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Risk-Neutral Pricing of Financial Instruments in Emission Markets: A Structural Approach

We present a novel approach to the pricing of financial instruments in emission markets, for example, the EU ETS. The proposed structural model is positioned between existing complex full equilibrium models and pure reduced form models. Using an exogenously specified demand for a polluting good it gives a causal explanation for the accumulation of CO2 emissions and takes into account the feedback effect from the cost of carbon to the rate at which the market emits CO2. We derive a forward-backward stochastic differential equation for the price process of the allowance certificate and solve the associated semilinear partial differential equation numerically. We also show that derivatives written on the allowance certificate satisfy a linear partial differential equation. The model is extended to emission markets with multiple compliance periods and we analyse the impact different intertemporal connecting mechanisms, such as borrowing, banking and withdrawal, have on the allowance price.


Introduction
Global warming has been recognised by policy makers as a key 21 st century problem. The phenomenon is widely believed to be the result of a greenhouse effect that is caused by increases in atmospheric gases such as carbon dioxide, methane, ozone and water vapour. Forced to address this issue, 37 countries ratified the Kyoto Protocol on 11 December 1997 in Kyoto, Japan. Under this agreement, binding limits, expressed in assigned amount units (AAU) and measured in tonnes of CO 2 equivalent greenhouse gas (GHG), 1 are imposed on the emissions of participating countries. To meet their obligations, countries may -amongst other mechanisms -draw upon any of the following three market-based mechanisms: (1) The Clean Development Mechanism (CDM), defined in Article 12 of the Kyoto Protocol, allows countries to implement emission-reduction projects in developing countries. For this they receive certified emission reduction (CER) credits, each worth one tonne of CO 2 equivalent, which can be used for meeting Kyoto targets. (2) The Joint Implementation (JI) mechanism, defined in Article 6 of the Kyoto Protocol, allows countries to earn emission reduction units (ERU), each worth one tonne of CO 2 equivalent, from establishing emission-reduction projects in other Kyoto countries. Like CERs, these units can be used to meet Kyoto targets.
(3) Emissions Trading as defined in Article 17 of the Kyoto Protocol, allows market participants that have AAUs, CERs or ERUs to spare, to sell their excess capacity to other participants. This creates the so-called carbon market.
The Kyoto Protocol merely constitutes a global framework that encourages participating countries to put in place platforms on which CERs, ERUs and AAUs can be traded. Subject to broad provisions, the market design of any local implementation of an emissions trading system is left to the hosting countries.
In this paper we present a simple model for emissions trading. Our model incorporates market design features which are commonly found in successful implementations of such trading schemes, for example the SO 2 and NOx trading programs in the US or the European Union Emissions Trading Scheme (EU ETS). Because of its pioneering role, we choose the latter example to illustrate the working principle of emissions trading.
1.1. Emissions Trading in the EU ETS. The limit on emissions during one compliance period, also referred to as the cap on emissions, is realised through an initial allocation of allowance certificates -each worth one EU Allowance Unit (EUA) and permitting its holder to emit one tonne of CO 2 equivalent 2 GHGsby the government to firms in the market. At the end of each compliance period, firms must offset their accumulated emissions by submitting an adequate number of certificates. If they fail to do so (the event of non-compliance), they must pay a monetary penalty for each unit of excess emissions. Throughout a compliance period allowances are traded actively and this leads to the formation of a price, which represents the cost of carbon.
In practice, an emissions trading scheme consists of multiple compliance periods, each with its own distinct cap and penalty. Subsequent periods are joined by connecting mechanisms, which regulate the transition from one compliance period to the next. The key mechanisms go by the names of banking, borrowing and withdrawal. The banking mechanism allows market participants to carry allowance certificates, allocated for compliance at the end of the current period, forward to the next compliance period. Similarly, borrowing enables firms to use next period's certificates for compliance at the end of the current trading period. The withdrawal mechanism constitutes additional punishment for non-compliance: it prescribes that in addition to the monetary penalty payment one allowance certificate from next period's allocation is withdrawn for each unit of excess emissions at the end of the current period.
Since the coming into force of the Linking Directive, the EU has been accepting credits from Clean Development-and Joint Implementation Mechanisms for compliance in its trading scheme. Because one EUA is equivalent to an AAU, the base unit of the Kyoto Protocol, CERs, ERUs and EUAs can all be traded within the same system straightforwardly. In practice, this takes mostly place on platforms such as the European Energy Exchange (EEX), where EUA and CER spot and future contracts are traded actively.
Emission reduction as part of a trading scheme occurs in two ways. The immediate consequence is to shift production within the existing fleet of resources to pollution friendlier ones -an effect we refer to as load shifting. The cost of carbon also makes it attractive for firms to invest in long-term abatement measures, if the cost of reducing their emissions by one unit lies below the value of an allowance certificate.
Even if a firm has sufficient allowances to cover its emissions it should make use of all available emission-reduction measures whose marginal abatement cost (MAC) lies below the value of the allowance certificate. It can then sell spare certificates to companies whose MAC is above the market price of allowances and make a profit. For this reason it has been argued that cap-and-trade schemes provide emission reduction at the lowest cost to society. However, there is also evidence which suggests that the implied cost of carbon to make long-term investment in renewables such as solar cells worthwhile is $196 per tonne of CO 2 -far above current allowance prices (cf. [1]).
1.2. Electricity Generation: a Pollution Intensive Process. The primary process that releases CO 2 emissions is the burning of fossil fuels. Since this is heavily used for the generation of electric power, electricity offers itself as an exemplary good for the academic study of emission markets. A wide spectrum of technologies, including nuclear fission, wind turbines, hydropower and the burning of fossil fuels, are used for the generation of electricity. Because these technologies differ substantially in their emission rates it is important to identify which generators are used in the market at any point in time. In principle, this can be deduced from the electricity bid stack.
