Carbon Derivatives Pricing - An Arbitrageable Market

E vidence of extreme volatility in the European permit markets suggests the urgent need for the development of selective hedging techniques such as futures contracts and option instruments. As a result, a valid price model is

required for pricing !nancial instruments or projects whose value derives from the future carbon dioxide (CO2) spot permit price. Due to the recent introduction in the market of option-like instruments for hedging purposes, various models were developed to approximate the dynamics of spot prices for CO2 emission allowances. Di"erent approaches to the modelling of European Union emission allowances(EUAs) #ourished in the literature, ranging from equilibrium models considering one trading period, to models derived from empirical studies based on the !rst two phase periods.

1 Modelling the dynamics of CO2 emissionsPresently, the relevant spot price history is still short and may be distorted by potential one-time e"ects due to the market’s immature state. Nonetheless, various authors considered the qualitative and quantitative properties of the data, based on time-series analysis and distribution analysis of the time series, to devise pricing models. However, as opposed to equilibrium models, all these empirical models do not yet consider the fundamental properties of permit contracts.

1.1 Abatement opportunitiesOne of the key understandings in economics is that in an e$cient market, the equilibrium price of emission allowances is equal to the marginal costs of the cheapest pollution abatement solution. Hence, at a given time t, a company emitting CO2 has to decide whether to invest in infrastructure to reduce fuel production or delay the investment to a future time, and instead buy emission allowances. However, the fact that a market-based approach leads to an e$cient allocation of abatement costs across di"erent polluters strongly depends on the assumption that any technological abatement

solution is perceived as a perfect substitute for emission allowances. Unfortunately, this is no longer true in the presence of uncertainty and Chao et al (1993) proved that most abatement technologies are perceived as durable and irreversible investments compared with emission allowances, which are seen to provide a greater #exibility in adapting to changing conditions.

Consequently, few options are available to the majority of companies a"ected by emission trading and even fewer fall into the list of short-term abatement possibilities. As a result, it is reasonable to assume that companies optimise their cost function by continuously adjusting their permit portfolio allocations and by choosing the optimal permit amount to purchase or sell, considering the payment of the penalty as an alternative to compliance (see Chesney et al (2008)).

1.2 An equilibrium approachIn 2006, when insu$cient price history for the European Union’s emissions trading scheme (EU ETS) existed, Seifert et al (2006) considered a theoretical equilibrium model that incorporated the key properties of the EU ETS, such as abatement and penalty costs, and deduced the main properties of the resulting spot price process.

%ese results should be accounted for when explicitly modelling a CO2 spot price process. Fehr et al (2007) showed that the success of projects including a carbon !nance component is determined by the correct valuation of their returns, whose cash#ows are equivalent to derivatives written on future carbon prices.

Further, they showed that the spot price must be positive and bounded by the penalty cost and the cost of delivering any lacking allowances. Later, this idea was pursued by other authors such as Chesney et al (2008) and Carmona et al (2009a) to name just a few. Assuming the existence of a single representative !rm, they considered a pollution process with an initial pollution level, initial permit endowment and with corresponding boundary conditions.

In CO2 equilibrium models, permit prices are positive and bounded by the penalty level. To obtain closed-form solutions to the pricing of CO2 derivatives, Daniel Bloch models the permit price as a function of a positive unbounded process, and shows that there is no equivalent probability measure whereby the discounted spot price is a martingale

64 EnergyRisk.com March 2011

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CARBON DERIVATIVES PRICING:AN ARBITRAGEABLE MARKET

March 2011 EnergyRisk.com 65

Cutting edge

Assuming that the company’s pollution dynamics are exogenous processes, they used that process to derive the price of the tradable permit by minimising the company’s total cost. %at is, the tradable permit is a function of the pollution process, which is the unique source of risk in that model. It is interesting to note that since the pollution process is not a tradable asset, its risk can not be hedged away, and Chesney et al (2008) concluded that there is no need to construct a risk-neutral probability measure for the pollution process. %ey are therefore implicitly assuming that the underlying asset is not a tradable asset and that consequently, its discounted price does not need to be a martingale under the risk-neutral measure.

