
the period 2005–2007, since a certain amount of allowances could be banked into the next period
2008–2012 and, more importantly, the penalty structure was different from what we modeled.
16 R. CARMONA AND M. FEHR AND J. HINZ
Indeed, at the end of the first EU ETS period, for each ton of CO2equivalent uncovered by a
EUA, one EUA from the second period was charged for non-compliance in the first period, in
addition to the fine of 40 EURO.
Another simplification is that we do not consider the impact of allowances gained from CDM-
and JI-projects. As mentioned earlier, in our model, the impact of long term projects is captured
by the anticipated short-term reduction demand (E{Γ|Ft})T
t=0 . Hence, the role of the stochastic
process (E{Γ|Ft})T
t=0 is to account for the aggregate impact of all uncertainty sources, including
ambiguity from emitting factors (weather, climate, business activity), the success of long and
middle term reduction projects, imperfections in information flow, etc. Because of this complexity,
one of the most difficult problems in the present framework is to find an appropriate quantitative
description for (E{Γ|Ft})T
t=0 . In the present study, we choose for this martingale, a discrete version
of a Brownian motion independent from the fuel switch processes. This choice is motivated by the
connection of (E{Γ|Ft})T
t=0 with the Emission-to-Cap Indicator listed by PointCarbon, see e.g. [1].
This index is designed to monitor the estimated emission savings required to meet the compliance.
Note however that our (E{Γ|Ft})T
t=0 is not the Emission-to-Cap Indicator since the latter does
not consider potential demand reduction from CDM projects and long term investments.
Finding out if (E{Γ|Ft})T
t=0 and (Ei
t)T
t=0 can be modeled by independent processes will require
an empirical correlation analysis between historical values of the Emission-to-Cap Indicator and
representative fuel switch price processes. Unfortunately, historical data for the Emission-to-Cap
Indicator is not available yet.
Remark. The problem of the correct choice for (E{Γ|Ft})T
t=0 is part of the model calibration which
amounts to choosing a specific parametric family of martingales, and tuning the parameters in such
a way that the prices produced within the model match the listed carbon prices. No-arbitrage
valuation of a European derivative written on EUA could then be obtained by computation of its
expected payoff within the calibrated model.
2) A further simplification is to consider a single fuel switch price process. The argument is
that if the cheapest technology is applied first, and if the switch capacity is small enough, other
technologies are rarely used. So, the decision to consider only one fuel switch price process does
not mean that we omit other short-term abatement measures. We merely suppose that their
impact on carbon price can be neglected, due to high capacity of cheaper abatement measures
from CCGT. Even tough there are numerous plants with different efficiencies, we aggregate all fuel
switch possibilities from CCGT to coal plants into one representative capacity. This approximation
seems acceptable, since according to [4], fuel switch prices vary within a 4 Euro range, which is
small compared to the changes of the fuel switch prices caused by gas and coal price fluctuations.
Based on [4] we assume, that the European yearly switch capacity is 52 Mega tonnes of carbon
dioxide. For the fuel switch, we suppose the efficiencies of 38% for coal and 52% for CCGT plants.
Next, we discuss the time resolution of our calculations. Since the model is set up in terms of
(Et)T−1
t=0 and (Γt=E{Γ|Ft})T
t=0 (4.1)
which represent the evolutions of the fuel switch cost and of expected demand respectively. These
processes are obtained as time discretizations of continuous time processes
(E(t))t∈[0,T],and (Γ(t))t∈[0,T](4.2)
for fuel switch price and expected demand evolutions. Note that we write the time parameter in
parenthesis instead using subscript, to indicate continuous-time processes. Moreover, the horizon
OPTIMAL STOCHASTIC CONTROL AND CARBON PRICE FORMATION 17
for continuous time is [0,T] , where we suppose that the time unit is one year. Given the fact that
a reasonable time step should not fall below the time required to re-schedule a CCGT turbine, we
sample the continuous time evolutions daily.
Fuel Switch Process.
-20
-10
0
10
20
30
40
50
60
70
01/01/03 07/01/03 01/01/04 07/01/04 01/01/05 07/01/05
Euro/MWh
Fig. 4.1.Historical fuel switch prices for CCGT technology calculated with (2.3) and based on historical data
from McCloskey Index and NBP natural gas spot prices.