The bid stack, introduced in [3,18,2,11] aggregates the bidding behaviour of firms that supply electricity. A bid is the amount of electricity a single generator is willing to supply at a specific price. Firms submit their Pareto-optimal bids for each hour of the next trading day to a central market administrator. An example would be a generator submitting bids (600MW,100e), (200MW,120e), (200MW,200e). This generator offers to sell its first 600MW for the specified hour at a price of 100e, the next 200MW at a price of 120e and a further 200MW at a price of 200e. Consequently, each firm submits an increasing simple (step) function that maps electricity supply to its marginal price. The market administrator aggregates the bids for each price level and arranges them in increasing order of price. Using cheapest bids first, electricity is supplied at the marginal price of the last unit of electricity that is needed in order to satisfy demand.
The bids of generators reflect their production costs (cf. [11]). In particular, firms consider fixed and variable costs when deciding upon their bid levels. In the absence of emissions trading -a scenario called business-as-usual -the latter costs consist predominantly of the price to be paid by a particular plant for the amount of fuel necessary to generate each unit of electricity (the plant's heat rate multiplied by the price of the utilised fuel). The introduction of a cap-and-trade system levies a cost on emissions. In principle, firms may remain idle and sell unused certificates to the market. Therefore, if they choose to produce and to utilize their certificates for compliance purposes, this forgone profit constitutes an opportunity cost, which leads to an increase in variable costs. As a consequence bids increase by an amount equal to the marginal emissions rate of the plant (measured in tonnes of CO 2 per MWh) multiplied by the allowance price.
Although not a requirement for the model that we propose, in the absence of emissions trading, pollution intensive fuels have historically gathered on the left end of the bid stack because they are cheaper to use, whereas environmentally friendlier technologies tend to be more expensive and are concentrated further to the right. 3 The rationale behind cap-and-trade is that for sufficiently high carbon 3 An exception to this rule of thumb constitute must-run bids, which are always placed on the left end of the bid stack and may, for example, contain bids from nuclear generators that do not emit.
costs, pollution-intensive technologies become more expensive than environmentally friendlier ones. The market administrator rearranges bids to preserve the increasing order and as a result environmentally friendly technologies are now called upon before pollution intensive ones, leading to cleaner production of electricity.
Example 1. Consider a simple market with one coal and one gas generator, who bid at only one price level. We illustrate the influence of emissions trading on the bid stack in Figure 1. Initially the cost of carbon is low and bids from coal generators are cheaper than those from gas generators. Accordingly, coal bids come first in the bid stack and the marginal emissions corresponding to bids further to the left in the stack are relatively higher than those corresponding to bids on the right. As emissions become more costly the bid levels from coal and gas generators increase, with coal bids doing so more than gas bids. This results in the market administrator rearranging the bid stack and placing gas first. The result of the higher allowance price are lower emissions as intended.  The price setting mechanism described above applies directly to day-ahead spot prices set by uniform auctions, as is the case at most exchanges today. For example, the power spot price for Germany, Austria, Switzerland and France is determined by such auctions organised by the EEX. Although, since the onset of electricity market deregulation in 1998, the auction based trading volume at the EEX has increased substantially over the last few years -from 49 TWh in 2003 to 279 TWh in 2010 (cf. [14]) -a large share of electricity in Europe is still traded overthe-counter or on a forward basis. However, we believe that in a competitive market with rational agents the day-ahead auction price also serves as the key reference point for real-time and over-the-counter prices (cf. [26]).
1.3. Literature Review. The first academic treatment of emission markets can be traced back to [13,25]. Early models of allowance trading in discrete and in continuous time were proposed in [12,22,24,28,29,30]. More recently, emission markets have been treated from two different angles. On the one hand are full equilibrium models that derive the price processes of allowances and goods (the production of which causes pollution) from the preferences of individual firms and additional sources of uncertainty. These have proved insightful but rather cumbersome in their complexity (cf. [8,9]). On the other hand are risk-neutral approaches which rely on the absence of arbitrage to specify the allowance price evolution directly in the form of a process that has the correct terminal properties and the correct bounds; then the parameters in the model are calibrated to market data. In these models, the event of non-compliance is described exogenously and no causal explanation is given for the accumulation of emissions in the economy. Within this class of models one can distinguish between models that ignore the feedback from the allowance price to the rate at which firms emit (cf. [10]) and those that take this feedback effect into account through an exogenously specified abatement function (cf. [4,6,19]).
In this paper we propose a structural model which draws upon elements of the equilibrium approach but still retains the simplicity of the risk-neutral approach. We take as a starting point an exogenously specified stochastic process representing demand for electricity and we regard allowances as derivatives on demand and cumulative emissions. The demand process is translated into an emissions process via the bid stack, which allows us to deduce which generators are active at any point in time. As noted above, the bid stack both influences and is influenced by the allowance price. This leads naturally to a formulation of the allowance price as the backward part of a forward-backward stochastic differential equation (FB-SDE). To solve the problem numerically, we derive a semilinear partial differential equation (PDE) for the allowance price as a function of demand and cumulative emissions. We extend our model to emission markets with multiple compliance periods and analyse the impact different intertemporal connecting mechanisms such as borrowing, banking and withdrawal have on the allowance price. The last section is devoted to the pricing of European derivatives written on the allowance certificate. Throughout the analysis we focus on the trading of AAUs only. The joining of multiple markets using CERs and ERUs in the present setting is left to future research and addressed from a different point of view for example in [7].
Although we have formulated the problem in terms of electricity generation (which is, indeed, responsible for a large proportion of the emissions covered by the Kyoto protocol), we may at least conceptually extend our model to all emissions covered by Kyoto if we view the formation of the equilibrium price for energy as equivalent to the action of the market regulator's arrangement of the bid stack in increasing price order.