However, the CO2 permit is a tradable asset – and to avoid arbitrage when estimating its value – one should construct an equivalent probability measure such that the discounted spot price is a martingale under that measure. In that setting, one can expect either a shortage or a surplus situation at the end of the trading period, meaning the company will either be holding worthless permits or paying the penalty costs. Hence, in a wait-and-see situation, the payout at maturity is that of an Asian call option with a #oating strike price so that emission allowances are option contracts. %e expected future permits net position is valued by minimising the total cost of the !rm, computed as the sum of the cash#ows at the initial time and the potential penalties at the end of the period. As a result, the spot price is a function of the penalty level and the probability of a permit-shortage situation

S t,T( ) = pe�r T�t( ) �P ce 0,T( ) > N Ft( )where ce(0,T) models the cumulative emissions in the trading period [0,T], N is the amount of emission allowances handed out by the regulator and p is the penalty cost. As the pollution process is uncertain, the emission allowance price lies between zero and the discounted penalty level StD [0, e–r(T–t)p]. Each author speci!es a di"erent process for the method of approximation of cumulative emissions ce(0, t). In order to compute the probability P(.), each author speci!es the cumulative emissions with a di"erent process ce(0, t) = 0 t0Y(s)ds where Y(.) is the emission rate following a particular process. %erefore, given the drift and di"usion terms of the emission rate one can deduce the CO2 spot price. For ease of computation, the emission rate is usually normally or lognormally distributed so that the probability of a permit-shortage situation is the cumulative distribution function of a standard normal random variable. However, permit prices are inherently prone to jumps – therefore, to enable equilibrium models to reproduce these jumps, the distribution of the emission rate must be modi!ed accordingly (see Grull et al (2010)).

1.3 An empirical approachVarious authors have presented empirical studies using spot and futures prices suggesting that CO2 emission allowances price levels are non-normally distributed with fat tails, and that the price dynamics are nonstationary and exhibit abrupt discontinuous shifts. For instance, Daskalakis et al (2006) considered various popular jump-di"usion processes from the equity market to approximate the dynamics of CO2 allowance contracts, using a maximum likelihood (ML) approach to estimate the model parameters. %ey showed that the Merton model (1976) had the best performance in terms of likelihood and parsimony. Further, Gagliardi (2009) stressed that volatility exhibits clustering over time and proposed a Heston model combined with jump components. %eir !ndings imply that the

EUAs have a proportional, non-mean reverting structure with jumps. In their articles, they did not check for no-arbitrage conditions, and considered the dynamics of the spot price under the historical measure without compensating the drift for the jump process. As a result, they performed analysis with an incorrect expected value of the spot price. Unfortunately, these models do not take into consideration the properties of the CO2 spot price described in section (1.2). More precisely, the processes involved do not consider the penalty function and as a result are not bounded, which is in contradiction with the characteristics obtained from the equilibrium models.

In addition, the analysis performed by the authors on futures prices were done under the historical measure and the market prices of risk were not considered. As we saw earlier, it can have tremendous implications in the dynamics of the spot process, resulting in misleading interpretation of market behaviour. To illustrate our comments, we refer the readers to Paul et al (2010) who presented another empirical study on the CO2 spot prices where they considered jump-di"usion models and in particular the normal-inverse Gaussian model. %ese models were calibrated to EUA prices ranging from January 2006 to March 2009 and model parameters summarised in a table. In view of the market behaviour of the spot prices during that period of time, the spot prices are clearly bounded by the penalty costs, and the excess of permits allocated, combined with a slower global economy, pushed prices down. It is therefore not surprising that an unbounded process such as a geometric Brownian motion exhibits a negative drift, with + = –0.29.

Similarly, in the case of geometric Brownian motion with mean reversion, the alpha term is negative so that the drift becomes0.009(+ – lnS), and for + = 1, the drift is negative for most values of the spot price. To conclude, in the geometric Brownian motion with jumps, the jump size is given by +saut= –2.28, so that in all cases, the model parameters of the unbounded processes are arti!cially constrained so that their dynamics resemble those of a bounded process. To remedy these drawbacks, we are going to consider a spot process that take into consideration the properties of the CO2 emission allowances.

