MANAGEMENT SCIENCEVol. 57, No. 8, August 2011, pp. 1453–1468issn 0025-1909 �eissn 1526-5501 �11 �5708 �1453 doi 10.1287/mnsc.1110.1358

© 2011 INFORMS

Risk-Neutral Models for Emission AllowancePrices and Option Valuation

René CarmonaBendheim 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 HinzDepartment 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 increasedliquidity of trading on futures contracts on CO2 emissions allowances, led naturally to the next step in

the development of these markets: These futures contracts are now used as underliers for a vibrant derivativemarket. In this paper, we give a rigorous analysis of a simple risk-neutral reduced-form model for allowancefutures prices, demonstrate its calibration to historical data, and show how to price European call optionswritten on these contracts.

Key words : emission derivatives; emissions markets; cap-and-trade schemes; environmental financeHistory : Received April 25, 2009; accepted March 7, 2011, by Haitao Li, guest editor, finance. Published online

in Articles in Advance June 20, 2011.

1. IntroductionGlobal warming and environmental problems con-tinue to challenge policy makers. In part because ofthe success of the U.S. Acid Rain Program, cap-and-trade systems are now considered to be one of themost promising market mechanism to reduce green-house gas emissions on an international scale. Thecore principle of such a mechanism is based on theallocation of fully tradable credits among emissionsources 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 putsa price on emissions and helps identify and exer-cise the cheapest emission abatement measures. Forthis reason, market-based mechanisms for emissionreduction are supposed to yield pollution control atthe lowest cost for the society. Notwithstanding therigorous equilibrium analyses from Carmona et al.(2009, 2010) that confirm that social optimality doesnot necessarily mean that the scheme is cheap for con-sumers, emission trading should be considered as acost-efficient and effective tool.

By its very nature, the regulatory framework of amandatory cap-and-trade system involves its partici-pants in a risky business, necessarily creating the needfor appropriate risk management. Trading of certifi-cates from a mandatory scheme is typically accom-panied by an active secondary market where diverseemission-related financial derivatives (e.g., futures)are traded. Options have been traded since 2006, and

as we explain below, on any given day, the volumeof European call and put options traded on forwardEuropean Union Allowance (EUA) contracts rangesbetween 15 and 25 million tons of CO2 equivalent, theshort end of the forward curve being clearly the mostactively traded.

In this work, we propose reduced-form models forthe risk-neutral dynamics of allowance prices, pro-viding a quantitative framework for pricing emissionderivatives.

Despite the large number of pieces in the popularpress and numerous speculative articles in magazines,the scientific literature on cap-and-trade systems israther limited, especially if we restrict ourselves toquantitative analysis of models including stochas-tic factors. For the sake of completeness, we brieflyreview the literature relevant to our contribution.The economic theory of allowance trading can betraced 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 Cronshawand Kruse (1996), Tietenberg (1985), Rubin (1996),Leiby and Rubin (2001), Schennach (2000), Stevensand Rose (2002), Maeda (2004), and in the literaturecited therein. Empirical evidence from existing markets isdiscussed in Daskalakis et al. (2007) and Paolella andTaschini (2008). This last paper suggests economicimplications and hints at several ways to modelspot and futures allowance prices, whose detailed

1453

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Carmona and Hinz: Risk-Neutral Models for Emission Allowance Prices and Option Valuation1454 Management Science 57(8), pp. 1453–1468, © 2011 INFORMS

interrelations are investigated in Uhrig-Homburg andWagner (2008, 2009). There, the demand for derivativeinstruments in emission markets is also addressed.In Benz and Trueck (2009), characteristic proper-ties for financial time series are observed for pricesof emission allowances from the mandatory Euro-pean Union Emission Trading Scheme (EU ETS) andseveral Markov and AR/GARCH–like models areproposed. Paolella and Taschini (2008) also considertail behavior and the heteroscedasticity of the returns.Dynamic price equilibrium and optimal market design areinvestigated in Carmona et al. (2009), which providesa mathematical analysis of market equilibriums anduses optimal stochastic control to show social opti-mality. Based on this approach, Carmona et al. (2010)discuss price formation for goods whose productionis affected by emission regulations. In this setting,an equilibrium analysis confirms the existence of theso-called windfall profits (see Sijm et al. 2006) andprovides quantitative tools to analyze alternative mar-ket designs, which are applied in Carmona et al.(2008) to optimize a cap-and-trade mechanism for aproposed Japanese emission trading scheme. Seifertet al. (2008) and Wagner (2006) also deal with risk-neutral allowance price formation within the EU ETS.Using equilibrium properties, the price evolution istreated in terms of marginal abatement costs and opti-mal stochastic control. Cetin and Verschuere (2010)use hidden Markov models and a filtering approachto capture the impact of news releases. Option pric-ing within the EU ETS was considered only recently.Chesney and Taschini (2008) rely on endogenousemission permit price dynamics within an equilib-rium setting to value European options on emissionallowances.

The present paper is organized as follows. Afteran introductory discussion of the various approachesto risk-neutral modeling in §2, we present a generalapproach to modeling of an emission market with onecompliance period. More realistic multiperiod mod-els are treated in §6. The mathematical treatment of§3 is based on the analysis of diffusion martingalesending with only two possible values. We identifyexplicit classes of such martingales, and we showhow simple deterministic time changes can providefamilies of versatile risk-neutral models for allowanceprices. In §4, we demonstrate how to calibrate one ofthese models to historical allowance price data. Wedevelop a historical calibration procedure, because theoption market has not yet matured to a point we cantrust more standard calibration procedures based onoption price data. The second part of the paper gen-eralizes the one-period setup to more realistic mul-tiperiods models incorporating important features ofreal-world markets, and §6 provides the necessary

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

Option Option Allowance Implied Settlementmaturity type Volume Strike price vol. (%) price

Dec-08 Cal 2001000 22000 23055 51025 5006Dec-08 Call 1501000 26000 23055 51025 3057Dec-08 Call 4501000 27000 23055 51025 3027Dec-08 Call 1001000 28000 23055 51025 2099Dec-08 Call 1251000 29000 23055 51025 2074Dec-08 Call 5251000 30000 23055 51025 2051Dec-08 Call 2501000 40000 23055 51025 1004Dec-08 Call 7001000 50000 23055 51025 0045Dec-08 Put 110001000 14000 23055 51025 0064Dec-08 Put 2001000 15000 23055 51025 0086Dec-08 Put 2001000 15000 23055 51025 0086Dec-08 Put 4001000 16000 23055 51025 1013Dec-08 Put 1001000 17000 23055 51025 1043Dec-08 Put 110001000 18000 23055 51025 1078Dec-08 Put 5001000 20000 23055 51025 2060Dec-08 Put 2001000 21000 23055 51025 3007Dec-08 Put 2001000 22000 23055 51025 3057

Note. ECX, European Climate Exchange.

changes needed to extend the pricing formula to thismore general setup.