From Electricity Markets to Carbon Emissions
In this section we develop our approach to modelling the interaction between electricity and emission markets. We introduce the random factors and the key parameters that are later shown to drive the price formation of allowance certificates. An important part is played by the merit order -the rule by which available resources with the lowest marginal costs of production are called upon first to supply electricity. We introduce the electricity bid stack, which is modelled as a continuous map from the supply of electricity to its marginal price and analogously define the emissions stack as a continuous map to the marginal emissions caused by the production of the last unit. Using an equilibrium assumption, we relate supply to demand; we show how this allows us to deduce which technologies are used to meet demand at any point in time and the total market emissions rate this production schedule implies. Lastly, we illustrate the impact the introduction of a cost of carbon has on the bid and emissions stacks. In the context of an emissions trading scheme the merit order assumption very naturally leads to load shifting, the reallocation of energy production from emission-intensive to pollution friendly resources.

2.1.
Market Set-up. We consider a finite time interval [0, T ], which, initially, corresponds to one compliance period; later we will consider multi-period markets. We denote by (Ω, F, (F t ) t∈[0,T ] , P) a filtered probability space satisfying all the usual assumptions, where (F t ) t∈[0,T ] is generated by a standard Brownian motion (W t ) t∈[0,T ] , the only source of randomness in the market. In order to simplify the notation we omit the subscript that restricts a stochastic process to the time interval [0, T ], from now on. We only deviate from this habit in §3.2, where it becomes important as part of a multi-period setting and in §4, where we discuss the pricing of derivatives.
Agents in our market demand a good, the production of which causes emissions; as discussed above, we take this good to be electricity. Firms can produce electricity using different technologies that vary in their costs of production and their emissions intensity. The market is subject to an emissions trading scheme, as follows. Each registered firm receives an initial allocation of allowances, which can be used to offset its cumulative emissions at the end of the compliance period. If a firm is unable to submit a sufficient number of certificates, its excess emissions are subject to the payment of a monetary penalty. Allowances are represented by printed certificates. Because their cost of carry is negligible, we consider them to be liquidly traded financial products in which long and short positions can be taken. Consequently, if a firm believes its initial allocation to be incorrect it can buy or sell allowances as needed. This leads to a liquid market and the formation of a price at which allowances are traded.
Analogous to the idea of a representative agent, we ignore the aggregation problem and instead take the point of view of the whole market. Our goal then becomes to determine the arbitrage-free price of emission permits as a function of the aggregate forces that act in the market. As will be shown in §3, this price directly and crucially depends on the accumulated emissions during the compliance period and on the aggregate demand for electricity.
The actions of consumers in the market result in an exogenously given F t -adapted demand process (D t ). Firms respond to this demand by generating electricity. In particular, at any time 0 ≤ t ≤ T , the aggregate of all firms supplies an amount (ξ t ) of electricity. We assume that the market uses only currently available information to decide on its production level and that this level is always non-negative and below a constant maximum production capacity ξ max ≥ 0. Therefore, ξ t is F t -adapted and 0 ≤ ξ t ≤ ξ max , for 0 ≤ t ≤ T.
Moreover, we assume that there are always sufficient resources in the market to meet demand; so that The demand process is assumed to be perfectly inelastic, as is frequently justifiable in electricity markets (cf. [8,11]), and demand and supply are related by a Walrasian equilibrium assumption (cf. [31]). This concept is realised by the market administrator, who ensures that aggregate demand for and aggregate supply of energy are matched on a daily basis, namely that Typically, spot data for demand and supply is quoted in megawatts. For example, a demand of 60MW for one hour is equivalent to 60MWh.
The production of electricity causes CO 2 emissions, in a way that we describe more precisely in §2.2 and §2.3. The total (cumulative) emissions during the time interval [0, t], are described by the process (E t ), which is measured in tonnes of CO 2 . Moreover, since emission intensive production resources are finite (E t ) is assumed to be bounded; i.e.
The regulator decides on an acceptable maximum level of cumulative emissions during the compliance period (the cap) and issues a corresponding amount of allowance certificates, 0 ≤ E cap ≤ E max , measured in tonnes of CO 2 . At the end of the compliance period, cumulative emissions in the market are offset against the initial allocation of allowances. Certificates that are not used for this purpose expire worthless in the case of the single-period setup, whereas unaccounted emissions are subject to monetary penalty payment at a rate π ≥ 0. Thus, an amount The allowance certificates constitute traded assets in the market. Their value is represented by the process (A t ). We shall also consider options written on the certificate and assume the existence of a riskless money market account with constant risk-free rate r ≥ 0.