1.4 A risk-neutral approachTo ensure no-arbitrage in order to compute option prices, we need to derive the dynamics of the spot price or that of the future price under a risk-neutral measure. Carmona et al (2009b) addressed the problem of risk-neutral modelling of emission permits and considered models based on di"usion martingales ending up with two values. %ey assumed a one-period setting with no banking allowed, and let the carbon price at the compliance date be a random variable taking only the values zero and p. Because of the digital nature of the terminal allowance price, they focused on the event of non-compliance and modelled the future price to match the terminal condition. It amounts to modelling a hypothetical positive-valued random variable K exceeding the boundary condition one at the end of the period. To match the recent allowance price and the observed instantaneous #uctuation intensity, they assumed a deterministic volatility function for the process Kt , leading to a local volatility di"usion process for the allowance price. Further, they devised a non-compliance process not hitting zero or one in a !nite time with probability one, and as a result they identi!ed classes of martingales taking values in the interval (0,1) for the normatised future price process. %en, they extended their model to a two-period setting by assuming that the cap-and-trade system is terminated at the end of the second period.


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2 An alternative approachWe propose to bridge the gap between theory and observed market price behaviours by considering a stochastic di"erential equation for the carbon permit price that satis!es the fundamental properties of the permit contracts. Rather than working directly with a bounded process and focusing on the event of non-compliance, we are going to consider a positive exogenous process bounded by the penalty cost. Similarly to Carmona et al (2009b), the dynamics of the normalised spot price take values in the interval (0,1) but it is not forced by a digital payout at the end of the trading period. In general, the valuation of forward and futures contracts in the commodity market can be divided into two groups. %e !rst group considers a risk premium to derive a model relating short-term and long-term prices, while the second group is closely linked to the cost and convenience of holding inventories. In order to de!ne our model in a non-arbitrage market, we are going to consider both approaches and try to compute an equivalent probability measure.

2.1 Accounting for the penalty costWe let [Ti–1, Ti ] be the ith trading period and consider the penalty costs K(Ti ) for lacking EUAs during the entire ith trading period. Assuming multi-period trading, allowing for unlimited banking and forbidding borrowing, we let the strike K(t) be piecewise constant given by K(t) =-n

i=1ki I [Ti–1, Ti ](t) where ki is a positive constant in the ith trading period. Taking into consideration the properties of the spot price described in section (1.2), we let (Xt)t * 0 be a positive process describing the unbounded spot price of emission allowances in the range [0,'), and de!ne the CO2 spot price St as

St =min Xt , K t( )( ) = Xt � Xt �K t( )( )+ = K t( )� K t( ) – Xt( )+ (2.1)

where, the spot price St is a positive process taking values in the interval [0,K(t)]. We !rst concentrate on modelling the spot price within the single period of time [0,T ]. If constant penalty costs p are paid at the end of the trading period, their values at time t are K(t)= e –r(T– t)p, while if they are paid at any time tD [0,T ], their values remain constant. Assuming the latter, the dynamics of the spot price are given by

dSt = �d K � Xt( )+

which is equivalent to the dynamics of a put option on the unbounded process. %e process Xt has the predictable representation property, which is not the case of the discontinuous process St , so that the market for options on the spot price is incomplete.

2.2 The dynamics of the spot priceGiven the unbounded spot price X=(Xt)tD [0,T ], we consider the derivative price St= f(t, Xt) written on the underlying X. We choose to work with semimartingales, since in that framework stochastic integration and nonlinear transformations are stable. Without loss of generality, and with the aim of presenting our idea clearly, we let the dynamics of the unbounded spot price X under the historical measure P be given by this stochastic di"erential equation (SDE)

dXtXt

=�X dt +� X dWX t( ) (2.2)

where +X is the historical drift and mX is the volatility of the process, and satisfying the usual conditions for the SDE to have a unique solution. We consider the convex function C(Xt) of the underlying price given by the call payout C(Xt)= (Xt–K)+ and apply Tanaka’s formula to get its dynamic

dC Xt( ) = I Xt�K{ } �XXtdt +

+ I Xt�K{ } �XXtdWX t( )+ 12� Xt �K( )� X

2 Xt2dt

where b(.) is the Dirac function. In the case where the strike is a function of time, we get the extra term I{Xt *K}(dK(t)/dt)/dt. So, given Equation (2.1), the dynamics of the spot price are

dSt = dXt � dC Xt( )