As a motivation for our derivations of option pric-ing formulas, we close this introduction with a shortdiscussion of the idiosyncrasies of the EUA optionmarkets. 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 onEUA futures contracts. Since 2006, trades of optionsmaturing in December of each year (prior to 2012)have produced a term structure of option prices. Onany given day, the volume varies from 5 to 25 mil-lion tons of CO2 equivalent, the short end of the curvebeing the most active with a good number of financialinstitutions involved, whereas the long end dependsmostly on a few energy companies. It is not clearhow these options are priced, and a persistent rumorclaims that traders plainly use Black’s formula. Thedata reproduced in Table 1 is an extract of quotespublished on January 4, 2008. Obviously, the impliedvolatility is perfectly flat, and the absence of skewor smile is consistent with the rumor. Whether ornot traders are using Black or Black–Scholes formulasto price options on EAUs and futures contracts, andgiven what we learned from equilibrium models, wethink that it is important to derive option price formu-las based on underlying martingales with binary ter-minal value, because Black–Scholes formula is basedon an underlying price martingale converging to zero!

2. Risk-Neutral Modeling ofEmission 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 ofstatistical aspects in price movements, possibly ignor-ing the underlying economic underpinnings, the equi-librium approach focuses on the mechanics of priceformation: Given incentives, strategies, uncertainty,and risk aversions, the market is described by thecumulative effect of individual actions. Investigationsare geared toward understanding market featuresappearing in steady realistic state, i.e., market equi-librium. Finally, the risk-neutral approach adopted inthis paper abstracts from the mechanism driving themarket to the equilibrium state and focuses on itsbasic properties. Starting from the absence of arbi-trage, the asset price evolution is introduced directly.This approach arose from the idea that although pricemovement is stochastic in nature, derivative valua-tion does not refer to the real-world probability. Thedescription of statistical issues is not a goal of risk-neutral models.

Absence of arbitrage is central to the risk-neutralapproach. Moreover, we want to the importantinsights gained from equilibrium theory (Carmonaet 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 allowanceprice drops to zero or it approaches the penaltylevel �. Indeed, the price must vanish at maturity ifthere is excess of allowances, whereas in the case ofshortage, the price will raise, reaching the level of thepenalty. It is reasonable to suppose that in reality, thedemand for allowances will coincidence with the sup-ply with zero probability. So the occurrence of suchan event will be disregarded.

(c) Allowance trading instantaneously triggers allabatement measures whose costs are below theallowance price because if an agent owns a tech-nology with lower reduction costs than the presentallowance price, then it is optimal to immediately useit to reduce his or her own pollution and profit fromselling allowances.

At this point, we distinguish between two types ofrisk-neutral approaches:

• the reduced-form risk-neutral approach, whichfocuses on (a) and (b);

• the detailed risk-neutral approach, which aims atall three properties (a), (b), and (c).

Let us explain, at least at a formal level, the maindifferences between these two classes of models. Forthe sake of concreteness, we focus on a continuous-time framework in which the risk-neutral evolution4At5t∈601T 7 of a futures contract on an allowance withcompliance date T for a penalty � ∈ 401�5 is beingmodeled. In this framework, the allowance price pro-cess 4At5t∈601T 7 is realized a filtered probability space

4ì1F1�1 4Ft5t∈601T 75, equipped with a distinct mea-sure �∼�, which is interpreted as the spot martingalemeasure.

The reduced-form risk-neutral approach focuses onthe following problem:

Model the noncompliance event N ∈FT ,which defines the �-martingale 4At5t∈601T 7

with terminal value AT =�1N .(1)

The noncompliance event N is the only object thatneeds to be described exogenously. However, toobtain a useful model, several requirements, rangingfrom computational tractability in derivatives valua-tion to diverse aspects of calibration, must be fulfilled.

Within the detailed risk-neutral approach, the non-compliance event N is obtained endogenously, interms of other quantities that in turn must be spec-ified exogenously. This is where the issue (c) comesinto play. Under natural equilibrium assumptions (seeCarmona et al. 2009), the abatement activity in themarket 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 valueof At . This is also known as the equilibrium allowanceprice equals to the marginal abatement costs in envi-ronmental economics. Hence in equilibrium, the totalabatement in the market can be described in terms ofallowance prices 4As5s∈601T 7 as

∫ T

0cs4As5 ds1

where cs4a54�5 stands for the total intensity of theabatement measures at time s ∈ 601T 7 available in themarket at price less than or equal to a ∈ 601�5 inthe market scenario � ∈ ì. In this context, the abate-ment volume function ct2 601�7 × ì → 601�5, t ∈

601T 7 must be specified exogenously. In practice, theabatement volume function can be be estimated frommarket data: Given a risk-neutral fuel price model,ct can be described by an appropriate B601�7 ⊗ Ft-measurable functions for each t ∈ 601T 7. In this con-text, the noncompliance event is given by

{

� ∈ì3 ET 4�5−∫ T

0cs4As54�5ds ≥ 0

}

1

where an exogenously specified FT —measurableallowance demand ET —is used for the numberof excess pollution units in the business-as-usualscenario (i.e., given zero penalty). Thus, the detailedrisk-neutral approach leads to a more complex math-ematical problem:

Determine 4ct5t∈601T 7 from market dataand model allowance demand ET to

obtain a �-martingale 4At5t∈601T 7 with theterminal value AT =�18ET −

∫ T0 ct 4At 5 dt≥09.

(2)

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Carmona and Hinz: Risk-Neutral Models for Emission Allowance Prices and Option Valuation1456 Management Science 57(8), pp. 1453–1468, © 2011 INFORMS

Although the detailed risk-neutral approach seemsappealing from a methodological perspective, it isnot obvious whether its higher complexity is justi-fied from the viewpoint of derivative valuation. Thepresent authors believe and show in this work thatthe reduced-form risk-neutral approach yields satisfac-tory results, at least in the area of pricing plain-vanillaEuropean options written on allowance futures. Asillustration we investigate the solution of a particularproblem of the type (2) and compare its results to aclass of solutions to (1) suggested in the present work.

The existence and uniqueness of 4At5t∈601T 7 solv-ing (2) requires a delicate discussion. The martin-gale Et = ƐQ4ET � Ft5 defined for t ∈ 601T 7 plays anessential role. Indeed, the analysis of the discrete-time framework shows that if the future incre-ments of 4Es5s∈6t1T 7 are independent of the presentinformation Ft at any time t, then a solution to(2) should be expected in the functional form At 2=�4t1Gt5 with an appropriate deterministic function�2 601T 7×� 3 4t1g5 7→ �4t1g5 ∈� and a state process4Gt5t∈601T 7 given by

Gt =Et −

∫ t

0cs4As5 ds1 t ∈ 601T 70

This insight helps guess a solution in the standarddiffusion framework, when there exists a process4Wt1Ft5t∈601T 7 of Brownian motion with respect to�∼�, in the simplest case dEt = �dWt with pre-specified � ∈ 401�5 and continuous, nondecreasing,and deterministic abatement function c2 401�5 → �.Under these conditions, Itô’s formula, applied to themartingale At = �4t1Gt5, leads, to a nonlinear partialdifferential equation for � on 601T 5×�:

¡t�4t1g5− ¡g�4t1g5c4�4t1g55+12¡

2gg�4t1g5�

2= 01 (3)

subject to the boundary condition

�4T 1g5=�1601�54g5 for all g ∈�1 (4)

justified by the digital nature of the terminalallowance price. Having obtained � in this way, oneconstructs the state process 4Gt5t∈601T 7 as a solution ofthe stochastic differential equation

dGt = dEt − c4�4t1Gt55 dt1 G0 =E01 (5)

from which we get a solution to (2) from At = �4t1Gt5.Once 4At5t∈601T 7 is determined, one applies standardintegration to value European options. Althoughclosed-form expressions are rare, option prices canbe calculated numerically. The only case that yieldsquasi-explicit expressions (involving only numericalintegrations) is that of linear abatement functions (seeSeifert et al. 2008, Wagner 2006).