The Bid and Emissions Stacks.
We turn to the modelling of the cumulative emissions. We begin with the business-as-usual market and analyse the impact of an emissions trading scheme in the next subsection.
Key to our analysis is the following assumption, which summarises the actions of the central market administrator as introduced above.
Assumption 1. The market administrator ensures that resources are used according to the merit order. This means that the cheapest production technologies are called upon first to satisfy a given demand and hence electricity is supplied at the lowest possible price.
As explained in §1.2, bid levels are mostly determined by variable costs. Therefore, these costs play an integral part in determining the merit order arrangement in Assumption 1. The resulting increasing map from market supply of electricity to marginal price forms the bid stack. As explained in the introduction, the bid stack is, strictly speaking, an increasing simple function. In practice, however, it consists of sufficiently many steps to be approximated by a smooth function. This leads us to the following definition.
Definition 1. The business-as-usual bid stack is given by the bounded function Here and throughout the rest of the paper, the variable ξ represents the supply of electricity (measured in MW). Correspondingly, b BAU (ξ) denotes the bid level of the marginal production unit (measured in e per MWh).
We note immediately that in reality business-as-usual bid levels are stochastic. Most importantly, fuel prices, which are key drivers of variable costs, fluctuate continuously. For our purposes, however, we are only interested in the relative position of the different technologies in the bid stack. Fluctuations in fuel prices only become important if they induce merit order changes. From historic data observations this is only relevant in the long-run and we prefer not to consider it for now. Hence we model the business-as-usual bid stack as a deterministic function (cf. [3]) allowing us to focus exclusively on the impact of emissions trading on variable costs and the merit order in §2.3.
Remark 1. As pointed out in the introduction, emission-intensive technologies tend to be cheaper than environmentally friendly ones as a means to produce electricity. Therefore, we find that bids associated with a small level of electricity supply stem mostly from emission-intensive generators, while bids at the right end of the interval [0, ξ max ] stem mostly from environmentally friendly ones (as remarked earlier an exception to this rule are nuclear plants, which do not cause any CO 2 emissions and are generally placed at the very left end in the bid stack). In between exists a spectrum in which a mixture of technologies contribute to bids. This assumption has been confirmed (cf. [11]) by analysing the correlation between production costs and bid levels.
Remark 2. In principle the model that we propose can be extended to include stochastic fuel prices as part of the variable costs that determine firms' bids. The business as usual bid stack b BAU would then become a function of additional independent variables (the prices of the fuels used in the production process) and the dimensionality of the allowance pricing problem (9) would increase. Such an extension should be considered when one is interested in pricing contracts such as, for example, clean spread options, which explicitly feature the prices of electricity, fuels and emissions in their payoff. In this case the subtle dependence of electricity spot prices on fuel prices becomes important. Since we are predominantly interested in the price formation of allowance certificates, we only mention the possibility of this extension and leave its investigation to future research.
Analogous to the bid stack we construct an emissions stack, by creating a map from the supply of electricity to the marginal emissions associated with the supply of the last unit.
Definition 2. The marginal emissions stack is given by the bounded function With the above definition, e(ξ) associates with a specific supply of electricity ξ the emissions rate of the marginal unit (measured in tonnes of CO 2 per MWh).
Proposition 1. The business-as-usual market emissions rate µ BAU e is given by where κ > 0.
Proof. The Walrasian equilibrium assumption (1) for our inelastic model implies that the market produces the exact amount of electricity consumers demand, and that -under business-as-usual -the generation capacity associated with the interval [0, D] is used for this purpose. The market emissions rate per hour is then obtained at any time by integrating over the marginal emissions stack up to the current level of demand. We rescale this rate using the strictly positive constant κ in order to obtain µ BAU e , the market emissions rate measured on the same time scale that characterises T .
Note that whereas the rate e is measured per hour, µ BAU e is rescaled to live on the same timescale as T . If, for example, T is measured in years κ would take the value of the number of hours per year and µ BAU e would now correspond to the market emissions rate per year, given an allowance price and a level of demand.

2.3.
Load Shifting -a Short-Term Abatement Measure. We now analyse the effects of emissions trading on the business-as-usual economy introduced above. As explained in the introduction, emissions trading puts a price on carbon and thereby increases the production costs of firms. In particular, it makes it more expensive for firms that rely on emission-intensive technologies to produce. For each unit of CO 2 that these firms emit in excess of their initial allocation, they must buy an allowance contract in order to avoid penalisation; the cost of carbon is a real cost. Alternatively, if a firm owns more allowances than it requires, it can sell spare ones in the market. In this case, the cost of carbon represents an opportunity cost.
We ignore the possibility that firms might invest in long-term abatement projects and only focus on the direct impact on the bid stack. We assume that, in order to maintain their profit margin, firms pass the emissions-related increase in production costs on to consumers. Because the cost of carbon is represented by the price of an allowance certificate, the business-as-usual bids of each firm increase by an amount equal to the allowance price multiplied by the marginal emissions rate of that firm. On an aggregate level this means that, for a given allowance price A the bid stack now becomes the function g, where For A = 0, (2) is equivalent to the business-as-usual bid stack. For positive certificate prices emissions trading may cause the mapping ξ → g(·, ξ) to lose its monotonicity. In particular, we observe that bids associated with large marginal emission rates become relatively more expensive, as the cost of carbon makes it relatively more costly for firms relying on dirty fuels, such as coal, to produce.
By the merit order assumption the market administrator calls upon generators in increasing order of their bid levels. We define the set of active generation units at a given allowance and electricity price P by By the definition of sublevel set, P → λ(S(·, P )), where λ denotes the Lebesgue measure, is strictly increasing; under the following assumption, it is also continuous and therefore invertible.
Using (3), for observed values of the allowance price, the market bid stack b is now defined by This immediately yields the market price of electricity P , which is given by Whereas under business-as-usual demand D is met using the generation capacity [0, D] (considered as a subset of the domain of the emissions stack e), emissions trading may shift this interval further to the right, or, depending on the shape of the marginal emissions stack, split it up into multiple sets -an effect we refer to as load shifting. We make the impact of load shifting on the market emissions rate µ e precise in the next proposition.