=�XXt I Xt< K{ }dt +� XXt I Xt< K{ }dWX t( )� 12� Xt �K( )� X

2 Xt2dt

where I{Xt<K} = 1– I{Xt *K} . %e volatility of the spot price mS=mXXt I{Xt<K}is bounded and equal to zero when the unbounded process X is either equal to zero or greater or equal to the strike, which is in accordance with the properties of the theoretical spot price given in section (1.2). However, it contradicts one of the fundamental properties of a tradable asset, which states that the volatility must be almost surely not zero to make the hedge possible (see Shreve (2004)). %at is, if the volatility vanishes, then the randomness of the Brownian motion does not enter the spot, but it may still enter the derivative security, making the spot price no-longer an e"ective hedging instrument. Hence, we can directly conclude that the model for carbon trading is incomplete.

2.3 A no-arbitrage model2.3.1 The risk premium approachTo avoid arbitrage, we want to define an equivalent probability measure Q to the real-world probability P such that the discounted stock price is a martingale under Q. Hence, we let S

_t = e –rtSt be the

discounted spot price, which we di"erentiate with respect to time to get its dynamics under the historical measure as

dSt = re–rt Xt I Xt�K{ }dt � re

–rtKI Xt�K{ }dt + e–rt I Xt�K{ }�XXtdt +

+ e–rt I Xt�K{ }� XXtdWX t( )� 12e–rt� Xt �K( )� X

2 Xt2dt

%e standard approach would be to compare the annualised rate of return of the spot price per unit of time +X to a risk-free investment and consider +X – r as the reference parameter. However, this is no-longer the case, and instead we must consider

�X = r+�S� X + rKXt

I Xt�K{ }

I Xt�K{ }+12� Xt – K( )I Xt�K{ }

� X2 Xt

so that the market price of risk becomes

�S =

�X – r – rKXt

I Xt�K{ }

I Xt�K{ }– 12� Xt – K( )I Xt�K{ }

�X2 Xt

� X

which is only de!ned if mX ȴ 0 and XD [¡,K– ¡] where ¡ > 0 , that is, mS ȴ 0. However, we saw in section (2.2) that mS vanishes on some part of the domain. In other words, for the no-arbitrage condition to apply, the market price of risk must satisfy the Novikov condition (see Shreve, (2004)). Hence, the no-arbitrage condition is not satis!ed when the unbounded spot price Xt is greater or equal to the penalty cost K. As a result, we cannot construct the dynamics for the spot price and use the change of measure theory to express it under an equivalent probability measure. Hence, in that setting there is no equivalent probability measure such that the discounted spot price S

_t is a martingale and we must therefore consider an alternative

approach. In the next section we are going to see if we can apply the convenience yield approach to solve the problem.


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2.3.2 The convenience yield approachAssuming the existence of a bene!t or a cost attached to holding one unit of the spot price, the dynamics of the spot price under the historical measure P aredSt = I Xt�K{ }�XXt dt + I Xt�K{ }� XXt dWX t( ) – 1

2� Xt – K( )� X

2 Xt2dt

where q(t, T) = -12m2X X 2

t is a convenience yield paid at all time t when Xt = K. %is is consistent with the results obtained by Borak et al(2006) where they found that a high fraction of the yields could be explained by the price level and volatility of the spot prices. Since the extra drift term q~(t, T) =-12 b(Xt –K)m 2

X X 2t is just a function of the

unbounded process Xt , there is no additional source of risk to that of the Brownian motion ŠX (t). Hence, a standard approach would be to compare the annualised rate of return of the spot price per unit of time +X to a risk-free investment and consider +X – r as the reference parameter. Once again, considering the weak form of the no-arbitrage condition, we get

I Xt�K{ }� XXt�S = I Xt�K{ }�XXt – rXt I Xt�K{ } – rKI Xt�K{ }

so that the market price of risk becomes

�S =

�X – r – rKXt

I Xt�K{ }

I Xt�K{ }

� X

which is only de!ned if mX ȴ 0 and XD [¡,K–¡] where ¡> 0 . %erefore, the no-arbitrage conditions do not apply, since the Novikov condition is not satis!ed. One possibility would be to modify the dynamics in Equation (2.2) by adding the drift term

r KXt

I Xt�K{ }

I Xt�K{ }.