Let us elaborate on this case, to give the reader afeeling of allowance option pricing in the frameworkof 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

Opt

ion’

s m

atur

ity

Example 1. Set the time to compliance date T totwo years and assume the diffusion coefficient � is4 and the penalty � is 100, and suppose that theabatement function c = ct for t ∈ 601T 7 is linear withc2 a 7→ 0002 · a. At time t = 0, we consider a family ofEuropean calls with the same strike price of K = 25but different maturity times � running through 601T 7.Suppose that the initial allowance price equals to thestrike price a = A0 = 25. Determine call prices C04�5at t = 0 for different maturity times � ∈ 601T 7. Inde-pendently of the model, the price of the call in frontof expiry date � = 0 must be equal to zero C0405 = 0,whereas the longest maturity call � = T must have aprice C04T 5=A04� −K5/� = 25 · 0075 = 18075. Becauseof the digital terminal value of the underlying, such acall is equivalent to 0075 allowances. Call prices mustincrease with call’s maturity from 0 to 18075. Thismust be true within any risk-neutral model. Figure 1illustrates the exact curve 4C04�55�∈601T 7 for the param-eters as above. Because the end points are model-independent, merely intermediate-maturity pricesexhibit model-dependent properties. Here we observeone remarkable issue: the so-called inverse S-shape.

In this work, we show that these features are sharedby significantly simpler option pricing schemes (seeFigure 4) based on the reduced-form approach. For thisreason, we believe that the reduced-form approachcan provide a reasonable pricing mechanism foremission-related financial products. However, we alsoagree that further development of detailed risk-neutral, econometric, and equilibrium modeling isneeded to help understand allowance price evolu-tion. Certainly, such models could be better suited toaddress the impact of information asymmetry, jumpsin the information flow, regulatory uncertainty, andmarket idiosyncracies.

3. Reduced-Form Model for a SingleCompliance Period

In this section, we introduce a simple model for anabstract emission market. We first restrict ourselvesto a single compliance period, say 601T 7. The morerealistic case of multiperiod models is treated in §6.

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In the one-period setting, credits are allocated atthe beginning of the period to enable allowance trad-ing until time T and to encourage agents to exer-cise efficient abatement strategies. At the compliancedate T , market participants cover their emissions byredeeming allowances or pay a penalty � per unitof pollution not offset by credits. In this one-periodmodel, unused allowances expire and are worthlessbecause we do not allow for banking into the nextperiod. Under natural assumptions, equilibrium anal-ysis shows that the allowance price AT at compliancedate T is a random variable taking only the values0 and � (see Carmona et al. 2009, 2010). More pre-cisely, if the market remains under the target pollutionlevel, then the price approaches zero. Otherwise, theallowance price tends to the penalty level �.

All the relevant asset price evolutions are assumedto be given by adapted stochastic processes on a fil-tered probability space 4ì1F1�1 4Ft5t∈601T 75 on whichwe fix an equivalent probability measure � ∼ �,which we call the spot martingale measure.

We denote by 4At5t∈601T 7 the price process of afuture contract with maturity date T written on theallowance price. Given the digital nature of the termi-nal allowance price AT , the central object of our studyis the event N ⊂ FT of noncompliance, which set-tles the 801�9-dichotomy of the terminal futures priceby AT = �1N . Furthermore, a standard no-arbitrageargument shows that the futures price 4At5t∈601T 7 needsto 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 forthe martingale

At =�Ɛ�41N �Ft51 t ∈ 601T 70

There are many candidates for such a process, but noobvious choice seems to be versatile enough for thepractical requirements described below. An importantrequirement is the need to match the observed volatil-ity structure. For a practitioner trying to calibrate attime � ∈ 601T 7 a model for the martingale 4As5s∈6�1T 7that finishes at 0 or �, the minimum requirementsare to match the price observed at time � as wellas the observed price fluctuation intensity up to thistime � . Further model requirements include the exis-tence of closed-form formulas, or at least fast valua-tion schemes for European options; a small number ofparameters providing sufficient model flexibility; andreliable 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 thetwo reduced-form approaches, we choose our startingpoint to be the noncompliance event N ∈ FT , whichwe describe as the event where a hypothetic positive-valued random variable âT exceeds the boundary 1,

say N = 8âT ≥ 19. If one denotes by ET the total pol-lution within the period 601T 7 that must be balancedagainst the total number � ∈ 401�5 of credits issuedby the regulator, then the event of noncomplianceshould be given by N = 8ET ≥ �9, which suggests thatâT should be viewed as the normalized total emissionET /�. However, in our modeling, we merely describethe noncompliance event. Strictly speaking, so anyrandom variable âT with

8âT ≥ 19= 8ET /� ≥ 19

would do as well. On this account, we do not claimthat âT represents the total normalized emission ET /�.So the allowance spot price is given by the martingale

At =�Ɛ�(

18âT ≥19 �Ft

)

1 t ∈ 601T 71

where the random variable âT is chosen from asuitable parameterized family of random variables8â �

T 2 � ∈ä9. For reasons of model tractability, we sup-pose that the filtered probability space supports a pro-cess 4Wt5t∈601T 7 of Brownian motion with respect to thespot martingale measure �, and we investigate para-metric families which give allowance prices

A�t =�Ɛ�

(

18â�T ≥19 �Ft

)

1 t ∈ 601T 7

with a Markovian stochastic evolution of the form

dA�t = v�4t1A�

t 5 dWt1

where the diffusion term v� captures the basic prop-erties of historical price observations. In particular,we will match exactly the observed initial allowanceprice and the initial instantaneous price fluctuationintensity.

Remark 1. We propose a consistent pricing schemefor emission-related financial instruments within theframework of diffusion processes. Although this rulesout discontinuity in allowance prices, we believethat this approach is reasonable. It has been arguedthat 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 playa significant role in mature emission markets. Anincreasing number of consultancies and market ana-lysts are carefully watching the European emissionmarket. Several agencies are providing news and peri-odical publications. Moreover, because energy gener-ation and consumption are publicity observable, oneshould not expect significant allowance price jumpsin a mature emissions market. On this account, a risk-neutral model based on continuous allowance priceevolution is reasonable.