Since during the pre-Kyoto period 2005 – 2007 at least half of the entire EU fuel switch capacity
is located within the United Kingdom, we decided to base fuel switch prices on the McCloskey
North-West Europe Steam Coal Index and on natural gas prices from NBP (National Balancing
Point, which specifies delivery location within the UK). The continuous–time fuel switch price
process is modeled by
E(t) = P(t) + X(t)t∈[0,T] (4.3)
where the deterministic part
P(t) = a+bt +
2
X
j=0
cjcos(2πϕjt+lj)t∈[0,T] (4.4)
accounts for a linear price increase superimposed onto seasonal price fluctuations. The stochastic
part (X(t))t∈[0,T]is modeled by an Ornstein-Uhlenbeck process whose evolution follows the stochastic
differential equation
dX(t) = γ(α−X(t))dt +σdW (t) (4.5)
with parameters γ, α, σ ∈R. Here, (W(t))t∈[0,T]is a Brownian motion process. After performing
estimation (see the Appendix at the end of the paper) based on the historical data shown in Figure
4.1, the process (4.3) is identified with the following parameters:
stochastic part (X(t))t∈[0,T]
γ α σ
31.82 −0.12 68.24
(4.6)
deterministic part (P(t))t∈[0,T]
a b c0ϕ0l0c1ϕ1l1c2ϕ2l2
21.42 6.19 7.62 1 5.95 0.55 2 1.14 1.11 3 3.24
(4.7)
18 R. CARMONA AND M. FEHR AND J. HINZ
Expected Allowance Demand. The continuous-time counterpart of the expected allowance
demand is described by
Γ(t) := m+vW ′(t)t∈[0,T] (4.8)
where (W′(t))t∈[0,T]is a Brownian motion independent of (W(t))t∈[0,T]. In this context, the
parameters mand vare interpreted as the mean and the standard deviation of the final allowance
demand. In accordance with [4] we set m= 30m for the total required abatement which equals to
60 −65m tonnes of nominal emissions minus 20 to 25 m tonnes savings which are automatically
effected (being at very low costs). The parameter v= 20 m is chosen to reflect the possible
deviations of the nominal emissions caused by dry/wet year, since the required abatement may
change by 18 to 20m tons, due to changes in hydro electric capacity.
Numerical Implementation. As mentioned earlier, both (E(t))t∈[0,T]and (Γ(t))t∈[0,T]are
interpreted as continuous time counterparts of fuel switch prices and expected allowance demand
respectively. For numerical purposes we used a standard trinomial tree discretization of each
component of the two-dimensional diffusion process (E(t),Γ(t))t∈[0,T]to solve the corresponding
dynamic optimization problem (3.7) through a backward induction method. Figure 4.2 graphically
Fig. 4.2.Backward induction optimal control schematic, as applied to the trinomial tree discretization to each
component of the processes (E(t),Γ(t))t∈[0,T].
illustrates this technique. At each node, we see a splitting into three vertical and three horizontal
directions, giving nine branches in all. The vertical direction describes the movement of the
fuel switch price, whereas the horizontal branches model the expected demand dynamics. At
maturity, paths finish either at positive realizations of the allowance demand or at non-positive.
The optimally controlled fuel switch process is calculated by backward induction: At each node
the maximum principle is applied to decide either to apply the fuel switch or not. If the fuel switch
is performed, then the state is changed due to the effectively reduced allowance demand, indicated
by a move to the next lower tree in the forest diagram.
We now discuss the impact of parameters on carbon price. The following numerical illustration
is based on discrete-time model (Et)T−1
t=0 ,(Γt)T
t=0 which corresponds to the parameters (4.6), (4.7)
estimated in appendix. Here, the time horizon T= 253 stands for the number of working days in
2005. The fuel switch process is based on the deterministic component fitted to that year’s data,
and the starting point E0is set at the value of the deterministic component at the beginning of
January 2005.
Present Values. For commensurability reasons, we decided to show the dependence of the
OPTIMAL STOCHASTIC CONTROL AND CARBON PRICE FORMATION 19
allowance price A∗
ton allowance demand Γtin terms of the relative demand
δt=Γt−Pt
s=0 ξs
λ(T−t), t = 0,...,T −1
which stands for the percentage of time steps at which the fuel switch at full intensity is needed in
order to meet the compliance. (Note that we have to take into account carbon Pt
s=0 ξssaved by
the previous fuel switches (ξs)t
s=0 ) The dependence illustrated in the Figure 4.3 is obvious. The
price A∗
tis increasing in δtand Et. Moreover, for δt→+∞, the allowance price approaches the
0
5
10
15
20
25
30
35
40
-10
103050
-150
-100
-50
0
50
100
150
200
250
0
5
10
15
20
25
30
35
40
At2
*
Et2
δt2
At2
*
0
5
10
15
20
25
30
35
40
-10
103050
-150
-100
-50
0
50
100
150
200
250
0
5
10
15
20
25
30
35
40
At2
*
Et2
δt2
At2
*
0
5
10
15
20
25
30
35
40
-10
10305070
-150
-100
-50
0
50
100
150
200
250
0
5
10
15
20
25
30
35
40
At1
*
Et1
δt1
At1
*
0
5
10
15
20
25
30
35
40
-10
10305070
-150
-100
-50
0
50
100
150
200
250
0
5
10
15
20
25
30
35
40
At1
*
Et1
δt1
At1
*
Fig. 4.3.The dependence of allowance price on the present expected demand δtand Etfor different times
(right: t=t1beginning of March, left: t=t2beginning of September).
