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MANAGEMENT SCIENCE

Vol. 57, No. 8, August 2011, pp. 1453–1468

issn 0025-1909 eissn 1526-5501 115708 1453 doi 10.1287/mnsc.1110.1358

© 2011 INFORMS

Risk-Neutral Models for Emission Allowance

Prices and Option Valuation

René Carmona

Bendheim Center for Finance, Program in Applied and Computational Mathematics, and Department of Operations Research and

Financial Engineering, Princeton University, Princeton, New Jersey 08544, rcarmona@princeton.edu

Juri Hinz

Department of Mathematics, National University of Singapore, Singapore 119076, Singapore, mathj@nus.edu.sg

The existence of mandatory emission trading schemes in Europe and the United States, and the increased

liquidity of trading on futures contracts on CO2emissions allowances, led naturally to the next step in

the development of these markets: These futures contracts are now used as underliers for a vibrant derivative

market. In this paper, we give a rigorous analysis of a simple risk-neutral reduced-form model for allowance

futures prices, demonstrate its calibration to historical data, and show how to price European call options

written on these contracts.

Key words: emission derivatives; emissions markets; cap-and-trade schemes; environmental ﬁnance

History : Received April 25, 2009; accepted March 7, 2011, by Haitao Li, guest editor, ﬁnance. Published online

in Articles in Advance June 20, 2011.

1. Introduction

Global warming and environmental problems con-

tinue to challenge policy makers. In part because of

the success of the U.S. Acid Rain Program, cap-and-

trade systems are now considered to be one of the

most promising market mechanism to reduce green-

house gas emissions on an international scale. The

core principle of such a mechanism is based on the

allocation of fully tradable credits among emission

sources and a penalty to be paid per unit of pollu-

tant that is not offset by a credit at the end of a pre-

determined period. The introduction of trading puts

a price on emissions and helps identify and exer-

cise the cheapest emission abatement measures. For

this reason, market-based mechanisms for emission

reduction are supposed to yield pollution control at

the lowest cost for the society. Notwithstanding the

rigorous equilibrium analyses from Carmona et al.

(2009, 2010) that conﬁrm that social optimality does

not necessarily mean that the scheme is cheap for con-

sumers, emission trading should be considered as a

cost-efﬁcient and effective tool.

By its very nature, the regulatory framework of a

mandatory cap-and-trade system involves its partici-

pants in a risky business, necessarily creating the need

for appropriate risk management. Trading of certiﬁ-

cates from a mandatory scheme is typically accom-

panied by an active secondary market where diverse

emission-related ﬁnancial derivatives (e.g., futures)

are traded. Options have been traded since 2006, and

as we explain below, on any given day, the volume

of European call and put options traded on forward

European Union Allowance (EUA) contracts ranges

between 15 and 25 million tons of CO2equivalent, the

short end of the forward curve being clearly the most

actively traded.

In this work, we propose reduced-form models for

the risk-neutral dynamics of allowance prices, pro-

viding a quantitative framework for pricing emission

derivatives.

Despite the large number of pieces in the popular

press and numerous speculative articles in magazines,

the scientiﬁc literature on cap-and-trade systems is

rather limited, especially if we restrict ourselves to

quantitative analysis of models including stochas-

tic factors. For the sake of completeness, we brieﬂy

review the literature relevant to our contribution.

The economic theory of allowance trading can be

traced back to Dales (1968) and Montgomery (1972),

authors who proposed a market model for the pub-

lic good environment described by tradable permits.

Dynamic allowance trading is addressed in Cronshaw

and Kruse (1996), Tietenberg (1985), Rubin (1996),

Leiby and Rubin (2001), Schennach (2000), Stevens

and Rose (2002), Maeda (2004), and in the literature

cited therein. Empirical evidence from existing markets is

discussed in Daskalakis et al. (2007) and Paolella and

Taschini (2008). This last paper suggests economic

implications and hints at several ways to model

spot and futures allowance prices, whose detailed

1453

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Carmona and Hinz: Risk-Neutral Models for Emission Allowance Prices and Option Valuation

1454 Management Science 57(8), pp. 1453–1468, © 2011 INFORMS

interrelations are investigated in Uhrig-Homburg and

Wagner (2008, 2009). There, the demand for derivative

instruments in emission markets is also addressed.

In Benz and Trueck (2009), characteristic proper-

ties for ﬁnancial time series are observed for prices

of emission allowances from the mandatory Euro-

pean Union Emission Trading Scheme (EU ETS) and

several Markov and AR/GARCH–like models are

proposed. Paolella and Taschini (2008) also consider

tail behavior and the heteroscedasticity of the returns.

Dynamic price equilibrium and optimal market design are

investigated in Carmona et al. (2009), which provides

a mathematical analysis of market equilibriums and

uses optimal stochastic control to show social opti-

mality. Based on this approach, Carmona et al. (2010)

discuss price formation for goods whose production

is affected by emission regulations. In this setting,

an equilibrium analysis conﬁrms the existence of the

so-called windfall proﬁts (see Sijm et al. 2006) and

provides quantitative tools to analyze alternative mar-

ket designs, which are applied in Carmona et al.

(2008) to optimize a cap-and-trade mechanism for a

proposed Japanese emission trading scheme. Seifert

et al. (2008) and Wagner (2006) also deal with risk-

neutral allowance price formation within the EU ETS.

Using equilibrium properties, the price evolution is

treated in terms of marginal abatement costs and opti-

mal stochastic control. Cetin and Verschuere (2010)

use hidden Markov models and a ﬁltering approach

to capture the impact of news releases. Option pric-

ing within the EU ETS was considered only recently.

Chesney and Taschini (2008) rely on endogenous

emission permit price dynamics within an equilib-

rium setting to value European options on emission

allowances.

The present paper is organized as follows. After

an introductory discussion of the various approaches

to risk-neutral modeling in §2, we present a general

approach to modeling of an emission market with one

compliance period. More realistic multiperiod mod-

els are treated in §6. The mathematical treatment of

§3 is based on the analysis of diffusion martingales

ending with only two possible values. We identify

explicit classes of such martingales, and we show

how simple deterministic time changes can provide

families of versatile risk-neutral models for allowance

prices. In §4, we demonstrate how to calibrate one of

these models to historical allowance price data. We

develop a historical calibration procedure, because the

option market has not yet matured to a point we can

trust more standard calibration procedures based on

option price data. The second part of the paper gen-

eralizes the one-period setup to more realistic mul-

tiperiods models incorporating important features of

real-world markets, and §6 provides the necessary

Table 1 ECX EUA Option Quotes (in Euros) on January 4, 2008

Option Option Allowance Implied Settlement

maturity type Volume Strike price vol. (%) price

Dec-08 Cal 2001000 22000 23055 51025 5006

Dec-08 Call 1501000 26000 23055 51025 3057

Dec-08 Call 4501000 27000 23055 51025 3027

Dec-08 Call 1001000 28000 23055 51025 2099

Dec-08 Call 1251000 29000 23055 51025 2074

Dec-08 Call 5251000 30000 23055 51025 2051

Dec-08 Call 2501000 40000 23055 51025 1004

Dec-08 Call 7001000 50000 23055 51025 0045

Dec-08 Put 110001000 14000 23055 51025 0064

Dec-08 Put 2001000 15000 23055 51025 0086

Dec-08 Put 2001000 15000 23055 51025 0086

Dec-08 Put 4001000 16000 23055 51025 1013

Dec-08 Put 1001000 17000 23055 51025 1043

Dec-08 Put 110001000 18000 23055 51025 1078

Dec-08 Put 5001000 20000 23055 51025 2060

Dec-08 Put 2001000 21000 23055 51025 3007

Dec-08 Put 2001000 22000 23055 51025 3057

Note. ECX, European Climate Exchange.

changes needed to extend the pricing formula to this

more general setup.