However, even though the market price of risk would be bounded, the dynamics of the unbounded process would no longer be de!ned.

2.4 An arbitrage modelVarious pricing methods exist, some of which are based on hedging arguments, on the law of large numbers or – as actuaries know it – on the standard deviation principle1, to name but a few. Recognising that the discounted CO2 stock price is not a martingale under an equivalent probability measure – that is, assuming the existence of arbitrage opportunities in the carbon market – we can therefore arbitrarily de!ne the drift in the CO2 spot price. We can either assume a market price of risk and ‘price in’ the corresponding measure or we can directly price in the historical measure. Accordingly, one must rely on the actuarial pricing approach of marked-to-model. In that sense, the pricing of carbon permits derivatives is very sensitive to the assumptions made and the choice of a model for the underlying process.

2.4.1 The Black-Scholes measureGiven the dynamics of the spot price in Section (2.3.2), together with its associated market price of risk, we assume that the growth rate of the spot price is not far from the risk-free rate. %erefore, following Black & Scholes (1973), the market price of risk of the spot price becomes

�S =�X – r� X

Hence, the dynamics of the spot price under the Qh-measure are

dSt = I Xt�K{ }rXtdt + I Xt�K{ }�S� XXtdt

+ I Xt�K{ }�XXtdWX t( ) – 12� Xt – K( )� X

2 Xt2dt .

Using the change of measure, the Brownian motion Wt given bydWt = dWt +�Sdt

is a Qh-Brownian motion, and the dynamics of the spot price under the measure Qh become

dSt = I Xt�K{ }rXtdt –12� Xt – K( )� X

2 Xt2dt+ I Xt�K{ }� XXt dWX t( ) (2.3)

One of the key assumptions in mathematical !nance is that the market price of risk is not speci!c to the traded asset but to its source of noise. In our particular example, the unique source of noise of the spot price is given by the Brownian motionŠX of the unbounded spot price. Even though the unbounded spot price is not a tradable, one can not freely specify its market price of risk with respect to a particular risk aversion, as it has already been de!ned for the traded spot price. Hence, the dynamics of the unbounded spot price under the measure Qh are

dXtXt

= rdt +� X dWX t( )

which is the classical geometric Brownian motion used within the Black-Scholes formula. In our setting, the dynamics of the discounted spot price S–t under the user-de!ned probability measure are

dSt = –re–rtKI Xt�K{ }dt –

12e–rt� Xt – K( )� X

2 Xt2dt

+ e–rt I Xt�K{ }� XXtdWX t( )with extra drift term –re–rtKI{Xt *K}dt. Since the spot rate and the strike are positive, in this model, the growth rate is lower than the risk-free rate. Using a deterministic setting, Rubin (1996) provided a continuous time trading model for carbon permits and found that the prices grow in equilibrium with risk-free interest rates. Later, as a result of introducing uncertainty in Rubin’s model, Schennach (2000) showed that the expected permit price growth rate was reduced, which is also what we get in our model with the extra drift term q~(t,T) .

2.5 The forward priceGiven the de!nition of the spot price in Equation (2.1), the forward price under either the historical measure P or a measure Qh becomes

F t,T( ) = E XT Ft�� – E XT – K T( )( )+ Ft��

��

= K T( ) – E K T( ) – XT( )+ Ft��

��

We see that as the CO2 spot price has a !xed upper bound given by the penalty costs, the resulting forward price is an embedded option on the unbounded price process equivalent to the strike minus a discounted put option, and is therefore model-dependent. %at is, the forward price is a function of a European option on the unbounded spot price and its associated volatility, and as such is no-longer linear. Note, when the strike K is constant or time-dependent, as expected in a single trading period, the option is a call or put option. However, when the strike is stochastic, which could happen in multi-periods, the option becomes an exchange option. Hence, given a process for Xt , we can compute the forward price at a given time, infer the dynamics of the call or put option on X, and as a result, determine 1. P(X) = E[X] +hm[X], where hm[X] corresponds to a risk premium