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To simplify the notation, we consider the normal-ized futures price process

at 2=1�At = Ɛ�

(

18âT ≥19 �Ft

)

1 t ∈ 601T 71

and we describe it under special assumptions on âT .Our goal is to identify classes of martingales 8at9t∈601T 7taking values in the interval 40115 and satisfying

�

{

limt↗T

at ∈ 80119}

= 10 (6)

We first identify a parametric family of such martin-gales by working backward from a simple model forthe random variable âT , motivated by intuitive under-standing of the final cumulative level of emissions.

3.1. Basic Modeling of the Compliance EventWe use the standard notation N4�1�25 for the normaldistribution with mean � and variance �2, and wewrite ê for the cumulative distribution function ofthe standard normal distribution.

Proposition 1. Suppose that

âT = â0e∫ T

0 �sdWs−41/25∫ T

0 �2s ds1 â0 ∈ 401�5 (7)

for some continuous and square-integrable deterministicfunction 401T 5 3 t ↪→ �t . Then the martingale

at = Ɛ�(

18âT ≥19 �Ft

)

1 t ∈ 601T 7 (8)

is given by

at =ê

(

ê−14a05√

∫ T

0 �2s ds +

∫ t

0 �sdWs√

∫ T

t�2s ds

)

1 (9)

and it solves the stochastic differential equation

dat =ê′4ê−14at55√zt dWt1 (10)

where the positive-valued function 401T 5 3 t ↪→ zt isgiven by

zt =�2t

∫ T

t�2u du

1 t ∈ 401T 50 (11)

Remark 2. Notice that even though the distribu-tion of âT depends only upon

∫ T

0 �2s ds, the noncompli-

ance event N depends upon the entire function 8�s9s .

Proof. A direct calculation shows

at = Ɛ�(

18âT ≥19 �Ft

)

=�8âT ≥ 1 �Ft9

= �8âte∫ Tt �sdWs−41/25

∫ Tt �2

s ds ≥ 1 �Ft9

= ê

(

ln ât − 41/25∫ T

t�2s ds

√

∫ T

t�2s ds

)

= ê

(

ln4â05− 41/25∫ T

0 �2s ds

√

∫ T

0 �2s ds

√

∫ T

0 �2s ds

√

∫ T

t�2s ds

+

∫ t

0 �s dWs√

∫ T

t�2s ds

)

1

and taking into account the initial condition

a0 =ê

(

ln â0 − 41/25∫ T

0 �2s ds

√

∫ T

0 �2s ds

)

1

we obtain the desired expression (9). To show (10), westart with at =ê4�t5, t ∈ 601T 7, where

�t =�01T +

∫ t

0 �s dWs√

∫ T

t�2s ds

for t ∈ 601T 71

with �01T = ln â0 −12

∫ T

0�2s ds (12)

and �0 = ê−14a05 with deterministic a0 ∈ 40115. Com-puting its Itô differential, we get

d�t =

(

∫ T

t�2s ds

)−1/2

�t dWt

+ 12

(

x0 +

∫ t

0�s dWs

)(

∫ T

t�2s ds

)−3/2

�2t dt

=

(

∫ T

t�2s ds

)−1/2

�tdWt +12�t

(

∫ T

t�2s ds

)−1

�2t dt

=√zt dWt +

12zt�t dt

d6�7t = zt dt0

Next, Itô’s formula gives the differential of the nor-malized allowance prices as

dat = ê′4�t5d�t +12ê

′′4�t5d6�7t

= ê′4�t5(√

zt dWt +12zt�t dt

)

+ 12ê

′′4�t5zt dt

= ê′4ê−14at55√zt dWt

because xê′4x5+ê′′4x5≡ 0. �We notice for later use that if t < � , �� is given

explicitly as a function of �t by

�� = e41/25∫ �t zs ds�t +

∫ �

te41/25

∫ �s zu du

√zs dWs0 (13)

3.2. Construction via Time ChangeThe stochastic differential Equation (10) can be inter-preted in the following way. Because of the factor

√zt

in front of dWt , at can be viewed as the time-changeof a martingale 8Yt9t∈601�5 given by the strong solutionof the stochastic differential equation

dYt =ê′4ê−14Yt55 dWt1 (14)

for t ∈ 601�5, with Y0 ∈ 40115. This solution stays inthe open interval 40115 and converges to the bound-aries 0 or 1 with certainty when t approaches �:

�

{

limt↗�

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 a40115-valued martingale 8Yt9t∈601�5 satisfying (15) andthen search for a time change bringing the half-axis601�5 onto the bounded interval 601T 5. With this inmind, it appears natural to consider the solutions ofthe stochastic differential equation

dYt =ä4Yt5 dWt1 Y0 ∈ 401151 t ≥ 01 (16)

where ä is a nonnegative continuous function on60117 satisfying ä405 = ä415 = 0. We can then useFeller’s classification (see, for example, Karatzas andShreve 1991, Karlin and Taylor 1981) to check thatsuch a diffusion is conservative, does not reach theboundaries 0 and 1 in finite time, and satisfies (15).This is the case if v40+5 = v41−5 = �, where v4x5 isdefined by

v4x5= 2∫ x

0054x− y5

dy

ä4y51 x ∈ 401150

Straightforward computations show that the solutionof the stochastic differential Equation (14) does indeedsatisfy these conditions; hence it does not hit 0 and 1in finite time with probability one.

Explicit families of such martingales can easily beconstructed. Case in point, a two-parameter familyof examples can be constructed from the analysis ofCarmona et al. (2001), which we learned from MikeTerhanchi (who extended the argument of Carmonaet al. 2001 to Lévy processes in Tehranchi 2009). Ifwe set

Xt = e−Wt+ct

(

X0 −

∫ t

0eWs−cs4ads + dBs5

)

1 X0 ∈�

for c > 0 and a ∈ �, where 8Wt9t∈601�5 and 8Bt9t∈601�5

are independent Wiener processes, then 8Xt9t∈601�5

satisfies

dXt =[(

c+ 12

)

Xt − a]

dt −XtdWt − dBt1 t ∈ 601�5

and

limt→�

Xt = −� on the set{

∫ �

0eWs−cs4ads + dBs5 >X0

}

1

limt→�

Xt = +� on the set{

∫ �

0eWs−cs4ads + dBs5 <X0

}

0

Now if we define the function G by G4x5=∫ x

−�g4y5dy

for all x ∈�, where the function g is

g4y5=Ce2a tan−1 y

41 + y25c+1/21 y ∈�1

with the constant C > 0 chosen so that∫ +�

−�g4y5dy = 1,

then it is easy to check that

12g

′4y541 + y25+ g4y5[(

c+ 12

)

y− a]

= 01

which in turn implies that Yt =G4Xt5 is a martingale.Clearly, this martingale satisfies the limits (15). More-over, a simple application of Itô’s formula shows that8Yt9t is a solution of the stochastic differential Equa-tion (16) with ä4y5= g4G−14y55

√

1 +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∈601�6 for

Xt = et/2

(

x0 +

∫ t

0e−s/2 dWs

)

