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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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