As a motivation for our derivations of option pric-

ing formulas, we close this introduction with a short

discussion of the idiosyncrasies of the EUA option

markets. The facts reported below were a determin-

ing factor in our decision to write the present paper.

European call and put options are actively traded on

EUA futures contracts. Since 2006, trades of options

maturing in December of each year (prior to 2012)

have produced a term structure of option prices. On

any given day, the volume varies from 5 to 25 mil-

lion tons of CO2equivalent, the short end of the curve

being the most active with a good number of ﬁnancial

institutions involved, whereas the long end depends

mostly on a few energy companies. It is not clear

how these options are priced, and a persistent rumor

claims that traders plainly use Black’s formula. The

data reproduced in Table 1 is an extract of quotes

published on January 4, 2008. Obviously, the implied

volatility is perfectly ﬂat, and the absence of skew

or smile is consistent with the rumor. Whether or

not traders are using Black or Black–Scholes formulas

to price options on EAUs and futures contracts, and

given what we learned from equilibrium models, we

think that it is important to derive option price formu-

las based on underlying martingales with binary ter-

minal value, because Black–Scholes formula is based

on an underlying price martingale converging to zero!

2. Risk-Neutral Modeling of

Emission Markets

To position our contribution within the existing lit-

erature, we brieﬂy review the different methodolo-

gies of quantitative ﬁnancial modeling. Whereas the

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econometric approach aims to give a description of

statistical aspects in price movements, possibly ignor-

ing the underlying economic underpinnings, the equi-

librium approach focuses on the mechanics of price

formation: Given incentives, strategies, uncertainty,

and risk aversions, the market is described by the

cumulative effect of individual actions. Investigations

are geared toward understanding market features

appearing in steady realistic state, i.e., market equi-

librium. Finally, the risk-neutral approach adopted in

this paper abstracts from the mechanism driving the

market to the equilibrium state and focuses on its

basic properties. Starting from the absence of arbi-

trage, the asset price evolution is introduced directly.

This approach arose from the idea that although price

movement is stochastic in nature, derivative valua-

tion does not refer to the real-world probability. The

description of statistical issues is not a goal of risk-

neutral models.

Absence of arbitrage is central to the risk-neutral

approach. Moreover, we want to the important

insights gained from equilibrium theory (Carmona

et al. 2009, 2010) which we summarize in the follow-

ing three points:

(a) There is no arbitrage from trading allowances.

(b) There are merely two ﬁnal outcomes for the

price of an allowance. Either the terminal allowance

price drops to zero or it approaches the penalty

level . Indeed, the price must vanish at maturity if

there is excess of allowances, whereas in the case of

shortage, the price will raise, reaching the level of the

penalty. It is reasonable to suppose that in reality, the

demand for allowances will coincidence with the sup-

ply with zero probability. So the occurrence of such

an event will be disregarded.

(c) Allowance trading instantaneously triggers all

abatement measures whose costs are below the

allowance price because if an agent owns a tech-

nology with lower reduction costs than the present

allowance price, then it is optimal to immediately use

it to reduce his or her own pollution and proﬁt from

selling allowances.

At this point, we distinguish between two types of

risk-neutral approaches:

• the reduced-form risk-neutral approach, which

focuses on (a) and (b);

• the detailed risk-neutral approach, which aims at

all three properties (a), (b), and (c).

Let us explain, at least at a formal level, the main

differences between these two classes of models. For

the sake of concreteness, we focus on a continuous-

time framework in which the risk-neutral evolution

4At5t∈601 T 7 of a futures contract on an allowance with

compliance date Tfor a penalty ∈4015is being

modeled. In this framework, the allowance price pro-

cess 4At5t∈601 T 7 is realized a ﬁltered probability space

4ì1 F11 4Ft5t∈601 T 7 5, equipped with a distinct mea-

sure ∼, which is interpreted as the spot martingale

measure.

The reduced-form risk-neutral approach focuses on

the following problem:

Model the noncompliance event N∈FT,

which deﬁnes the -martingale 4At5t∈601 T 7

with terminal value AT=1N.

(1)

The noncompliance event Nis the only object that

needs to be described exogenously. However, to

obtain a useful model, several requirements, ranging

from computational tractability in derivatives valua-

tion to diverse aspects of calibration, must be fulﬁlled.

Within the detailed risk-neutral approach, the non-

compliance event Nis obtained endogenously, in

terms of other quantities that in turn must be spec-

iﬁed exogenously. This is where the issue (c) comes

into play. Under natural equilibrium assumptions (see

Carmona et al. 2009), the abatement activity in the

market is driven by the allowance price in the follow-

ing way: At any time t, given the allowance price At,

the market exercises exactly those abatement mea-

sures whose costs are less than or equal to the value

of At. This is also known as the equilibrium allowance

price equals to the marginal abatement costs in envi-

ronmental economics. Hence in equilibrium, the total

abatement in the market can be described in terms of

allowance prices 4As5s∈601 T 7 as

ZT

0

cs4As5 ds1

where cs4a545 stands for the total intensity of the

abatement measures at time s∈601 T 7 available in the

market at price less than or equal to a∈6015in

the market scenario ∈ì. In this context, the abate-

ment volume function ct2 601 7 ×ì→6015,t∈

601 T 7 must be speciﬁed exogenously. In practice, the

abatement volume function can be be estimated from

market data: Given a risk-neutral fuel price model,

ctcan be described by an appropriate B601 7 ⊗Ft-

measurable functions for each t∈601 T 7. In this con-

text, the noncompliance event is given by

∈ì3 ET45 −ZT

0

cs4As545ds ≥01

where an exogenously speciﬁed FT—measurable

allowance demand ET—is used for the number

of excess pollution units in the business-as-usual

scenario (i.e., given zero penalty). Thus, the detailed

risk-neutral approach leads to a more complex math-

ematical problem:

Determine 4ct5t∈601 T 7 from market data

and model allowance demand ETto

obtain a -martingale 4At5t∈601 T 7 with the

terminal value AT=18ET−RT

0ct4At5 dt≥09.

(2)

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Although the detailed risk-neutral approach seems

appealing from a methodological perspective, it is

not obvious whether its higher complexity is justi-

ﬁed from the viewpoint of derivative valuation. The

present authors believe and show in this work that

the reduced-form risk-neutral approach yields satisfac-

tory results, at least in the area of pricing plain-vanilla

European options written on allowance futures. As

illustration we investigate the solution of a particular

problem of the type (2) and compare its results to a

class of solutions to (1) suggested in the present work.

The existence and uniqueness of 4At5t∈601 T 7 solv-

ing (2) requires a delicate discussion. The martin-

gale Et=ƐQ4ETFt5deﬁned for t∈601 T 7 plays an

essential role. Indeed, the analysis of the discrete-

time framework shows that if the future incre-

ments of 4Es5s∈6t1T 7 are independent of the present

information Ftat any time t, then a solution to

(2) should be expected in the functional form At2=

4t1 Gt5with an appropriate deterministic function

2 601 T 7 ×34t1 g 5 7→ 4t1 g 5 ∈and a state process

4Gt5t∈601 T 7 given by

Gt=Et−Zt

0

cs4As5 ds1 t ∈601 T 70

This insight helps guess a solution in the standard

diffusion framework, when there exists a process

4Wt1Ft5t∈601 T 7 of Brownian motion with respect to

∼, in the simplest case dEt=dWtwith pre-

speciﬁed ∈4015and continuous, nondecreasing,

and deterministic abatement function c2 4015→.