Proposition 2.
In the presence of cap-and-trade and given an allowance price A and demand level D, the market emissions rate µ e is given by Proof. Immediate from the discussion above.
We note that the business-as-usual market emissions rate is of course a special case of (4), which is obtained by setting Remark 3. As described earlier, in reality the bid and marginal emissions stack are step functions, whose finitely many constant values correspond to firms' bids and their corresponding marginal emissions. To model the impact of a positive allowance price on the bid stack in this case, one would add the cost of carbon to bids as usual, then the resulting step function is rearranged in increasing order. Because of the discrete nature of the problem, the rearrangement induces a permutation ν on the bids, which is then applied to the marginal emissions stack. Instantaneous emissions are now obtained by integrating the rearranged emissions stack over the closed interval [0, D]. We prefer to work with the continuous limit of the bid and marginal emissions stack. In this case the permutation ν cannot be defined explicitly and we identify active firms with the set S p .
In the following lemma we prove some technical properties of µ e , which show that the model we propose for the market emissions rate makes intuitive sense and leads to a suitably regular function.
, the result follows from the observation that, for a given 0 can be written as the union of a finite number of intervals. As A 1 increases to A 2 , there are three possibilities: (a) existing intervals grow or shrink; (b) new intervals appear or existing ones disappear (by Assumption 2 this always happens at a point) (c) the intervals remain unchanged. Differentiating the level curves g(A, ξ) = b(A, D) with respect to ξ, for a given level of demand, we find that By Assumption 2 the right hand side is bounded by a constant, 4 say C 2 ≥ 0. Therefore, as A changes, in each case (a) − (c), the sides of the intervals describing ∆ A S p (A 2 , A 1 ) do not move faster than

Risk-Neutral Pricing of Allowance Certificates
In this section we address the problem of determining the arbitrage-free price of an allowance certificate given the current demand for electricity and the cumulative emissions to date in the economy. We do this initially in the setting of an emissions market with one compliance period; subsequently we generalise the model to deal with markets that consist of multiple consecutive compliance periods and examine the impact that connecting mechanisms, namely banking, borrowing and withdrawal have on the certificate price.
Assumption 3. There exists an equivalent martingale measure Q ∼ P, under which, for 0 ≤ t ≤ T , the discounted price of any tradable asset is a martingale. We refer to Q as the risk-neutral measure.
As long as Assumption 3 is satisfied, the First Fundamental Theorem of Asset Pricing guarantees that our market is arbitrage-free.  We begin by making some additional assumptions about the demand and cumulative emissions processes (D t ) and (E t ). We assume that at time t = 0 demand for electricity is known. Thereafter, it evolves according to an Itô diffusion; i.e. for 0 ≤ t ≤ T , under the measure Q, demand for electricity is given by the stochastic process where (W t ) is F t -adapted and a Q-Brownian motion. We postpone the discussion of the relevance of the regularity of the coefficients to later in §3.1.
The assumption that demand is perfectly inelastic is reflected in the fact that both coefficients are functions of demand only. In practice demand for electricity exhibits seasonal periodicity, an attribute that would cause µ d to depend on time explicitly. For simplicity we choose to ignore this feature.
Cumulative emissions are measured from the beginning of the compliance period when time t = 0, so that E 0 = 0. Subsequently, they are determined by integrating over the market emissions rate µ e derived in Proposition 2. Consequently, the cumulative emissions process is represented by a bounded variation process; i.e. for 0 ≤ t ≤ T , Note that with this definition the process (E t ) is non-decreasing, which makes intuitive sense considering that it represents a cumulative quantity.

One Compliance Period.
To formulate the pricing model, it remains to characterise the allowance certificate price process (A t ). This is different from the specification of (D t ) and (E t ), because its value at time t = 0 is unknown. An arbitrage argument, however, allows us to determine its value at the end of the compliance period. The event of non-compliance is {E T ≥ E cap }; then the value of the allowance certificate at time t = T , is given by the terminal condition From Assumption 3 we know that the discounted allowance price is a martingale under the measure Q. Therefore, the allowance price is given as the discounted conditional expectation of its terminal condition under this measure; i.e.
which shows that the allowance price process (A t ) takes values in [0, π] only.
Proposition 3. For 0 ≤ t ≤ T , the price of an allowance certificate (A t ) in a market with one compliance period is described by the following FBSDE: Proof. Because the filtration (F t ) is natural, it is a consequence of the Martingale Representation Theorem (cf. [21]) that the discounted allowance price can be represented as an Itô integral with respect to the Brownian motion (W t ). It follows that for some F t -adapted process (Z t ).
Combining the processes (5) and (6) for demand and cumulative emissions and equation (10) together with the terminal condition (7), the pricing problem becomes that described by equation (9).
Remark 4. The existence and uniqueness of a solution to the FBSDE (9) is a delicate and partially still open question, whose investigation has only recently begun. The non-standardness of this kind of equation arises from the degeneracy of one of its forward components (the emissions process (E t ) in our case) combined with the singularity of the terminal condition. Together, these features conspire to cause the random variable E T to develop a point mass at the cap E cap , as shown in [5].
In the same paper it is also shown that under the assumption that µ d and σ d are Lipschitz continuous and exhibit at most linear growth and that µ e is Lipschitz continuous and strictly decreasing in A, a unique solution to equation (9) exists satisfying the initial conditions D 0 = d, E 0 = 0 and the relaxed terminal condition Moreover, it is shown that there exists a continuous function α such that A t = α(t, D t , E t ) for 0 ≤ t < T . Under considerably more restrictive conditions on the coefficients but preserving the distinctive features of the problem (degeneracy of the forward component and a singularity in the terminal condition) the value function α is actually smooth (cf. [6]). Both papers highlight the close connection between FBSDEs of the kind (9) and the theory of conservation law PDEs. The smoothness result in [6] together with existence results for conservation law PDEs that allow for less regular coefficients (cf. [20]) make it feasible to also hope for existence results for equation (9) under less strict conditions on the coefficients and leading to improved regularity of the value function α.
where U := (0, ξ max ) × (0, E max ), U T := [0, T ) × U and Notice that µ e takes α as its first argument; hence the PDE belongs to the class of semilinear ones. In addition to the terminal condition suitable boundary conditions have to be supplied. These depend on the specification of the coefficients of the PDE and we postpone the issue to section §5, where we discuss the numerical solution of the problem.
Remark 5. The intuition behind equation (11) is simple. We simply assume that, under Q, A t , being a traded asset, has a drift equal to the risk-neutral rate (cf. the last equation of (9)). Then we apply Itô's formula to A t = α(t, D t , E t ) using the first two equations of (9), and take expectations to derive (11). This procedure is purely formal, because it assumes the existence of a classical solution to the PDE (11).