1 x0 ∈�1 for all t ∈ 601�50

and 4zs5s∈601T 5 is a positive-valued, continuous func-tion, then the 40115-valued process

at = Y∫ t0 zs ds

1 t ∈ 601T 51

satisfies

dat =ê′4ê−14at55√ztdW̃t1 t ∈ 601T 5

for the process 4W̃t5t∈601T 5 defined by

dW̃t =1

√zt

dMt1(

Mt =W∫ t0 zs ds

)

t∈601T 51

which follows a Brownian motion with respect to thetime changed filtration

F̃t =F∫ t0 zs ds

1 t ∈ 601T 50

Moreover,

limt→T

∫ t

0zs ds = +� implies that �

{

limt↗T

at ∈ 80119}

= 10

4. Model Parametrization andCalibration

We now show how to calibrate the basic modelintroduced in §3.1. As explained above, historicaloption prices are not appropriate for calibration inthe present state of the market, and we limit our-selves to historical calibration of the model. Note thataccording to Proposiotion 1, the choice of the function401T 5 3 t ↪→ �s affects only the time-change 4zt5t∈401T 5.Moreover, Proposition 1 shows that when modelingthe random variable âT by (7), we must assume thatthe function 401T 5 3 t ↪→ �s is not constant. Indeed,a constant volatility

�s ≡ �̄ ∈ 401�5 for all s ∈ 601T 7

would give, independently on the choice of �̄ , thesame process

at =ê

(

ê−14a05√T +Wt

√T − t

)

(17)

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with dynamics

dat =ê′4ê−14at551

√T − t

dWt0 (18)

Thus, with a constant and deterministic �̄ it is impos-sible to match both observed allowance prices andtheir observed (instantaneous) fluctuation intensity.Indeed, the entire process is completely determinedby the value of a0. This suggests that we introduceextra degrees of freedom in (18). In this paper, wechoose to work with the model

dat =ê′4ê−14at55√

�4T − t5−�dWt1 (19)

parameterized by � ∈ � and � ∈ 401�5. This leads toa parametric family of functions 4�s5s∈601T 7, which wedenote by

4�s4�1�55s∈401T 51 �≥ 11 �> 01 (20)

and we show how to calibrate the parameterized fam-ily (20) to historical data.

As seen from (7), the function 401T 5 3 s ↪→ �s entersthe dynamics of 4at5t∈601T 7 indirectly through the time-change function 401T 5 3 t ↪→ zt defined in (11). Thecorrespondence between the functions � and z is elu-cidated in the following lemma.

Lemma 1. (a) Given any square-integrable continu-ous and positive function 401T 5 3 s ↪→ �s , the function401T 5 3 t ↪→ zt defined by

zt =�2t

∫ T

t�2u du

1 t ∈ 401T 51 (21)

is positive and continuous and satisfies

limt↗T

∫ t

0zu du= +�0 (22)

(b) Conversely, if the positive and continuous function401T 5 3 t ↪→ zt satisfies (22) then the function 401T 5 3

s ↪→ �s defined by

�t =

√

zte−∫ t

0 zu du1 t ∈ 401T 51

is positive and continuous and satisfies (21).

Proof. (a) Let us write (21) as zt�t = �2t for t ∈

401T 5, where

�t =

∫ T

t�2u du for all t ∈ 601 T 70

Then, �̇t = −�2t for t ∈ 401T 5 and � satisfies the differ-

ential equation zt�t = −�̇t for t ∈ 401T 5. Its solution isgiven by

�t = �0e−∫ t

0 zu du1 t ∈ 601T 50

From the terminal condition �T =∫ T

T�2u du = 0, we

get (22).

(b) Let us now suppose that 4zt5t∈401T 5 is positiveand continuous and satisfies (22), and let us definethe positive and continuous function 4�t5t∈601T 5 by

�t = e−∫ t

0 zu du1 t ∈ 601T 50 (23)

Clearly, it satisfies �̇t = −zt�t for t ∈ 401T 5, andbecause the divergence of the integral implies that�T = 0, we have

�t = −

∫ T

t�̇u du1 t ∈ 401T 5 and − zt = −

�̇t∫ T

t�̇u du

0

(24)Setting �2

t = −�̇t for t ∈ 401T 5, (21) is satisfied. More-over, this function is positive, continuous in the openinterval, and integrable because 1 = �405= −

∫ T

0 �̇u du,which follows from (23) and (24). Consequently, thefunction defined by �t 2=

√

�2t for t ∈ 401T 5 is square

integrable, continuous and positive and is related to4zt5t∈401T 5 by (21), as required. �

We return to the expression (7) for âT , using nowthe targeted family (20) to determine the stochas-tic differential Equation (19). In light of the previouslemma, the function

4zt4�1�5= �4T − t5−�5t∈401T 5 (25)

must satisfy (22), implying the following restrictionson the parameters � and �:

�> 0 and �≥ 10 (26)

However, we will let � vary freely over � for calibra-tion purposes, interpreting the fitted values in light ofthese conditions.

Remark 3. If we use the parametric familyzt4�1�5 = �4T − t5−�, then the actual time change isgiven by the integral

∫ t

0zs ds =

�4log4T 5− ln4T − t55 if �= 11

�

1 −�6T 1−�

− 4T − t51−�7 otherwise0

Notice that � is a multiplicative parameter in thesense that zt4�1�5 = �zt4�115. Also, the emissionvolatility 4�t4�1�55t∈401T 5 associated to the parameteri-zation 4zt4�1�55t∈601T 7 is given by

�t4�1�52

= zt4�1�5e−∫ t

0 zu4�1�5du (27)

=

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

�4T − t5�−1T −� if �= 10(28)

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4.1. Historical CalibrationConsider historical observations of the futures prices4At5t∈601T 7, recorded at times t1 < t2 <1 · · · 1< tn0 result-ing in a data set �11 0 0 0 1 �n, where

�ti =ê−14ati 5=ê−1

(

1�Ati

)

1 i = 11 · · · 1n (29)

The objective measure � governing the statistics ofthe observations can be recovered from the spot mar-tingale measure � via its Radon–Nikodym density

d�

d�= e

∫ T0 HtdWt−41/25

∫ T0 H2

t dt0

For the sake of simplicity, we follow the time-honoredapproach assuming that the market price of risk pro-cess 4Ht5t∈601T 7 is constant and deterministic, Ht ≡ hfor t ∈ 601T 7, for some fixed h ∈ �. According toGirsanov’s theorem, the process 8W̃t9t∈601T 5 defined asW̃t =Wt − ht for t ∈ 601T 5 is a Brownian motion withrespect to the objective measure �; under this mea-sure �, �t satisfies

d�t =(

12zt�t +h

√zt)

dt +√ztdW̃t

and the analog of (13) reads

�� = e41/25∫ �t zs ds�t +h

∫ �

te41/25

∫ �s zu du

√zs ds

+

∫ �

te41/25

∫ �s zu du

√zs dWs (30)

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

11 0 0 0 1n, the conditional distribution of �ti given �ti−1

is Gaussian with mean �i and variance �2i given by

�i4h1�1�5=e41/25∫ titi−1

zs ds�ti−1+h

∫ ti

ti−1

e1/2∫ tis zu du

√zs ds1 (31)

�2i 4h1�1�5=

∫ ti

ti−1

zse∫ tis zu du ds1 (32)

Figure 2 History of Futures Prices on EUA with Maturity 2012

35

30

25

20

15

10

Time in months

Pric

e

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 t0 and �0 by convention. So for agiven realization 8�ti9

ni=1 ∈�n, the log-likelihood is

L�110001�n4h1�1�5

=

n∑

i=1

(

−4�ti −�i4h1�1�55

2

2�2i 4�1�5

−log(

√

2��2i 4�1�5

)

)

(33)

for all h1�1� ∈ �. Although there is no closed-formestimate for this parameter, the maximum of the like-lihood function can be determined numerically. Let usillustrate this procedure.