Under these conditions, Itô’s formula, applied to the

martingale At=4t1 Gt5, leads, to a nonlinear partial

differential equation for on 601 T 5 ×:

¡t4t1 g 5 −¡g4t1 g 5c44t1 g 55 +1

2¡2

gg 4t1 g 52=01(3)

subject to the boundary condition

4T 1 g 5 =160154g5 for all g∈1(4)

justiﬁed by the digital nature of the terminal

allowance price. Having obtained in this way, one

constructs the state process 4Gt5t∈601 T 7 as a solution of

the stochastic differential equation

dGt=dEt−c44t 1 Gt55 dt1 G0=E01(5)

from which we get a solution to (2) from At=4t1 Gt5.

Once 4At5t∈601 T 7 is determined, one applies standard

integration to value European options. Although

closed-form expressions are rare, option prices can

be calculated numerically. The only case that yields

quasi-explicit expressions (involving only numerical

integrations) is that of linear abatement functions (see

Seifert et al. 2008, Wagner 2006).

Let us elaborate on this case, to give the reader a

feeling of allowance option pricing in the framework

of detailed risk-neutral modeling.

Figure 1 Price of a European Call Option as a Function of Its Maturity

0.0 0.5 1.0 1.5 2.0

15

10

5

Option’s price

Option’s maturity

Example 1. Set the time to compliance date Tto

two years and assume the diffusion coefﬁcient is

4 and the penalty is 100, and suppose that the

abatement function c=ctfor t∈601 T 7 is linear with

c2 a 7→0002 ·a. At time t=0, we consider a family of

European calls with the same strike price of K=25

but different maturity times running through 601 T 7.

Suppose that the initial allowance price equals to the

strike price a=A0=25. Determine call prices C04 5

at t=0 for different maturity times ∈601 T 7. Inde-

pendently of the model, the price of the call in front

of expiry date =0 must be equal to zero C0405=0,

whereas the longest maturity call =Tmust have a

price C04T 5 =A04 −K5/ =25 ·0075 =18075. Because

of the digital terminal value of the underlying, such a

call is equivalent to 0075 allowances. Call prices must

increase with call’s maturity from 0 to 18075. This

must be true within any risk-neutral model. Figure 1

illustrates the exact curve 4C0455∈601T 7 for the param-

eters as above. Because the end points are model-

independent, merely intermediate-maturity prices

exhibit model-dependent properties. Here we observe

one remarkable issue: the so-called inverse S-shape.

In this work, we show that these features are shared

by signiﬁcantly simpler option pricing schemes (see

Figure 4) based on the reduced-form approach. For this

reason, we believe that the reduced-form approach

can provide a reasonable pricing mechanism for

emission-related ﬁnancial products. However, we also

agree that further development of detailed risk-

neutral, econometric, and equilibrium modeling is

needed to help understand allowance price evolu-

tion. Certainly, such models could be better suited to

address the impact of information asymmetry, jumps

in the information ﬂow, regulatory uncertainty, and

market idiosyncracies.

3. Reduced-Form Model for a Single

Compliance Period

In this section, we introduce a simple model for an

abstract emission market. We ﬁrst restrict ourselves

to a single compliance period, say 601 T 7. The more

realistic case of multiperiod models is treated in §6.

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Carmona and Hinz: Risk-Neutral Models for Emission Allowance Prices and Option Valuation

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In the one-period setting, credits are allocated at

the beginning of the period to enable allowance trad-

ing until time Tand to encourage agents to exer-

cise efﬁcient abatement strategies. At the compliance

date T, market participants cover their emissions by

redeeming allowances or pay a penalty per unit

of pollution not offset by credits. In this one-period

model, unused allowances expire and are worthless

because we do not allow for banking into the next

period. Under natural assumptions, equilibrium anal-

ysis shows that the allowance price ATat compliance

date Tis a random variable taking only the values

0 and (see Carmona et al. 2009, 2010). More pre-

cisely, if the market remains under the target pollution

level, then the price approaches zero. Otherwise, the

allowance price tends to the penalty level .

All the relevant asset price evolutions are assumed

to be given by adapted stochastic processes on a ﬁl-

tered probability space 4ì1 F11 4Ft5t∈601 T 75on which

we ﬁx an equivalent probability measure ∼,

which we call the spot martingale measure.

We denote by 4At5t∈601 T 7 the price process of a

future contract with maturity date Twritten on the

allowance price. Given the digital nature of the termi-

nal allowance price AT, the central object of our study

is the event N⊂FTof noncompliance, which set-

tles the 801 9-dichotomy of the terminal futures price

by AT=1N. Furthermore, a standard no-arbitrage

argument shows that the futures price 4At5t∈601T 7 needs

to be a martingale for the spot martingale mea-

sure . Hence, the problem of allowance price mod-

eling reduces to the appropriate choice of a model for

the martingale

At=Ɛ41NFt51 t ∈601 T 70

There are many candidates for such a process, but no

obvious choice seems to be versatile enough for the

practical requirements described below. An important

requirement is the need to match the observed volatil-

ity structure. For a practitioner trying to calibrate at

time ∈601 T 7 a model for the martingale 4As5s∈6 1 T 7

that ﬁnishes at 0 or , the minimum requirements

are to match the price observed at time as well

as the observed price ﬂuctuation intensity up to this

time . Further model requirements include the exis-

tence of closed-form formulas, or at least fast valua-

tion schemes for European options; a small number of

parameters providing sufﬁcient model ﬂexibility; and

reliable and fast parameter identiﬁcation from data.

The goal of this paper is to present and analyze sim-

ple models satisfying these requirements.

In accordance with our earlier discussion of the

two reduced-form approaches, we choose our starting

point to be the noncompliance event N∈FT, which

we describe as the event where a hypothetic positive-

valued random variable âTexceeds the boundary 1,

say N=8âT≥19. If one denotes by ETthe total pol-

lution within the period 601 T 7 that must be balanced

against the total number ∈4015of credits issued

by the regulator, then the event of noncompliance

should be given by N=8ET≥9, which suggests that

âTshould be viewed as the normalized total emission

ET/. However, in our modeling, we merely describe

the noncompliance event. Strictly speaking, so any

random variable âTwith

8âT≥19=8ET/ ≥19

would do as well. On this account, we do not claim

that âTrepresents the total normalized emission ET/.

So the allowance spot price is given by the martingale

At=Ɛ18âT≥19Ft1 t ∈601 T 71

where the random variable âTis chosen from a

suitable parameterized family of random variables

8â

T2 ∈ä9. For reasons of model tractability, we sup-

pose that the ﬁltered probability space supports a pro-

cess 4Wt5t∈601 T 7 of Brownian motion with respect to the

spot martingale measure , and we investigate para-

metric families which give allowance prices

A

t=Ɛ18â

T≥19Ft1 t ∈601 T 7

with a Markovian stochastic evolution of the form

dA

t=v4t1 A

t5 dWt1

where the diffusion term vcaptures the basic prop-

erties of historical price observations. In particular,

we will match exactly the observed initial allowance

price and the initial instantaneous price ﬂuctuation

intensity.