Multiple Compliance Periods.
We now consider the pricing problem in an emission market with two compliance periods. In principle, the results presented in this section can easily be extended to an arbitrary number of periods. To ease the presentation, however, we choose to present the canonical case. Taking 0 = T 0 ≤ T 1 ≤ T 2 = T , we consider the two compliance periods [0, T 1 ], [T 1 , T ]. For simplicity we assume that each period corresponds to one year. As previously, the F t -adapted process (D t ) t∈[0,T ] represents the aggregate demand for electricity. For i ∈ {1, 2} the F t -adapted process (E t ) t∈[Ti−1,Ti] measures the cumulative emissions from the beginning of the ith compliance period up to time t and (A i t ) t∈[Ti−1,Ti] represents the price of an allowance certificate for compliance at time T i . Also, we denote by E 1 the cumulative emissions at the end of the first compliance period. Each year, the regulator issues a number E i cap ≥ 0 of allowance certificates and sets the penalty π i ≥ 0.
Demand for electricity is given at time t = 0 and thereafter evolves continuously throughout the trading period [0, T ]. Further, we assume that cumulative emissions are measured from the beginning of each compliance period, so that (12) E Ti−1 := 0, i ∈ {1, 2}.
Lastly, we note that each process (A i t ) t∈[Ti−1,Ti] corresponds to a different vintage of allowance certificates. If we disregard mechanisms that connect compliance periods, a certificate issued during the first period is for compliance at time T 1 only. However, we now wish to consider mechanisms that connect compliance periods and permit allowances to be transferred between periods. In this case both vintages of certificates have a more complex dependence. In particular, the second period allowance price depends on cumulative emissions not only during the second period, but also on the ones during the previous period, as we describe below. The connecting mechanism is now expressed through the terminal condition at time T 1 ; for now, we do not determine it explicitly and denote it by φ 1 (·), for some (possibly singular) function φ 1 .
Corollary 1. In a market with two compliance periods, the price (A t ) t∈[Ti−1,Ti] of an allowance certificate during the ith period, i ∈ {1, 2}, is described by the following FBSDE: for some F t -adapted process (Z i t ) t∈[Ti−1,Ti] and where φ 1 := φ 1 (E T1 ) and φ 2 := φ 2 (E T2 ; E 1 ) respectively denote the terminal condition at the end of the first and the second compliance period.
Proof. Follows immediately from Proposition 3 and the discussion above.
As in §3.1, we assume the existence of suitably regular functions α i : where U Ti := [T i−1 , T i ) × U .

Banking and Withdrawal.
Banking and withdrawal are two mechanisms that connect compliance periods and are implemented in most emission markets. Both affect the supply of certificates during the second compliance period. This leads us to introduceÊ 2 cap to denote the aggregate supply of certificates during the second compliance period. The implementation of banking offers an additional incentive for reducing emissions, since it specifies that spare allowance certificates, for compliance at the end of the first period, become perfect substitutes for certificates issued during the second compliance period. This means that in the event of compliance, a number (E 1 cap − E 1 ) of certificates with price A 1 T1 are exchanged for certificates valid during the next compliance period, with price A 2 T1 . This incentive to reduce emissions is strengthened by the withdrawal mechanism, which constitutes additional punishment for firms that exceed their emission limit. Under this mechanism excess emissions at the end of the first compliance period are not only penalised at the rate π 1 , but moreover a corresponding number of certificates are withdrawn from the subsequent allocation. Whereas any amount of certificates can be banked, at most next period's allocation can be withdrawn from the market. Therefore, in the event of non-compliance, a number min(E 1 − E 1 cap , E 2 cap ) of certificates with price A 2 T1 are subtracted from E 2 cap . In the event that the entire allocation of the second period has been withdrawn and there remain unaccounted for emissions (at the end of the first period), we specify that these are penalised at the combined rate of the first period penalty π 1 and -to compensate for the lack of certificates that can be withdrawn -an additional penaltyπ 1 ≥ A 2 T1 . These features imply that during the second period the aggregate supply of certificates now stems from two sources. First, the regulator issues a number of permits E 2 cap at the beginning of the period. Second, as explained above, a number of certificates are banked or withdrawn. The aggregate supply of certificates during the second period is then given by Figure 3 illustrates the banking and withdrawal mechanisms in the two-period market under consideration. In Figure 3a compliance at t = T 1 leads to the banking of a number (E 1 cap − E 1 ) of certificates. The market is in compliance at t = T 2 because of this additional supply of certificates. In Figure 3b non-compliance at t = T 1 leads to the withdrawal of a number (E 1 − E 1 cap ) of certificates. This leads to non-compliance at t = T 2 because of the decreased supply of certificates during the second compliance period, even though cumulative emissions during the period [T 1 , T ] are below the second-period cap.
The terminal condition φ 1 for the allowance price in an emissions market with two compliance periods connected by the mechanisms of banking and withdrawal now follows. Banking implies that in the event of compliance, that is if E T1 < E 1 cap , the value of the first period allowance at time T 1 equals the value of the second period allowance at time T 1 . In the event of non-compliance at time T 1 with E 1 cap ≤ E T1 < E 1 cap + E 2 cap , the penalisation of excess emissions and the withdrawal of certificates lead to the first period allowance certificate taking the value of the sum of the second period certificate and the penalty. In the event of non-compliance at time T 1 with E T1 ≥ E 1 cap + E 2 cap , the double penalisation rule implies that the value of the first period allowance certificate equals the sum of π 1 andπ 1 . Therefore, φ 1 is given by  At time T 2 , the terminal condition φ 2 for the allowance price is now the same as in the one period case with the exception that the aggregate supply of certificateŝ E 2 cap is used: Note that the terminal condition φ 2 uses the aggregate supply of certificatesÊ 2 cap , as defined in (15), and is hence path-dependent. In particular, it depends on E 1 as mentioned earlier. In the context of pricing futures contracts on allowance certificates in a two-period market a similar terminal condition was introduced in [10].