Cutting edge

the dynamics of the forward price. One can choose the model of his choice, and for simplicity of exposition, following the example given in section (2.3.2) where Xt is a geometric Brownian motion, we can apply Itō’s lemma to the function V(t, Xt) and obtain the dynamics

dV t, Xt( ) = �tV t, Xt( )dt + 12� X2 Xt

2�xxV t, Xt( )dt +�xV t, Xt( )dXtwith the instantaneous volatility of the European option (the forward price on S) being

�xV t, Xt( )V t, Xt( )

Xt� X

which is bigger than the volatility mX of the spot price. Hence, in that model, the holder of a forward contract must manage extra volatility risk compared with empirical models that do not take into consideration the penalty costs. Further, call option prices are positive, convex functions of the underlying and are bounded by

V t, T ; x, K( ) � x – Ke–r T –t( )( )+.

As a result, the forward prices in our model also exhibit these properties. To conclude, in the forward market on carbon permits, there is a single !xed strike (the penalty cost) leading to an implied term structure of volatility to be calibrated.

2.6 The option priceSince 2005, a CO2 option market is slowly growing and attracting a wide variety of industrials, utilities and !nancial institutions of various nature. %is market satis!es the primary need of risk transfer from those wishing to reduce the risk of permit shortage situation, to those willing to accept it. As the tradable permit is an option in disguise, an option on emission allowances should resemble a compound option. %at is, on the !rst expiration date T1, the holder has the right to buy a new call using the strike price K1 where the new call has expiration date T2 and strike price K2. If we let the current time be 0, the spot price is S, and C(S, o;K) denotes the value of a call with time to expiry o and strike price K, on the !rst expiration date T1, the value of a call on a call ismax K1, C S, T2 –T1; K2( )�� =max C S, T2 –T1 ; K2( ) – K1, 0�� +K1Letting P

_(S,o;K) be the discounted put price and setting K1<K2 ,

the payout at maturity T1 of a call option on the forward price is max F T1, T2( ) – K1, 0( ) =max K2 – K1 – P X, T2 –T1;K2( ), 0��

which we rewrite in terms of compound option payo" asK – min K ,P X, T2 –T1;K2( )��

where K_=K2 –K1 . Similarly, in the case of a call option on the permit

price, as the option strike cannot be higher than the penalty cost, we must have the constraint K1< K2 , and the payout at maturity T is

max ST – K1, 0( ) =max XT – XT – K2( )+ – K1, 0��

��

=max K2 – K1 K2 – XT( )+ , 0��

��

which simpli!es tomax ST – K1, 0( ) =max XT – K1, 0( ) – max XT – K2, 0( )

%is is the payo" of a call spread on the unbounded process X. %erefore, in our model, the call price on the CO2 spot price is lower than that of an empirical model, not taking into consideration the

penalty costs. In the case of a put option the payout at maturity ismax K1 – ST , 0( ) = K1 – XT( )+

which is equivalent to a put option on the unbounded process. Again, given the dynamics of the unbounded process X we can then price the European options in the risk-neutral measure. In the call-option case, the payo" is a convex and concave function such that its price depends on two di"erent volatility levels and such that the notion of skew becomes important. It is interesting to note that vanilla options on emission allowances become exotic options when considering the unbounded process. As a result, the choice of a model for the unbounded process Xt is important, and one should consider empirical results to infer its dynamics. All price series analysed in the literature present non-zero skewness and excess kurtosis with summary statistics of the data revealing fat-tailed leptokurtic distributions and non-normal returns. Hence, for tractability reasons, an a$ne jump-di"usion model could be a plausible candidate.

2.7 ResultsTo illustrate our purpose, we consider the underlying given in section (2.4.1) where Xt is a geometric Brownian motion in the Qh measure – that is, the Black-Scholes model. We compute both a call option and a call spread option on Xt with maturity T=1, where the latter is a call option on St in our model. As emission allowance prices are characterised in the literature by high historical volatility, we let the volatility and drift be respectively mX = 0.4 and r = 0.05. %is example con!rms that, assuming the same drift, the call prices obtained with a bounded spot process are lower than those obtained with an unbounded process. Hence, to recover the same option prices behaviour with a model, not taking into consideration the penalty costs, one must decrease the drift term, possibly obtaining negative drift.