Remark 1. We propose a consistent pricing scheme

for emission-related ﬁnancial instruments within the

framework of diffusion processes. Although this rules

out discontinuity in allowance prices, we believe

that this approach is reasonable. It has been argued

that because of jumps in the information ﬂow, sud-

den allowance price changes must be included (see,

for example, Cetin and Verschuere 2010). However,

based on our experience in the energy sector, pos-

sible allowance price jumps are not likely to play

a signiﬁcant role in mature emission markets. An

increasing number of consultancies and market ana-

lysts are carefully watching the European emission

market. Several agencies are providing news and peri-

odical publications. Moreover, because energy gener-

ation and consumption are publicity observable, one

should not expect signiﬁcant allowance price jumps

in a mature emissions market. On this account, a risk-

neutral model based on continuous allowance price

evolution is reasonable.

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Carmona and Hinz: Risk-Neutral Models for Emission Allowance Prices and Option Valuation

1458 Management Science 57(8), pp. 1453–1468, © 2011 INFORMS

To simplify the notation, we consider the normal-

ized futures price process

at2=1

At=Ɛ18âT≥19Ft1 t ∈601 T 71

and we describe it under special assumptions on âT.

Our goal is to identify classes of martingales 8at9t∈601 T 7

taking values in the interval 40115and satisfying

lim

t%Tat∈80119=10(6)

We ﬁrst identify a parametric family of such martin-

gales by working backward from a simple model for

the random variable âT, motivated by intuitive under-

standing of the ﬁnal cumulative level of emissions.

3.1. Basic Modeling of the Compliance Event

We use the standard notation N 41 25for the normal

distribution with mean and variance 2, and we

write êfor the cumulative distribution function of

the standard normal distribution.

Proposition 1. Suppose that

âT=â0eRT

0sdWs−41/25RT

02

sds1 â0∈4015(7)

for some continuous and square-integrable deterministic

function 401 T 5 3t ,→t. Then the martingale

at=Ɛ18âT≥19Ft1 t ∈601 T 7 (8)

is given by

at=ê ê−14a05qRT

02

sds +Rt

0sdWs

qRT

t2

sds !1(9)

and it solves the stochastic differential equation

dat=ê04ê−14at55√ztdWt1(10)

where the positive-valued function 401 T 5 3t ,→ztis

given by

zt=2

t

RT

t2

udu1 t ∈401 T 50 (11)

Remark 2. Notice that even though the distribu-

tion of âTdepends only upon RT

02

sds, the noncompli-

ance event Ndepends upon the entire function 8s9s.

Proof. A direct calculation shows

at=Ɛ18âT≥19Ft=8âT≥1Ft9

=8âteRT

tsdWs−41/25RT

t2

sds ≥1Ft9

=ê ln ât−41/25RT

t2

sds

qRT

t2

sds !

=ê ln4â05−41/25RT

02

sds

qRT

02

sds qRT

02

sds

qRT

t2

sds +Rt

0sdWs

qRT

t2

sds!1

and taking into account the initial condition

a0=ê ln â0−41/25RT

02

sds

qRT

02

sds !1

we obtain the desired expression (9). To show (10), we

start with at=ê4t5,t∈601 T 7, where

t=01 T +Rt

0sdWs

qRT

t2

sds

for t∈601 T 71

with 01 T =ln â0−1

2ZT

0

2

sds (12)

and 0=ê−14a05with deterministic a0∈40115. Com-

puting its Itô differential, we get

dt=ZT

t

2

sds−1/2

tdWt

+1

2x0+Zt

0

sdWsZT

t

2

sds−3/2

2

tdt

=ZT

t

2

sds−1/2

tdWt+1

2tZT

t

2

sds−1

2

tdt

=√ztdWt+1

2zttdt

d67t=ztdt0

Next, Itô’s formula gives the differential of the nor-

malized allowance prices as

dat=ê04t5dt+1

2ê004t5d67t

=ê04t5√ztdWt+1

2zttdt+1

2ê004t5ztdt

=ê04ê−14at55√ztdWt

because xê04x 5 +ê004x5 ≡0.

We notice for later use that if t < ,is given

explicitly as a function of tby

=e41/25R

tzsdst+Z

t

e41/25R

szudu√zsdWs0(13)

3.2. Construction via Time Change

The stochastic differential Equation (10) can be inter-

preted in the following way. Because of the factor √zt

in front of dWt,atcan be viewed as the time-change

of a martingale 8Yt9t∈6015given by the strong solution

of the stochastic differential equation

dYt=ê04ê−14Yt55 dWt1(14)

for t∈6015, with Y0∈40115. This solution stays in

the open interval 40115and converges to the bound-

aries 0 or 1 with certainty when tapproaches :

lim

t% Yt∈80119=10(15)

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This construction is in fact a special case of a gen-

eral program where the martingale 8at9t∈601T 7 satisfy-

ing (6) is constructed in two steps: First determine a

40115-valued martingale 8Yt9t∈6015satisfying (15) and

then search for a time change bringing the half-axis

6015onto the bounded interval 601 T 5. With this in

mind, it appears natural to consider the solutions of

the stochastic differential equation

dYt=ä4Yt5 d Wt1 Y0∈401151 t ≥01(16)

where äis a nonnegative continuous function on

60117satisfying ä405=ä415=0. We can then use

Feller’s classiﬁcation (see, for example, Karatzas and

Shreve 1991, Karlin and Taylor 1981) to check that

such a diffusion is conservative, does not reach the

boundaries 0 and 1 in ﬁnite time, and satisﬁes (15).

This is the case if v40+5=v41−5= , where v4x5 is

deﬁned by

v4x5 =2Zx

005

4x −y5 dy

ä4y51 x ∈401150

Straightforward computations show that the solution

of the stochastic differential Equation (14) does indeed

satisfy these conditions; hence it does not hit 0 and 1

in ﬁnite time with probability one.

Explicit families of such martingales can easily be

constructed. Case in point, a two-parameter family

of examples can be constructed from the analysis of

Carmona et al. (2001), which we learned from Mike

Terhanchi (who extended the argument of Carmona

et al. 2001 to Lévy processes in Tehranchi 2009). If

we set

Xt=e−Wt+ct X0−Zt

0

eWs−cs 4a ds +dBs51 X0∈

for c > 0 and a∈, where 8Wt9t∈6015and 8Bt9t∈6015

are independent Wiener processes, then 8Xt9t∈6015

satisﬁes

dXt=c+1

2Xt−adt −XtdWt−dBt1 t ∈6015

and

lim

t→ Xt= − on the setZ

0

eWs−cs 4a ds +dBs5 > X01

lim

t→ Xt= + on the setZ

0

eWs−cs 4a ds +dBs5 < X00

Now if we deﬁne the function Gby G4x5 =Rx

− g4y 5 dy

for all x∈, where the function gis

g4y5 =Ce2atan−1y

41+y25c+1/21 y ∈1

with the constant C > 0 chosen so that R+

− g4y 5 dy =1,

then it is easy to check that

1

2g04y541+y25+g4y5c+1

2y−a=01

which in turn implies that Yt=G4Xt5is a martingale.

Clearly, this martingale satisﬁes the limits (15). More-

over, a simple application of Itô’s formula shows that

8Yt9tis a solution of the stochastic differential Equa-

tion (16) with ä4y5 =g4G−14y55p1+G−14y52.

It is now plain to see that the basic model of

Proposition 1 is a particular case of this construction.