boundary πof 40 EURO, whereas for δt→ −∞ it tends to 0 . Further, A∗
tchanges significantly
with moderate deviations in δt, which is inline with the high correlation of allowance prices and
values of the Emission-To-Cap indicator observed by [1]. On the contrary, the impact of the present
fuel switch price Etis weak due to the distinct mean reversion. This poor correlation between
instantaneous fuelswitch price and allowance price is accurate also in reality as can be observed in
Figure 2.1. Despite the low degree of dependence of A∗
ton Et, we suppose though that fuel switch
price is a significant factor, whose impact is effected through the expected long term fuel switch
prices (to be deduced from fuel futures, whose price dynamics is not modeled here).
Dependence on Model Parameters. In accordance, the left picture in Figure 4.4 shows
high sensitivity of allowance price on α, which settles the level of expected long term fuel prices.
For this reason, we decided to visualize the effect of αby a plot of A∗
0against E{PT−1
t=0 Et}/T .
Moreover, this figure shows a weak dependence of the allowance price on σ, which we illustrate
by a plot of A∗
0against the stationary fuel switch price variance σ2/(2γ). The right picture in
the Figure 4.4 shows that the dependence of A∗
0with respect to changes in E{Γ}/(λT ) is higher
than in v. In other words, the dependence of allowance price on the need for emission reductions
is high whereas the uncertainty about necessitative emission reduction is of secondary importance.
Remark. Note that Γ(t) = m+vW (t) stands for the market expectation on allowance demand
occurring without short-term abatement. Thus, vis more related to market uncertainty about
future emissions than to the actual emissions dynamics.
Remark. We deliberately set the parameter intervals in Figure 4.4 larger than their physical
range in order to show boundaries where allowances price approaches its limits of 0 and 40 EURO.
For instance, we estimate that a realistic value for the expected percentage switching E(Γ)/(λT )
is between 0.4 and 0.9. Note that for this range, the price dependence on vis very weak (right
picture in Figure 4.4). This observation indicates that the ambiguity about vis not crucial and
that min (4.8) could be an appropriate parameter for the implicit calibration. Such a calibration
seems easily possible, since the allowance price is monotone in m.
20 R. CARMONA AND M. FEHR AND J. HINZ
Let us summarize our findings. The allowance price should be significantly correlated to the
expected long term fuel prices and to the expected need for emission reductions. These are the
main price drivers, since the remaining factors (recent fuel switch prices, their volatility, uncertainty
on the required emission reduction amount) have minor effects on the carbon price formation.
5
10
15
20
25
30
35
40
-10 10 30 50 70
10
30
50
70
90
0
5
10
15
20
25
30
35
40
A0
*
P
E( Es)/T
s
σ2/(2γ)
A0
*
5
10
15
20
25
30
35
40
-10 10 30 50 70
10
30
50
70
90
0
5
10
15
20
25
30
35
40
A0
*
P
E( Es)/T
s
σ2/(2γ)
A0
*
0
5
10
15
20
25
30
35
40
-40 0 40 80 120
10
20
30
40
50
0
5
10
15
20
25
30
35
40
A0
*
Γ0/(λT)
v
A0
*
0
5
10
15
20
25
30
35
40
-40 0 40 80 120
10
20
30
40
50
0
5
10
15
20
25
30
35
40
A0
*
Γ0/(λT)
v
A0
*
Fig. 4.4.On the left: the impact of αand σon allowance price, expressed for γ= 31.82 through long-term
fuel switch price mean E(PT−1
t=0 Et)/T and variance σ2/(2γ)respectively. On the right: the impact of vand
E(Γ)/(λT ).
Regulatory Controls. Designing a legally binding scheme, one of the main concerns of
regulatory authorities is on the one hand, to fulfill environmental targets (at least with a certain
probability) and, on the other hand, to achieve this emission reduction at the lowest possible
costs for the final consumer. Thus, we have studied the dependence of compliance probability and
allowance price on the penalty level and on the initially expected allowance demand (note that
this value is controlled by the total amount of allocated allowances). The diagrams in Figure 4.5
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
10 30 50 70 90
30
50
70
90
110
0
0.2
0.4
0.6
0.8
1
P(Γ>Π(ξ*))
Γ0/(λT)
π
P(Γ>Π(ξ*))
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
10 30 50 70 90
30
50
70
90
110
0
0.2
0.4
0.6
0.8
1
P(Γ>Π(ξ*))
Γ0/(λT)
π
P(Γ>Π(ξ*))
0
10
20
30
40
50
60
70
10 30 50 70 90
30
50
70
90
110
0
10
20
30
40
50
60
70
80
90
A0
*
Γ0/(λT)
π
A0
*
0
10
20
30
40
50
60
70
10 30 50 70 90
30
50
70
90
110
0
10
20
30
40
50
60
70
80
90
A0
*
Γ0/(λT)
π
A0
*
Fig. 4.5.The probability of non-compliance and the initial allowance price depending on penalty size and fuel
switch demand.