Consider historical daily prices of a futures contractwritten 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 thefollowing integral approximations:

�i4h1�1�5 ∼ e41/254ti−ti−15zti−1 �ti−1

+h4ti − ti−15√

zti−1e41/254ti−ti−15zti−1 1 (34)

�2i 4h1�1�5∼ 4ti − ti−15zti−1

e4ti−ti−15zti−1 0 (35)

Starting with initial parameter h 2= 0, �= 005, �= 005,a numerical method for unconstrained maximizationreturned the maximizer

�∗= 003321 �∗

= 001611 h∗= −000780 (36)

To confirm the validity of our procedure, we deter-mine the residuals

wi =�ti −�i4h1�1�5√

�2i 4h1�1�5

1 i = 11 0 0 0 1n0 (37)

Under the model assumptions, this series must be arealization of independent standard normal random

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Figure 3 Statistical Analysis of Series 4wi 5ni=1

25 321

0

1

2

3

0–1

–1

–2–3

–3

2015

Lag Theoretical quantiles

Time in months

01/0

7

02/0

7

04/0

7

05/0

7

06/0

7

07/0

7

08/0

7

09/0

7

10/0

7

11/0

7

12/0

7

01/0

8

02/0

8

04/0

8

05/0

8

06/0

8

07/0

8

08/0

8

09/0

8

10/0

8

Sam

ple

quan

tiles

0

1

2

3

–1

–3

AC

FW

1050

0.0

0.2

0.4

0.6

0.8

1.0

variables. Thus, standard statistical analysis of theseresiduals can be applied to verify the the quality ofthe model fit. In Figure 3 we show this series, itsempirical autocorrelation function (ACF), and its QQ-plot. The empirical properties of the residuals are notcompletely in line with model assumptions, but thereader should keep in mind that risk-neutral modelsare 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 theprice evolution with respect to the risk premium cor-rected pricing measure.

Remark 4. The original rationale for the choice ofour basic model was based on equilibrium considera-tions and the assumption that the terminal allowanceprice was binary. However, the real market EU ETS(second phase), whose data are used for historical cal-ibration, operates under uncertainty. One of the majorprice determinant here is the unknown impact of theinternational credits, the so-called Certified EmissionReductions, or CERs. Most likely, market participantsbelieve that a significant number of cheap interna-tional credits will be used to fulfill the compliancewithin the EU ETS if needed, and that noncompliancebecause of a shortage of certificates will not occurat compliance time. Under such condition, the dis-tribution of the terminal allowance price should notbe 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 penaltyof 100 euros. It is interesting to see that historical dataseem to reflect this concern, suggesting a value for theparameter � below 1 which would yield a martingalewith a nondigital terminal value because the integralgiving the time change does not diverge when �< 1!

Remark 5. The above maximum likelihood calibra-tion from historical data used strongly the explicitform (8) of the normalized allowance price and theGaussian property of the diffusion 4�t5t∈601T 7 givenby 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 ämore general than the specific ä =ê′ �ê−1, the max-imum likelihood estimates of �, � and h can becomputed using Aït-Sahalia (2002) approach.

4.2. Option PricingNow we turn our attention to the valuation ofEuropean call options written on allowance futuresprice 4At5t∈601T 7. The payoff of a European call withmaturity � ∈ 601T 7 and strike price K ≥ 0 is givenby 4A� −K5+. Under the assumption that the savingsaccount 8Bt9t∈601T 7 is given by Bt = e

∫ t0 rsds for t ∈ 601T 7

for some deterministic short rate 8rs9s∈601T 7, this pricecan be computed in the model proposed in this paper.

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Proposition 2. In a one-period 601T 7 compliancemodel, with risk-neutral parameters � ∈ � and � > 0, theprice of a European call with maturity � ∈ 601T 7 and strikeK ≥ 0 written on an allowance futures maturing at the endT of the compliance period is given at time t ∈ 601 �7 by

Ct = e−∫ �t rsds

∫

�4�ê4x5−K5+N4�t1�1�

2t1�5 4dx51 (38)

with �t1� and �2t1� given by formulas (39) and (40) below.

This result is obtained by a straightforward calcu-lation because the conditional distribution of �� given�t is Gaussian with mean

�t1�4�1�5

=

�t

(

T − t

T − �

)�/2

if �= 11

�t exp[

�

241 −�564T − t51−�

− 4T − �51−�7

]

if � 6= 1

(39)

and variance

�2t1�4�1�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 � onoption prices. In the following example, we fix �= 1and assume that the penalty is � = 100. Further-more, we suppose that at the initial time t = 0 fouryears prior to the compliance date T = 4, the price ofa 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 withstrike price of K = 25 and varying maturity date � ∈

601T 7. The option price is calculated from (38) at timet = 0. In Figure 4, we also illustrate the dependence ofthe option price upon the parameter � (recall Figure 1for a plot for fixed �). Comparing three cases �= 005,�= 008, and �= 101, Figure 4 shows that the call priceis increasing in �. Less surprisingly, the dependenceon � shows that longer-maturity calls (with the samestrike) are more valuable than their short-maturitycounterparts.

Let us stress that although there are no closed-formformulas for call prices, their numerical evaluationscan be performed very efficiently.

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

0 1 2 3 4

2

4

6

8

10

12

14

Time in years

Cal

l pric

e

Notes. The graphs marked by �, Î, and È stand for � = 005, � = 008,and � = 101, respectively. The values of the other parameters are given inthe text.

5. Implied CalibrationBecause emission futures prices are supposed to fol-low a bounded martingale in 6011007, the option pricemust vanish for strike prices exceeding the penaltylevel 100. At this point, our valuation technique(38)–(40) differs from the traditional Black 76 formulafrequently applied to price options on futures. How-ever, it turns out that the difference is rather minor,for parameter values relevant to the current situationof the EU ETS and for low strike prices. As an illus-tration, we plotted in Figure 5 the call option pricewith maturity �o = 3044 depending on the strike K,calculated at t = 0 for the underlying futures priceof A04�f 5= 17054, supposing that the futures contractmatures at �f = 3046. The calculation is done for twocases, the Black 76 model with volatility � = 0045 andour pricing formula (38)–(40) for the maximum likeli-hood parameters

�∗≈ 003321 �∗

≈ 00161 (41)

from (36). In all three cases, we have supposed thatthe time to compliance date is T = 5 and set theshort rate at r = 0007. Figure 5 shows that for lowstrikes K in the range 601207, Black 76 call pricesare very similar to those given by our model withmaximum-likelihood parameters. Although for strike