Indeed, if 4Yt=ê4Xt55t∈6016for

Xt=et/2x0+Zt

0

e−s/2dWs1 x0∈1for all t∈60150

and 4zs5s∈601T 5 is a positive-valued, continuous func-

tion, then the 40115-valued process

at=YRt

0zsds1 t ∈601 T 51

satisﬁes

dat=ê04ê−14at55√ztd˜

Wt1 t ∈601 T 5

for the process 4˜

Wt5t∈601T 5 deﬁned by

d˜

Wt=1

√zt

dMt1Mt=WRt

0zsdst∈601 T 5 1

which follows a Brownian motion with respect to the

time changed ﬁltration

˜

Ft=FRt

0zsds1 t ∈601 T 50

Moreover,

lim

t→TZt

0

zsds =+ implies that lim

t%Tat∈80119=10

4. Model Parametrization and

Calibration

We now show how to calibrate the basic model

introduced in §3.1. As explained above, historical

option prices are not appropriate for calibration in

the present state of the market, and we limit our-

selves to historical calibration of the model. Note that

according to Proposiotion 1, the choice of the function

401 T 5 3t ,→saffects only the time-change 4zt5t∈401 T 5 .

Moreover, Proposition 1 shows that when modeling

the random variable âTby (7), we must assume that

the function 401 T 5 3t ,→sis not constant. Indeed,

a constant volatility

s≡ ¯∈4015for all s∈601 T 7

would give, independently on the choice of ¯, the

same process

at=êê−14a05√T+Wt

√T−t(17)

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with dynamics

dat=ê04ê−14at55 1

√T−tdWt0(18)

Thus, with a constant and deterministic ¯it is impos-

sible to match both observed allowance prices and

their observed (instantaneous) ﬂuctuation intensity.

Indeed, the entire process is completely determined

by the value of a0. This suggests that we introduce

extra degrees of freedom in (18). In this paper, we

choose to work with the model

dat=ê04ê−14at55p4T −t5−dWt1(19)

parameterized by ∈and ∈4015. This leads to

a parametric family of functions 4s5s∈601 T 7 , which we

denote by

4s41 55s∈401 T 5 1 ≥11 > 01(20)

and we show how to calibrate the parameterized fam-

ily (20) to historical data.

As seen from (7), the function 401 T 5 3s ,→senters

the dynamics of 4at5t∈601 T 7 indirectly through the time-

change function 401 T 5 3t ,→ztdeﬁned in (11). The

correspondence between the functions and zis elu-

cidated in the following lemma.

Lemma 1. (a) Given any square-integrable continu-

ous and positive function 401 T 5 3s ,→s, the function

401 T 5 3t ,→ztdeﬁned by

zt=2

t

RT

t2

udu1 t ∈401 T 51 (21)

is positive and continuous and satisﬁes

lim

t%TZt

0

zudu =+0(22)

(b) Conversely, if the positive and continuous function

401 T 5 3t ,→ztsatisﬁes (22) then the function 401 T 5 3

s ,→sdeﬁned by

t=qzte−Rt

0zudu1 t ∈401 T 51

is positive and continuous and satisﬁes (21).

Proof. (a) Let us write (21) as ztt=2

tfor t∈

401 T 5, where

t=ZT

t

2

udu for all t∈601 T 70

Then, ˙t=−2

tfor t∈401 T 5 and satisﬁes the differ-

ential equation ztt= − ˙tfor t∈401T 5. Its solution is

given by

t=0e−Rt

0zudu1 t ∈601 T 50

From the terminal condition T=RT

T2

udu =0, we

get (22).

(b) Let us now suppose that 4zt5t∈401 T 5 is positive

and continuous and satisﬁes (22), and let us deﬁne

the positive and continuous function 4t5t∈601 T 5 by

t=e−Rt

0zudu1 t ∈601 T 50 (23)

Clearly, it satisﬁes ˙t= −zttfor t∈401 T 5, and

because the divergence of the integral implies that

T=0, we have

t=− ZT

t˙udu1 t ∈401 T 5 and −zt= − ˙t

RT

t˙udu0

(24)

Setting 2

t= − ˙tfor t∈401 T 5, (21) is satisﬁed. More-

over, this function is positive, continuous in the open

interval, and integrable because 1 =405=−RT

0˙udu,

which follows from (23) and (24). Consequently, the

function deﬁned by t2=p2

tfor t∈401 T 5 is square

integrable, continuous and positive and is related to

4zt5t∈401 T 5 by (21), as required.

We return to the expression (7) for âT, using now

the targeted family (20) to determine the stochas-

tic differential Equation (19). In light of the previous

lemma, the function

4zt41 5 =4T −t5−5t∈401 T 5 (25)

must satisfy (22), implying the following restrictions

on the parameters and :

> 0 and ≥10(26)

However, we will let vary freely over for calibra-

tion purposes, interpreting the ﬁtted values in light of

these conditions.

Remark 3. If we use the parametric family

zt41 5 =4T −t5−, then the actual time change is

given by the integral

Zt

0

zsds =

4log4T 5 −ln4T −t55 if =11

1−6T 1−−4T −t51−7otherwise0

Notice that is a multiplicative parameter in the

sense that zt41 5 =zt41 15. Also, the emission

volatility 4t41 55t∈401 T 5 associated to the parameteri-

zation 4zt41 55t∈601 T 7 is given by

t41 52

=zt41 5e−Rt

0zu415 du (27)

=

4T −t5−e−4/41−556 T 1−−4T −t51−7if 6=11

4T −t5−1T−if =10

(28)

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4.1. Historical Calibration

Consider historical observations of the futures prices

4At5t∈601T 7, recorded at times t1< t2<1 ···1 < tn0result-

ing in a data set 110001n, where

ti=ê−14ati5=ê−11

Ati1 i =11···1 n (29)

The objective measure governing the statistics of

the observations can be recovered from the spot mar-

tingale measure via its Radon–Nikodym density

d

d=eRT

0HtdWt−41/25RT

0H2

tdt0

For the sake of simplicity, we follow the time-honored

approach assuming that the market price of risk pro-

cess 4Ht5t∈601 T 7 is constant and deterministic, Ht≡h

for t∈601 T 7, for some ﬁxed h∈. According to

Girsanov’s theorem, the process 8˜

Wt9t∈601 T 5 deﬁned as

˜

Wt=Wt−ht for t∈601 T 5 is a Brownian motion with

respect to the objective measure ; under this mea-

sure ,tsatisﬁes

dt=1

2ztt+h√ztdt +√ztd˜

Wt

and the analog of (13) reads

=e41/25R

tzsdst+hZ

t

e41/25R

szudu√zsds

+Z

t

e41/25R

szudu√zsdWs(30)

for 0 ≤t≤≤T. Consequently, for each i=

110001n, the conditional distribution of tigiven ti−1

is Gaussian with mean iand variance 2

igiven by

i4h115=e41/25Rti

ti−1zsdsti−1+hZti

ti−1

e1/2Rti

szudu√zsds1 (31)

2

i4h1 1 5 =Zti

ti−1

zseRti

szudu ds1 (32)

Figure 2 History of Futures Prices on EUA with Maturity 2012

35

30

25

20

15

10

Time in months

Price

01/07 04/07 06/07 08/07 10/07 12/07 02/08 05/08 07/08 09/08 11/08 01/09 04/09 06/09 08/09 10/09 12/09

provided we ﬁx t0and 0by convention. So for a

given realization 8ti9n

i=1∈n, the log-likelihood is

L110001n4h115

=

n

X

i=1−4ti−i4h11552

22

i415 −log q22

i415(33)

for all h1 1 ∈. Although there is no closed-form

estimate for this parameter, the maximum of the like-

lihood function can be determined numerically. Let us

illustrate this procedure.