show the corresponding calculations. Again we show this influence in terms of the relative demand
δ0=E{Γ}/(λT ), which stands for the percentage of time steps at which the fuel switch at full
intensity is needed in order to meet the initially expected allowance demand. One concludes
that up to the relative demand of 50% the penalty can be increased without a notable increase
of the allowance price, giving, however a strong increase of the compliance probability. If the
relative demand is far above 50%, then the situation changes. The moderate increase of compliance
probability is reached only at the expense of a high allowance price. Note that the initial allowance
OPTIMAL STOCHASTIC CONTROL AND CARBON PRICE FORMATION 21
-40
-30
-20
-10
0
10
20
30
40
-20 -15 -10 -5 0 5 10 15 20
∆X
X
Fig. A.1.Scatter plot of (Y(t∆)(ω), X(t∆)(ω))n−1
t=1 calculated by (A.4) and based on historical fuel switch
prices from the Figure 4.1. The straight line depicts the estimated linear regression.
price is directly related to the consumers costs since EUAs are added to electricity prices as an
extra consumed commodity.
5. Conclusion. In this work, emission allowance price formation is investigated by equilibrium
methodology. The mathematical model reflects a generic cap and trade system, naturally generalized
to a fully stochastic framework, where both, short-term abatement prices and the total emission
volumes evolve randomly. Under mild assumptions, the existence of market equilibrium is proved.
We show that the calculation of allowance prices reduces to the solution of an optimal control
problem. By implementing a stylized model of the first phase of EU ETS, we demonstrate that
important calculations become tractable. We elaborate on the sensitivity with respect to model
parameters and initial data to identify main price drivers for the EUA. Since problems of this type
frequently arise in risk management, we hope that our findings are useful. Furthermore, we believe
that quantitative modeling in the spirit of our approach can help regulatory authorities to optimally
design market rules, when a new emission trading mechanism is being established. At this stage,
we have to address to future research the question of fair pricing and efficient hedging of allowance
derivatives. One pathway is through our structural modeling, which has to be accomplished by an
implicit model calibration. The other possibility is by means of a reduced–form approach, targeted
on direct modeling for the martingale process of allowance prices. Both directions entail interesting
challenges for future research which we hope to encourage by this contribution.
Appendix. Parameters of the Fuel Switch Price Process. Our estimation is based on a
series of n= 758 daily observations
(E(t∆)(ω))n
t=1
(where ∆ = 1/253 corresponds to one day), which are shown in Figure 4.1. The deterministic
harmonics (4.4) in the fuel switch price process are identified with parameters (4.7) obtained from
peaks in the Fourier transform. After removing the deterministic part (P(t·∆)(ω))n
t=1 (smooth
line in this figure) the residual component
X(t∆)(ω) = E(t∆)(ω)−P(t∆)(ω), t = 1,...,n (A.1)
is modeled as a realization of the Ornstein-Uhlenbeck process (4.5) whose parameters γ, α, σ are
estimated from the data (A.1) by a standard linear regression method applied as follows: From
22 R. CARMONA AND M. FEHR AND J. HINZ
the formulas for conditional mean and variance
E{X(t)|Fs}=X(s)e−γ(t−s)+α(1 −e−γ(t−s))s≤t(A.2)
Var{X(t)|Fs}=σ2
2γ(1 −e−2γ(t−s))s≤t(A.3)
we obtain the regression
Y(t∆) := X((t+ 1)∆) −X(t∆) = β0+β1X(t∆) + β2ǫtt= 1,...,n−1 (A.4)
where (ǫt)n−1
t=1 are independent, standard Gaussian random variables and β0, β1, β2are connected
to α, γ, σ by
α=−β0
β1
γ=−1
∆ln(1 + β1)
σ=r2γβ2
1−e−2γ∆
Figure A.1 shows a scatter plot of (Y(t∆)(ω), X (t∆)(ω))n−1
t=1 . Maximum likelihood parameter
estimation gave β0=−0.0147 , β1=−0.1182 , β2= 16.2708 from which we computed the original
parameters α, β, σ displayed in (4.6).
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