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

0

0

5

Opt

ion

pric

es 10

15

20 40 60 80

Black 76 price with sigma = 0.45Theoretical price with � = 0.332, � = 0.161

k100

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Figure 6 Evidence that European Calls Quoted on ECX Are MostLikely Priced Using Black 76 Formula

0 2 4 6 8

B76

10

8

6

4

2

0

Mar

ket

Market prices vs Black 76at 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 traditionalBlack 76 formula is considered to be useful for EUAoption pricing. As mentioned at the beginning, theEuropean Climate Exchange (ECX) provides beyondcall option quotas also the implied volatilities associ-ated with these quotas via Black 76 formula. In mostcases, the implied volatility remains in the interval60040100657. Based on a representative history of dailycall option prices quoted at the ECX during the periodJanuary 4, 2007–September 30, 2009, we plot in Fig-ure 6 the theoretical Black 76, where we have adoptedthe quoted volatility and set the short rate at r = 0005,against market price of the option. Although we couldnot reach a complete coincidences due to unknownhistorical interest rates entering these option prices,this figure strongly suggests that the traders polledby 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 volatilitiesquoted by the ECX, and the same short rate of r =

0005. Clearly, these data are more scattered, but theconcentration of price points across a straight lineshows that the option pricing practice of the ECX canbe approximatively described by the Black 76 formulawith a constant volatility of 0045.

Figure 7 Black 76 Formula with Volatility 0045 Explains MarketCall Prices

0 2 4 6 8

10

8

6

4

2

0

B76

Mar

ket

Market prices vs Black 76at volatility of 0.45

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

0

0

2

2

4

4Mar

ket

6

6

Theoretical call price8

8

10

10

Market pricesvs model price

To compare our technique to the market optionprices, we decided to fit our model to actual optionprices. To do so, we suggest determining those param-eters � and � that minimize the sum of squareddeviations between historical market prices and theirtheoretical values, based on our model (38)–(40). Hav-ing implemented the function describing the sum ofsquared deviations depending on model parameters �and �, we applied a numerical procedure based on theNelder–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 andimplied calibration are usually interpreted as modelvalidation. However, the deviations of market callprices from their theoretical values plotted in Figure 8are still strong. This will change when the optionmarket matures, the market participants realize thedifferences between allowance price evolutions andmodel assumptions underlying Black 76 formula, andthe ECX finally changes its way of reporting optionprices.

6. Multi–Compliance Periods MarketsSo far, we focused on a generic cap-and-trade schememodeled after the first phase of the EU ETS, namely,limited to one compliance period and without bank-ing in the sense that unused allowances becomeworthless at the end of the period. This is a strongsimplification because as already mentioned above,real-world markets are operating in a multiperiodframework. Furthermore, subsequent periods are con-nected by market specific regulations. In what fol-lows, we consider an abstract but generic model ofsuch a market and focus on most natural rules for theperiod interconnection.

Presently, three regulatory mechanisms connectsuccessive compliance periods in a cap-and-tradescheme. 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 thepresent one.

• Banking allows for the transfer of a (limited)number of (unused) allowances from the presentperiod into the next.

• Withdrawal penalizes firms that fail to comply intwo ways: by penalty payment for each unit of pollu-tant not covered by credits and by withdrawal of themissing allowances from their allocation for the nextperiod.

From the nature of the existing markets and thedesigns touted for possible implementation, it seemsthat policy makers tend to favor unlimited bankingand forbid borrowing. Furthermore, the withdrawalrule 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 becauseeach success (respectively, failure) in the previousperiod results in stronger (respectively, weaker) abate-ment in the subsequent periods.

6.1. Market ModelFor the remainder of this section, we consider a two-period market model without borrowing but withwithdrawal and unlimited banking. We denote thetwo periods by 601T 7 and 6T 1T ′7 and consider astochastic basis 4ì1F1�1 4Ft5t∈601T ′75 with a distinctmeasure �∼�, which we view as the spot martingalemeasure. Further, we introduce processes 4At5t∈601T 7,4A′

t5t∈601T ′7 for the futures contracts with maturities atcompliance dates T , T ′ written on allowance pricesfrom the first and the second period, respectively.To exclude arbitrage, we suppose that the prices4At5t∈601T 7 and 4A′

t5t∈601T ′7 are martingales with respectto the spot martingale measure �. Noncompliance inthe first and second periods occurs on events N ∈FT

and N ′ ∈FT ′ , respectively. As before, we assume thatthe savings account 4Bt5t∈601T ′7 is given by

Bt = e∫ t

0 rs ds1 t ∈ 601T ′7 (43)

for some deterministic short rate 4rs5s∈601T ′7. The resultsof the previous section imply that in the case ì\N ofthe first-period compliance, the allowance price drops

AT 1ì\N = �A′

T 1ì\N 1 (44)

where � ∈ 401�5 stands for discount factor describingthe interest rate effect

�= BTB−1T ′ = e−

∫ T ′

T rs ds0

The relation (44) is justified by considering spotprices. The random variable �A′

T is nothing but thespot price at time T of the second-period allowance.Because futures and spot price agree at maturity, AT

must be the spot price of the first-period allowanceat T . In the case of compliance in the first period, theunused allowances can be banked; hence we have theequality in (44).

In the case of noncompliance at the end of the firstperiod, the withdrawal regulation implies that

AT 1N = �A′

T 1N +�1N 0 (45)

Namely, the noncompliance in one pollutant unit attime T costs a penalty � in addition to one allowancefrom the next period, which must be withdrawn atthe spot price �A′

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

that the difference is

At−�A′

t = Ɛ�4AT −�A′

T �Ft5= Ɛ�4�1N �Ft51 t ∈ 601T 7

and must be modeled as 801�9-valued martingale. Wesuggest using the same methodology as in the one-period model

At −�A′

t =�ê4�1t 51 t ∈ 601T 71 (46)

where the Gaussian process 4�1t 5t∈601T 7 is introduced as

previously in (12), with 4�415s 5s∈601T 7 in parameterized

form (28) and driven by a process 4W 1t 1Ft5t∈601T ′7 of

Brownian motion.To model the second-period allowance futures

price, a continuation of the cap-and-trade systemmust be specified. If there is no agreement on long-term regulatory framework (as is the case for most ofthe existing emission markets), the process 4A′

t5t∈601T 7should be specified exogenously. The simplest choicewould be a geometric Brownian motion with constantvolatility. 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. Inthis case,

A′

t = Ɛ�4�1N ′ �Ft51 t ∈ 601T ′7

can also be modeled as in the one-period model

A′

t =�ê4�2t 51 t ∈ 601T ′70 (47)

Again, 4�2t 5t∈601T 7 is introduced as in (12), with a pro-

cess 8� 425s 9s∈601T 7 chosen in parameterized form (28) and

driven by another Brownian motion 4W 2t 1Ft5t∈601T ′7.

6.2. Option PricingAs an application of our two-period model, we con-sider pricing of European calls. Consider Europeancall option with strike price K ≥ 0 and maturity� ∈ 601T 7 written on futures price of allowance fromthe first period. This contract yields a payoff

C� = 4A� −K5+ at time � ∈ 601T 7.