Consider historical daily prices of a futures contract

written on the EUA with maturity in December 2012,

which are depicted in Figure 2. Based on these data,

the function (33) is calculated numerically, using the

following integral approximations:

i4h1 1 5 ∼e41/254ti−ti−15zti−1ti−1

+h4ti−ti−15pzti−1e41/254ti−ti−15zti−11(34)

2

i4h1 1 5 ∼4ti−ti−15zti−1e4ti−ti−15zti−10(35)

Starting with initial parameter h 2=0, =005, =005,

a numerical method for unconstrained maximization

returned the maximizer

∗=003321 ∗=001611 h∗=−000780(36)

To conﬁrm the validity of our procedure, we deter-

mine the residuals

wi=ti−i4h1 1 5

p2

i4h1 1 5 1 i =110001n0 (37)

Under the model assumptions, this series must be a

realization of independent standard normal random

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Figure 3 Statistical Analysis of Series 4wi5n

i=1

25 321

0

1

2

3

0–1

–1

–2–3

–3

2015

Lag Theoretical quantiles

Time in months

01/07

02/07

04/07

05/07

06/07

07/07

08/07

09/07

10/07

11/07

12/07

01/08

02/08

04/08

05/08

06/08

07/08

08/08

09/08

10/08

Sample quantiles

0

1

2

3

–1

–3

ACF W

1050

0.0

0.2

0.4

0.6

0.8

1.0

variables. Thus, standard statistical analysis of these

residuals can be applied to verify the the quality of

the model ﬁt. In Figure 3 we show this series, its

empirical autocorrelation function (ACF), and its QQ-

plot. The empirical properties of the residuals are not

completely in line with model assumptions, but the

reader should keep in mind that risk-neutral models

are not designed to capture all the statistical particu-

larities of the underlying ﬁnancial time series. In fact,

the target of risk-neutral modeling is to describe the

price evolution with respect to the risk premium cor-

rected pricing measure.

Remark 4. The original rationale for the choice of

our basic model was based on equilibrium considera-

tions and the assumption that the terminal allowance

price was binary. However, the real market EU ETS

(second phase), whose data are used for historical cal-

ibration, operates under uncertainty. One of the major

price determinant here is the unknown impact of the

international credits, the so-called Certiﬁed Emission

Reductions, or CERs. Most likely, market participants

believe that a signiﬁcant number of cheap interna-

tional credits will be used to fulﬁll the compliance

within the EU ETS if needed, and that noncompliance

because of a shortage of certiﬁcates will not occur

at compliance time. Under such condition, the dis-

tribution of the terminal allowance price should not

be binary any more. Namely, in the case of national

allowance shortage, it would reach a level deter-

mined by supply and demand for international cred-

its, which is likely to fall below the EU ETS penalty

of 100 euros. It is interesting to see that historical data

seem to reﬂect this concern, suggesting a value for the

parameter below 1 which would yield a martingale

with a nondigital terminal value because the integral

giving the time change does not diverge when < 1!

Remark 5. The above maximum likelihood calibra-

tion from historical data used strongly the explicit

form (8) of the normalized allowance price and the

Gaussian property of the diffusion 4t5t∈601T 7 given

by the explicit form of the solution given by (13).

For general models of one-dimensional diffusion pro-

cesses with volatility given by a more function ä

more general than the speciﬁc ä=ê0ê−1, the max-

imum likelihood estimates of ,and hcan be

computed using Aït-Sahalia (2002) approach.

4.2. Option Pricing

Now we turn our attention to the valuation of

European call options written on allowance futures

price 4At5t∈601 T 7 . The payoff of a European call with

maturity ∈601 T 7 and strike price K≥0 is given

by 4A−K5+. Under the assumption that the savings

account 8Bt9t∈601 T 7 is given by Bt=eRt

0rsds for t∈601 T 7

for some deterministic short rate 8rs9s∈601 T 7 , this price

can be computed in the model proposed in this paper.

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Proposition 2. In a one-period 601 T 7 compliance

model, with risk-neutral parameters ∈and > 0, the

price of a European call with maturity ∈601 T 7 and strike

K≥0written on an allowance futures maturing at the end

Tof the compliance period is given at time t∈601 7 by

Ct=e−R

trsds Z

4ê4x5 −K5+N 4t1 1 2

t1 5 4dx51 (38)

with t1 and 2

t1 given by formulas (39) and (40) below.

This result is obtained by a straightforward calcu-

lation because the conditional distribution of given

tis Gaussian with mean

t1 41 5

=

tT−t

T−/2

if =11

texp

241−564T −t51−−4T − 51−7

if 6=1

(39)

and variance

2

t1 41 5

=

T−t

T−

−1 if =11

exp

1−64T −t51−−4T − 51−7−1

if 6=10

(40)

Let us illustrate the role of the parameter on

option prices. In the following example, we ﬁx =1

and assume that the penalty is =100. Further-

more, we suppose that at the initial time t=0 four

years prior to the compliance date T=4, the price of

a futures contract, written on allowance price at T,

is A0=25. For constant and deterministic continu-

ously compounded interest rate r=0005, we con-

sider European calls written on the forward price with

strike price of K=25 and varying maturity date ∈

601 T 7. The option price is calculated from (38) at time

t=0. In Figure 4, we also illustrate the dependence of

the option price upon the parameter (recall Figure 1

for a plot for ﬁxed ). Comparing three cases =005,

=008, and =101, Figure 4 shows that the call price

is increasing in . Less surprisingly, the dependence

on shows that longer-maturity calls (with the same

strike) are more valuable than their short-maturity

counterparts.

Let us stress that although there are no closed-form

formulas for call prices, their numerical evaluations

can be performed very efﬁciently.

Figure 4 Plots of the Prices C04 5 at Time t=0as Functions of Option

Maturity

01234

2

4

6

8

10

12

14

Time in years

Call price

Notes. The graphs marked by ,Î, and Èstand for =005, =008,

and =101, respectively. The values of the other parameters are given in

the text.

5. Implied Calibration

Because emission futures prices are supposed to fol-

low a bounded martingale in 6011007, the option price

must vanish for strike prices exceeding the penalty

level 100. At this point, our valuation technique

(38)–(40) differs from the traditional Black 76 formula

frequently applied to price options on futures. How-

ever, it turns out that the difference is rather minor,

for parameter values relevant to the current situation

of the EU ETS and for low strike prices. As an illus-

tration, we plotted in Figure 5 the call option price

with maturity o=3044 depending on the strike K,

calculated at t=0 for the underlying futures price

of A04f5=17054, supposing that the futures contract

matures at f=3046. The calculation is done for two

cases, the Black 76 model with volatility =0045 and

our pricing formula (38)–(40) for the maximum likeli-

hood parameters

∗≈003321 ∗≈00161 (41)

from (36). In all three cases, we have supposed that

the time to compliance date is T=5 and set the

short rate at r=0007. Figure 5 shows that for low

strikes Kin the range 601207, Black 76 call prices

are very similar to those given by our model with

maximum-likelihood parameters. Although for strike

Figure 5 Option Prices (Expressed in Euro) Against Their Strikes

0

0

5

Option prices

10

15

20 40 60 80

Black 76 price with sigma = 0.45

Theoretical price with = 0.332, = 0.161

k

100

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Figure 6 Evidence that European Calls Quoted on ECX Are Most

Likely Priced Using Black 76 Formula

02468

B76

10

8

6

4

2

0

Market

Market prices vs Black 76

at quoted volatility

prices in the range 6201807, the difference is consider-

able, it becomes negligible again at strikes above 80.