2.7.1 Phase IAs a way of demonstrating the e"ect of neglecting the penalty cost, K2= 40 in Phase I, we price a series of call options on St and use the results to infer the historical drift that calibrate a call option on Xt

in the historical measure around the strike K1= 32. We present the results in table 1 where we let the strike vary in the range [20, 40], assume the spot price to be S(0) = 20 and obtain the historical drift

+ = –0.05, which is in line with the results found by Paul et al (2010).

In the Black-Scholes model, the probability P(Xt *K2) depends on the initial spot price, the maturity, the drift and the volatility, so that depending on the parametrisation of the model, the di"erences between the call prices and the call spread prices vary in magnitude. Fixing the model parameters and allowing for the initial spot price to vary in the range [5; 35], we repeat the

T1. Call options & call spread options in Phase I with varying strikeStrike Call Call spread Histor. call

20 3.6045 3.3565 2.7639

22 2.8008 2.5527 2.0739

24 2.1611 1.9131 1.5468

26 1.6589 1.4109 1.1490

28 1.2687 1.0207 0.8512

30 0.9679 0.7198 0.6298

32 0.7373 0.4892 0.4658

34 0.5612 0.3132 0.3447

36 0.4272 0.1792 0.2553

38 0.3254 0.0773 0.1894

40 0.2480 0.0 0.1407

Historical drift is + = –0.05 Source: Author


experiment by computing di"erent call options in both models – that is, on Xt and St under the Qh measure. For comparison, we list the call prices on Xt in the historical measure with drift + = –0.01.

On one hand we compute at-the-money call options, while on the other, call options with !xed strike K1= 20 are considered. %e results are presented in table 2 where in both cases the di"erences between the call prices and the call-spread prices increase as the initial spot price tends to the boundary of the domain – that is, the penalty costs. %is is an important result when pricing options on carbon, but more importantly, it is crucial when hedging call options, as observed CO2spot prices are presently low and the discrepancy small, but in the event of an increase in spot prices, the hedging strategy constructed with an unbounded process will become misleading.

2.7.2 Phase IIIn Phase II, the penalty cost has been increased to K2 = 100 and from 2006 to 2009 we saw that observed carbon prices decreased due to an excess of permits allocated and a slower global economy. %erefore, keeping the model parametrisation as in Phase I but with the new penalty cost, the probability P(XT *K2) in Phase II is smaller than that in Phase I, and the di"erences between the call and call-spread options are greatly reduced. However, the latest report from the World Meteorological Organization (2010) stated that the main greenhouse gases have reached their highest concentration levels since pre-industrial times despite the economic slowdown, increasing the likelihood of a reduction in the permits allocated by governments, which in turn would result in an increase in CO2 spot prices.

ConclusionWe proposed a CO2 permit price model consistent with the features exhibited by the CO2 equilibrium models. To remain close to classical option pricing theory and obtain closed-form solutions, we directly modelled CO2 permit prices as a function of an exogenous and positive unbounded process and introduced one-period contingent claims in terms of such a dynamics. In that setting, the permit price is not a martingale under an equivalent probability measure, which is consistent with empirical !ndings, but implies the existence of arbitrage opportunities. Following a marked-to-model approach, we considered arbitrarily chosen growth rates for the CO2spot permit price, and computed European call option prices. ■

Daniel Bloch, Université Paris VI Pierre et Marie Curie, France Email: [email protected] author is grateful to Nicole El Karoui, Monique Jeanblanc, Mark Davis, Paul Mills as well as to the referee for their useful comments and suggestions

Cutting edge

T2. Call options & call spread options in Phase I with varying spotSpot ATM call Call spread Histo. call CallK=20 Callspread Histo.call

5 0.9011 0.9011 0.7716 0.0004 0.0004 0.0002

10 1.8022 1.8014 1.5433 0.1240 0.1231 0.0888

15 2.7034 2.6716 2.3150 1.1382 1.1064 0.9119

20 3.6045 3.3565 3.0867 3.6045 3.3565 3.0867

25 4.5057 3.5840 3.8584 7.2441 6.3224 6.4701

30 5.4068 3.1303 4.6301 11.5745 9.2980 10.6267

35 6.3080 1.9187 5.4018 16.2575 11.8682 15.2060

Historical drift is + = –0.01 Source: Author

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Carbon Derivatives Pricing - An Arbitrageable Market

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