3.2.2.
Borrowing, Banking and Withdrawal. In addition to banking and withdrawal, consecutive compliance periods may also be connected by the borrowing mechanism. The effect of banking and withdrawal at time T 1 is to increase the value of the first period allowance certificate from zero to A 2 T1 in the event of compliance (due to the banking mechanism) and from π 1 to (π 1 + A 2 T1 ) or (π 1 +π 1 ) in the event of non-compliance (due to the withdrawal mechanism). In contrast, the borrowing mechanism decreases the probability with which non-compliance occurs.
In an emissions market in which borrowing is allowed, firms may bring forward certificates from the second allocation E 2 cap and use them for compliance at time T 1 . This does not affect the aggregate supply of certificates during the first compliance period, but whereas previously non-compliance occurred when E T1 ≥Ê 1 cap , this is no longer the case, as certificates from the second period may be borrowed to supplement the aggregate supply during the first period. Therefore non-compliance now only occurs if E T1 ≥ E 1 cap + E 2 cap , in which case the entire allocation of the second period must be borrowed and additional units of excess emissions are penalised at the combined rate of π 1 andπ 1 .
The borrowing mechanism is illustrated in Figure 4. Compliance at time T 1 is only possible by borrowing a number (E 1 − E 1 cap ) of certificates from the second compliance period. As a result these certificates are deducted from E 2 cap . In our example this leads to non-compliance at time T 2 .
Whereas φ 2 continues to be given by (17), the terminal condition φ 1 in an emissions market, which connects subsequent compliance periods with the banking, borrowing  Figure 4. Borrowing mechanism in an emissions market with two periods. and withdrawal mechanisms, now follows and is given by

Risk-Neutral Pricing of European Derivatives
We now turn to the arbitrage-free pricing of European derivatives written on the allowance certificate within our model. For this purpose we work in the emissions market of §3.1 with one compliance period.
Our example of choice is a European call (C t (τ )) t∈[0,τ ] with maturity τ , where 0 ≤ τ ≤ T , and strike K ≥ 0, so that its payoff is We know from Assumption 3 that, for 0 ≤ t ≤ τ ≤ T , the discounted call price (e −rt C t ) t∈[0,τ ] is a martingale under the measure Q. Therefore, it is given as the discounted conditional expectation of its terminal condition under this measure; i.e. C t = e −r(τ −t)Ẽ (C τ − K) + F t , for 0 ≤ t ≤ τ.
As we argued previously for the allowance certificate, the discounted call price can be represented as an Itô integral with respect to the Brownian motion (W t ) t∈[0,τ ] . It follows that d e −rt C t = Z t dW t , for 0 ≤ t ≤ τ and some F t -adapted process (Z t ) t∈[0,τ ] .
The key difference between the allowance certificate and the option pricing problem is that the allowance price (representing the cost of carbon) has an impact on the rate at which firms emit. This is reflected in the fact that the drift of the cumulative emissions process depends on the price of the allowance certificate, but not on that of the option. Consequently, the FBSDE (9) is coupled and the partial differential equation (11), which describes the allowance price, is nonlinear, whereas (19) is linear.

Numerical Analysis
This section is dedicated to the numerical analysis of the model. We illustrate the dependency of allowance prices on demand and the cumulative emissions and compare prices in the setting of a single-period market to those implied by multiperiod markets. Further, we demonstrate the dependence structure of a European option written on the allowance certificate.

5.1.
Concretising the Model. We begin by specifying the functions and parameters in the model.

Functional Form of the Bid and the Emissions Stack.
We take the businessas-usual bid stack to be of the form where b, b ≥ 0 and 2 < θ 1 < ∞. With this choice b BAU is strictly convex and strictly increasing on its domain of definition. The parameters b and b correspond to the minimum and maximum prices of electricity the model can produce. Because the range of allowed bids and typically observed market prices in many auction based electricity markets is well known, these are relatively easy to infer in practice. The parameter θ 1 controls the steepness of the stack and in particular how quickly marginal costs of generators increase.
Similarly, we take the marginal emissions stack to be of the form where e, e ≥ 0 and 0 ≤ θ 2 < 1. With this definition also e is strictly convex and decreasing on its domain of definition. The parameters e and e correspond to the minimum and maximum marginal emissions rate in the market. In a market with coal and gas generators only and under the reasonable assumption that coal is the more emission intensive technology than gas, e would represent the marginal emissions rate of coal and e that of gas. The parameter θ 2 controls the fuel mix in the market. The smaller the value of θ 2 , the smaller the proportion of the market capacity that is served by the pollution intensive technology.
Clearly b BAU and e satisfy the assumptions in Definitions 1 and 2. Further, since a linear combination of strictly convex functions is also strictly convex, so is the function g. Therefore, also Assumption 2 is satisfied. Moreover, we note that for this choice of bid and emissions stack the set S p (·, ·) is always of the form [ξ 1 , ξ 2 ] for 0 ≤ ξ 1 ≤ ξ 2 ≤ ξ max .