This approximation may explain why the traditional

Black 76 formula is considered to be useful for EUA

option pricing. As mentioned at the beginning, the

European Climate Exchange (ECX) provides beyond

call option quotas also the implied volatilities associ-

ated with these quotas via Black 76 formula. In most

cases, the implied volatility remains in the interval

60040100657. Based on a representative history of daily

call option prices quoted at the ECX during the period

January 4, 2007–September 30, 2009, we plot in Fig-

ure 6 the theoretical Black 76, where we have adopted

the quoted volatility and set the short rate at r=0005,

against market price of the option. Although we could

not reach a complete coincidences due to unknown

historical interest rates entering these option prices,

this ﬁgure strongly suggests that the traders polled

by ECX priced EUA options using Black 76 formula.

In Figure 7 we used a ﬁxed approximative volatil-

ity value of 0045, instead of the implied volatilities

quoted by the ECX, and the same short rate of r=

0005. Clearly, these data are more scattered, but the

concentration of price points across a straight line

shows that the option pricing practice of the ECX can

be approximatively described by the Black 76 formula

with a constant volatility of 0045.

Figure 7 Black 76 Formula with Volatility 0045 Explains Market

Call Prices

02468

10

8

6

4

2

0

B76

Market

Market prices vs Black 76

at volatility of 0.45

Figure 8 Market Prices vs. (38) and (39) with ??from (42)

0

0

2

2

4

4

Market

6

6

Theoretical call price

8

8

10

10

Market prices

vs model price

To compare our technique to the market option

prices, we decided to ﬁt our model to actual option

prices. To do so, we suggest determining those param-

eters and that minimize the sum of squared

deviations between historical market prices and their

theoretical values, based on our model (38)–(40). Hav-

ing implemented the function describing the sum of

squared deviations depending on model parameters

and , we applied a numerical procedure based on the

Nelder–Mead method to determine the minimizer

?≈003181 ?≈001440(42)

These values are very close to the maximum likeli-

hood estimates in (41). Coincidences of historical and

implied calibration are usually interpreted as model

validation. However, the deviations of market call

prices from their theoretical values plotted in Figure 8

are still strong. This will change when the option

market matures, the market participants realize the

differences between allowance price evolutions and

model assumptions underlying Black 76 formula, and

the ECX ﬁnally changes its way of reporting option

prices.

6. Multi–Compliance Periods Markets

So far, we focused on a generic cap-and-trade scheme

modeled after the ﬁrst phase of the EU ETS, namely,

limited to one compliance period and without bank-

ing in the sense that unused allowances become

worthless at the end of the period. This is a strong

simpliﬁcation because as already mentioned above,

real-world markets are operating in a multiperiod

framework. Furthermore, subsequent periods are con-

nected by market speciﬁc regulations. In what fol-

lows, we consider an abstract but generic model of

such a market and focus on most natural rules for the

period interconnection.

Presently, three regulatory mechanisms connect

successive compliance periods in a cap-and-trade

scheme. Their rules go under the names of borrowing,

banking, and withdrawal.

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• Borrowing allows for the transfer of a (limited)

number of allowances from the next period into the

present one.

• Banking allows for the transfer of a (limited)

number of (unused) allowances from the present

period into the next.

• Withdrawal penalizes ﬁrms that fail to comply in

two ways: by penalty payment for each unit of pollu-

tant not covered by credits and by withdrawal of the

missing allowances from their allocation for the next

period.

From the nature of the existing markets and the

designs touted for possible implementation, it seems

that policy makers tend to favor unlimited banking

and forbid borrowing. Furthermore, the withdrawal

rule is most likely to be included. Banking and with-

drawal seem to be reasonable rules to reach an emis-

sion target within a ﬁxed number of periods because

each success (respectively, failure) in the previous

period results in stronger (respectively, weaker) abate-

ment in the subsequent periods.

6.1. Market Model

For the remainder of this section, we consider a two-

period market model without borrowing but with

withdrawal and unlimited banking. We denote the

two periods by 601 T 7 and 6T 1 T 07and consider a

stochastic basis 4ì1 F11 4Ft5t∈601 T 075with a distinct

measure ∼, which we view as the spot martingale

measure. Further, we introduce processes 4At5t∈601T 7 ,

4A0t5t∈601 T 07for the futures contracts with maturities at

compliance dates T,T0written on allowance prices

from the ﬁrst and the second period, respectively.

To exclude arbitrage, we suppose that the prices

4At5t∈601 T 7 and 4A0

t5t∈601 T 07are martingales with respect

to the spot martingale measure . Noncompliance in

the ﬁrst and second periods occurs on events N∈FT

and N0∈FT0, respectively. As before, we assume that

the savings account 4Bt5t∈601 T 07is given by

Bt=eRt

0rsds1 t ∈601 T 07(43)

for some deterministic short rate 4rs5s∈601 T 07. The results

of the previous section imply that in the case ì\Nof

the ﬁrst-period compliance, the allowance price drops

AT1ì\N=A0

T1ì\N1(44)

where ∈4015stands for discount factor describing

the interest rate effect

=BTB−1

T0=e−RT0

Trsds0

The relation (44) is justiﬁed by considering spot

prices. The random variable A0

Tis nothing but the

spot price at time Tof the second-period allowance.

Because futures and spot price agree at maturity, AT

must be the spot price of the ﬁrst-period allowance

at T. In the case of compliance in the ﬁrst period, the

unused allowances can be banked; hence we have the

equality in (44).

In the case of noncompliance at the end of the ﬁrst

period, the withdrawal regulation implies that

AT1N=A0

T1N+1N0(45)

Namely, the noncompliance in one pollutant unit at

time Tcosts a penalty in addition to one allowance

from the next period, which must be withdrawn at

the spot price A0

T.

Combining the results (44) and (45), we ﬁnd out

that the difference is

At−A0

t=Ɛ4AT−A0

TFt5=Ɛ41NFt51 t ∈601 T 7

and must be modeled as 801 9-valued martingale. We

suggest using the same methodology as in the one-

period model

At−A0

t=ê41

t51 t ∈601 T 71 (46)

where the Gaussian process 41

t5t∈601 T 7 is introduced as

previously in (12), with 4415

s5s∈601 T 7 in parameterized

form (28) and driven by a process 4W 1

t1Ft5t∈601 T 07of

Brownian motion.

To model the second-period allowance futures

price, a continuation of the cap-and-trade system

must be speciﬁed. If there is no agreement on long-

term regulatory framework (as is the case for most of

the existing emission markets), the process 4A0

t5t∈601 T 7

should be speciﬁed exogenously. The simplest choice

would be a geometric Brownian motion with constant

volatility. Another idea to handle the uncertain con-

tinuation is to suppose that the cap-and-trade sys-

tem will be terminated after the second period. In

this case,

A0

t=Ɛ41N0Ft51 t ∈601 T 07

can also be modeled as in the one-period model

A0

t=ê42

t51 t ∈601 T 070 (47)

Again, 42

t5t∈601 T 7 is introduced as in (12), with a pro-

cess 8425

s9s∈601 T 7 chosen in parameterized form (28) and

driven by another Brownian motion 4W 2

t1Ft5t∈601 T 07.