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Carmona and Hinz: Risk-Neutral Models for Emission Allowance Prices and Option Valuation1466 Management Science 57(8), pp. 1453–1468, © 2011 INFORMS

Under the assumptions of the previous section, westart with the computation of the price C0

C0 = e−∫ �

0 rs dsƐ�44A� −K5+5

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

4A� −K5+ = 4A� −�A′

� +�A′

� −K5+1

we utilize our modeling of 801�9-valued martingales(46) and (47) to express the terminal payoff as

4A� −K5+ = 4�ê4�1� 5+��ê4�2

� 5−K5+

with expectation

C0 = e−∫ �

0 rsdsƐ�44A� −K5+5

= e−∫ �

0 rsdsƐ�44�ê4�1� 5+��ê4�2

� 5−K5+5

= e−∫ �

0 rsds∫

�24�ê4x15+��ê4x25−K5+

·N4��1 ��54dx11dx251 (48)

where N4��1 ��5= F�1� 1 �

2�

stands for joint normal distri-bution of �1

� and �2� .

Let us derive the mean �� and the covariancematrix �� under the standing assumption �1 = �2 = 1,for �1 > 0, �2 > 0. We have

�1� = ê−1

(

A0 −�A′0

�

)

√

(

T

T − �

)�1

+�1/21

∫ �

0 4T −u54�1−15/2W 1u du

4T − �5�1/21

�2� = ê−1

(

�A′0

�

)

√

(

T ′

T ′ − �

)�2

+�1/22

∫ �

0 4T′ −u54�2−15/2W 2

u du

4T ′ − �5�2/20

Denoting by � the correlation of the two Brownianmotions 4W 1

t 5t∈601T ′7 and 4W 2t 5t∈601T ′7

6W 11W 27 dt = �dt1 � ∈ 6−11171

we can apply the same argumentation to obtain themeans

�1� = Ɛ4�1

� 5=ê−1

(

A0 −�A′0

�

)

√

(

T

T − �

)�1

1

�2� = Ɛ4�2

� 5=ê−1

(

�A′0

�

)

√

(

T ′

T ′ − �

)�2

3

the variances

�111� = Var4�1

� 5=

(

T

T − �

)�1

− 11

�212� = Var4�2

� 5=

(

T ′

T ′ − �

)�2

− 13

and the covariance as

�112� = �211

� = Cov4�1� 1 �

2� 5

= �1/21 �1/2

2

∫ �

0 4T −u54�1−15/24T ′ −u54�2−15/2�du

4T − �5�1/24T ′ − �5�2/20

At times t ∈ 601 �7 prior to maturity, the price Ct of thecall is obtained similarly.

Proposition 3. In a two–compliance periods model asabove, with parameters �11�2 > 0, and � ∈ 4−1115, theprice of the European call with strike price K ≥ 0 and matu-rity � ∈ 601T 7 written on first-period allowance futuresprice is given at time t ∈ 601 �7 by

Ct = e−∫ �t rs ds

∫

�24�ê4x15+��ê4x25−K5+

·N4�t1�1 �t1�54dx11dx25 (49)

with mean �t1�

�1t1� =ê−1

(

At −�A′t

�

)

√

(

T − t

T − �

)�1

1 (50)

�2t1� =ê−1

(

�A′t

�

)

√

(

T ′ − t

T ′ − �

)�2

1 (51)

and covariance matrix �t1�

�111t1 � = Var4�1

� 5=

(

T − t

T − �

)�1

− 11 (52)

�212t1 � = Var4�2

� 5=

(

T ′ − t

T ′ − �

)�2

− 11 (53)

�112t1 � = �211

t1 � = �1/21 �1/2

2

∫ �

t4T −u54�1−15/24T ′ −u54�2−15/2�du

4T − �5�1/24T ′ − �5�2/20

(54)

If we take a closer look at the computations in-volved in the valuation of the call price

Ct =Ct4�1T 1T′1A01A

′

01K1 r1�11�21�5

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

Ct4�1T 1T′1A01A

′

01K1 r1�11�21�5

=C04� − t1 T − t1 T ′− t1A01A

′

01K1 r1�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 ofthe two-dimensional normal distribution. To ease thenotation, let us skip t1 � to write �i = �i

t1 � , �i1 j = �i1 jt1 �

for i1 j = 112. It holds

N4�1�54dx11dx25

=N4�11 c4x251 �1111 c54dx15N 4�21 �21254dx251 (55)

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

−4�21152

�2120

With factorization (55), the inner integral is calculatedexplicitly in the following cases:∫

�4�ê4x15+��ê4x25−K5+N4�11 c4x251 �

1111 c54dx15

=

0 if K −��ê4x25≥�1

�ê

(

�11 c4x25√

1 + �1111 c

)

+��ê4x25−K

if K −��ê4x25≤ 00

That is, the numerical valuation is required only inthe case 0 <K −��ê4x25 < �, where

∫ �

ê−14K/�−�ê4x2554�ê4x15+��ê4x25−K5

·N4�11 c4x251 �1111 c54dx15

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 �15 7→ C04� 1 �15 asFunction of Maturity � and �1 for Correlation � = 008 (Top)and �= −008 (Bottom)

1

2

3

0.6

0.8

1.01.2

2

4

6

8

Maturity

Maturity

1

2

3

Beta

Beta

0.6

0.8

1.01.2

Cal

l pric

eC

all p

rice

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 notencounter any problem implementing this formula.

For the sake of completeness, we illustrate thedependence of the call price on �1 and maturity ofthe call. To make the results comparable with the one-period example given above, we chose the followingparameters: four years to the first-period compliancedate T = 4, eight years to the second-period compli-ance date T2 = 8, initial first-period allowance futuresprice A0 = 25, initial second-period allowance futuresprice A′

0 = 15, strike price of the European call K = 25,interest rate r = 0005, and �2 = 002. Figure 9 depictsthe dependence of the call price on the value of �1 forthe first period and of the call maturity � .

7. ConclusionMandatory emission markets are being establishedthroughout the world. In the most mature market,the EU ETS, beyond physical allowances, a largevolume of allowance futures is traded. Furthermore,European options written on these futures have beenintroduced and traded although no theoretical foun-dation for their pricing is available yet.

The goal of this work is to fill this gap. In ouranalysis, we gradually move from a one-period mar-ket model to a more realistic situation of two-periodmarkets (covering the present EU ETS regulations)and show that martingales finishing at two-valuedrandom 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 historicalor implied volatility of the underlying future. Froma hedging perspective, this issue could be one ofthe most desirable model properties. Other practi-cal aspects like ease of calibration and simple optionvaluation schemes are also fulfilled in our approach.We show how parameters can be estimated from his-torical price observation and suggest efficient optionvaluation schemes. Although option price formulasare not available in a closed form, a simple and fastnumerical integration can be applied.

AcknowledgmentsThe research of René Carmona was partially supported byNSF-FRG-DMS-0455982 and DMS-0806591. The research ofJuri Hinz was partially supported by WBS R-703-000-020-720/C703000 of the Risk Management Institute and by WBSR-146-000-107-133 of the National University of Singapore.

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