6.2. Option Pricing

As an application of our two-period model, we con-

sider pricing of European calls. Consider European

call option with strike price K≥0 and maturity

∈601 T 7 written on futures price of allowance from

the ﬁrst period. This contract yields a payoff

C=4A−K5+at time ∈601 T 7.

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1466 Management Science 57(8), pp. 1453–1468, © 2011 INFORMS

Under the assumptions of the previous section, we

start with the computation of the price C0

C0=e−R

0rsdsƐ44A−K5+5

of this option at time t=0. Using the decomposition

4A−K5+=4A−A0

+A0

−K5+1

we utilize our modeling of 801 9-valued martingales

(46) and (47) to express the terminal payoff as

4A−K5+=4ê41

5+ê42

5−K5+

with expectation

C0=e−R

0rsdsƐ44A−K5+5

=e−R

0rsdsƐ44ê41

5+ê42

5−K5+5

=e−R

0rsds Z24ê4x15+ê4x25−K5+

·N 41 54dx11 dx251 (48)

where N 41 5=F1

1 2

stands for joint normal distri-

bution of 1

and 2

.

Let us derive the mean and the covariance

matrix under the standing assumption 1=2=1,

for 1>0, 2>0. We have

1

=ê−1A0−A0

0

sT

T−1

+1/2

1R

04T −u541−15/2W1

udu

4T −51/21

2

=ê−1A0

0

sT0

T0−2

+1/2

2R

04T 0−u542−15/2W2

udu

4T 0−52/20

Denoting by the correlation of the two Brownian

motions 4W 1

t5t∈601 T 07and 4W 2

t5t∈601 T 07

6W 11 W 27 dt = dt1 ∈6−11171

we can apply the same argumentation to obtain the

means

1

=Ɛ41

5=ê−1A0−A0

0

sT

T−1

1

2

=Ɛ42

5=ê−1A0

0

sT0

T0−2

3

the variances

111

=Var41

5=T

T−1

−11

212

=Var42

5=T0

T0−2

−13

and the covariance as

112

=211

=Cov41

1 2

5

=1/2

11/2

2R

04T −u541−15/24T 0−u542−15/2 du

4T −51/24T 0−52/20

At times t∈601 7 prior to maturity, the price Ctof the

call is obtained similarly.

Proposition 3. In a two–compliance periods model as

above, with parameters 11 2>0, and ∈4−1115, the

price of the European call with strike price K≥0and matu-

rity ∈601 T 7 written on ﬁrst-period allowance futures

price is given at time t∈601 7 by

Ct=e−R

trsds Z24ê4x15+ê4x25−K5+

·N 4t1 1 t1 54dx11dx25(49)

with mean t1

1

t1 =ê−1At−A0

t

sT−t

T−1

1(50)

2

t1 =ê−1A0

t

sT0−t

T0−2

1(51)

and covariance matrix t1

111

t1 =Var41

5=T−t

T−1

−11(52)

212

t1 =Var42

5=T0−t

T0−2

−11(53)

112

t1 =211

t1 =1/2

11/2

2R

t4T −u541−15/24T 0−u542−15/2 du

4T −51/24T 0−52/20

(54)

If we take a closer look at the computations in-

volved in the valuation of the call price

Ct=Ct4 1 T 1 T 01 A01 A0

01 K1 r 1 11 21 5

given by the formulas (49)–(54), we see that be-

cause of

Ct4 1 T 1 T 01 A01 A0

01 K1 r 1 11 21 5

=C04 −t1 T −t 1 T 0−t1 A01 A0

01 K1 r 1 11 21 5

for all t∈601 7, it sufﬁces to consider the case t=0.

The numerical evaluation of two-dimensional inte-

gral is easily performed by using a decomposition of

the two-dimensional normal distribution. To ease the

notation, let us skip t1 to write i=i

t1 ,i1 j =i1 j

t1

for i1 j =112. It holds

N 41 54dx11 dx25

=N 411 c 4x251 1111 c 54dx15N 421 21254dx251 (55)

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Carmona and Hinz: Risk-Neutral Models for Emission Allowance Prices and Option Valuation

Management Science 57(8), pp. 1453–1468, © 2011 INFORMS 1467

where the conditional mean and the conditional vari-

ance are given by

11 c4x25=1+211

2124x2−251

1111 c =111−421152

2120

With factorization (55), the inner integral is calculated

explicitly in the following cases:

Z

4ê4x15+ê4x25−K5+N 411 c 4x251 1111c 54dx15

=

0 if K−ê4x25≥1

ê11c 4x25

√1+1111 c + ê4x25−K

if K−ê4x25≤00

That is, the numerical valuation is required only in

the case 0 < K −ê4x25 < , where

Z

ê−14K/ −ê4x255

4ê4x15+ê4x25−K5

·N 411 c 4x251 1111 c 54dx15

needs to be calculated.

Having obtained the inner integral, the numerical

evaluation of the outer integral is straightforward.

Because the density of the normal distribution decays

Figure 9 Surface Plots of the Initial Call Price 4 1 157→ C04 1 15as

Function of Maturity and 1for Correlation =008(Top)

and =−008(Bottom)

1

2

3

0.6

0.8

1.0

1.2

2

4

6

8

Maturit

y

Maturity

1

2

3

Beta

Beta

0.6

0.8

1.0

1.2

Call price

Call price

0

2

4

6

8

Note. The values of the other parameters are given in the text.

sufﬁciently fast, we expect neither numerical difﬁcul-

ties nor long computation times. In fact, we did not

encounter any problem implementing this formula.

For the sake of completeness, we illustrate the

dependence of the call price on 1and maturity of

the call. To make the results comparable with the one-

period example given above, we chose the following

parameters: four years to the ﬁrst-period compliance

date T=4, eight years to the second-period compli-

ance date T2=8, initial ﬁrst-period allowance futures

price A0=25, initial second-period allowance futures

price A0

0=15, strike price of the European call K=25,

interest rate r=0005, and 2=002. Figure 9 depicts

the dependence of the call price on the value of 1for

the ﬁrst period and of the call maturity .

7. Conclusion

Mandatory emission markets are being established

throughout the world. In the most mature market,

the EU ETS, beyond physical allowances, a large

volume of allowance futures is traded. Furthermore,

European options written on these futures have been

introduced and traded although no theoretical foun-

dation for their pricing is available yet.

The goal of this work is to ﬁll this gap. In our

analysis, we gradually move from a one-period mar-

ket model to a more realistic situation of two-period

markets (covering the present EU ETS regulations)

and show that martingales ﬁnishing at two-valued

random variables can be considered as basic build-

ing blocks for risk-neutral futures price dynamics.

We suggest a model for two-valued martingales, ﬂex-

ible in terms of time- and space-changing volatil-

ity and capable of matching the observed historical

or implied volatility of the underlying future. From

a hedging perspective, this issue could be one of

the most desirable model properties. Other practi-

cal aspects like ease of calibration and simple option

valuation schemes are also fulﬁlled in our approach.

We show how parameters can be estimated from his-

torical price observation and suggest efﬁcient option

valuation schemes. Although option price formulas

are not available in a closed form, a simple and fast

numerical integration can be applied.

Acknowledgments

The research of René Carmona was partially supported by

NSF-FRG-DMS-0455982 and DMS-0806591. The research of

Juri Hinz was partially supported by WBS R-703-000-020-

720/C703000 of the Risk Management Institute and by WBS

R-146-000-107-133 of the National University of Singapore.

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