The Demand Process.
We specify that underP the process (D t ) follows the stochastic differential equation where D, η,σ d > 0. With this definition (D t ) is a Jacobi diffusion process; it has a linear, mean-reverting drift component and degenerates on the boundary. Moreover, subject to D ∈ (0, ξ max ) and min(D, ξ max − D) ≥ ξ maxσd , the process remains within the interval (0, ξ max ); its stationary distribution is a beta distribution and its mean is given byD (cf. [16]).

5.1.3.
Choice of Parameters. Tables 1, 2, 3 summarise the parameter values used for the numerical study that follows. We note that they do not correspond to a particular example of an electricity market. But they can be considered representative of a medium sized market whose fuel mix predominantly consists of coal and gas generators. Table 1 specifies the parameters for the bid and the emissions stack. Using (6) now with A t = 0 and D t = ξ max , for 0 ≤ t ≤ T , and with the assumption that there are 24 × 365 production hours in the year, we find that E max = 1.6519 × 10 8 .  The parameters relating to demand are given in Table 2.
ηDσ d r 10 21000 0.05 0.05 Table 2. Parameters for the demand process and the risk-free rate.
Calculating the cumulative emissions now for A t = 0 and demand at its mean level D t =D, for 0 ≤ t ≤ T , we find that E T = 1.2961 × 10 8 . This leads us to choose the cap slightly below this level, in order to incentivise a reduction in emissions. The parameters characterising the emissions trading scheme are given in Table 3. We note that here time is measured in years.

The Allowance Price Value Function.
We present the necessary boundary conditions for the allowance price valuation equation and discuss its solution. First, this is done in the setting of an emissions market with one compliance period, second we consider two periods connected by either banking and withdrawal or banking, borrowing and withdrawal.
With regards to the problem (11) (in the case of one period) or (14) (in the case of two periods) the question arises, at which points of the boundary we need to specify boundary conditions in addition to the terminal condition and what conditions make sense given the original stochastic problem (9) or (13). The former question is answered by considering the Fichera function f (cf. [27]). Defining n := (n d , n e ) to be the inward normal vector to the boundary, Fichera's function for the operator N (and L) reads t, D, E), D)n e , on ∂U T (or ∂U Ti ). In the case when the coefficients µ d and σ d are of the form prescribed in (20) we find that Given values for the demand for electricity and the cumulative emissions, the valuation equation representing the allowance pricing problem determines the arbitragefree price of an allowance certificate. We illustrate this dependency by solving the partial differential equation numerically, using the finite difference scheme explained in Appendix A.
5.2.1. One Compliance Period. The boundary condition at E = E max takes the form The condition (21) follows from the fact that, as soon as the cumulative emissions surpass the cap, every additional tonne of CO 2 is penalised at a rate π at time T .
The numerical results are displayed in Figure 5. At time t = T /2, the allowance price depends on the cumulative emissions to date and the current level of demand, as shown in Figure 5a. For each fixed level of emissions E = E T /2 , α(T /2, D, E T /2 ) is increasing in D. This makes intuitive sense, since for higher levels of demand, the corresponding market emissions rate is greater and consequently it is more likely that the cap will be reached. Similarly, fixing D = D T /2 results in α(T /2, D T /2 , E) being an increasing function of E. In particular, we can think of the current level of cumulative emissions determining an interval for the allowance price and the   demand for electricity setting the exact price within this interval. Further, we notice that the allowance price equals the discounted penalty, if cumulative emissions exceed the cap. At the end of the compliance period, α is given by the terminal condition (7). Figure 5b reflects the digital nature of the price at this time and its independence of D.

Multiple Compliance Periods -Banking and Withdrawal.
We illustrate the framework introduced in this section for the case of two consecutive compliance periods. We determine the prices of the first and second period allowance certificate as functions of demand and cumulative emissions. For i = 2, we solve the partial differential equation (14) with the terminal condition φ 2 given by (17). Further, the boundary condition at E = E max takes the form The problem is equivalent to the one-period pricing problem with the exception that the aggregate supply of certificatesÊ 2 cap depends on E 1 , the level of the cumulative emissions at the end of the first compliance period. As a result, the price of the second period allowance certificate depends not only on the current values of t, D and E but also on E 1 , i.e. α 2 = α 2 (·, ·, ·; E 1 ). We then solve (14) for i = 1 with the terminal condition φ 1 of the form (16), where A 2 T1 = α 2 (T 1 , D, 0; E). The boundary condition at E = E max now reads  Figure  6b. Concerning the price behaviour of the second period certificate, we note that it mirrors the one-period model with the initial allocation E cap replaced byÊ 2 cap .

5.2.3.
Multiple Compliance Periods -Borrowing, Banking and Withdrawal. As in the previous section, we analyse the prices of allowance certificates in this market, in the two-period setting. During the second compliance period the problem is equivalent to the market that only uses the banking and withdrawal mechanisms as described in §3.2.1. This is the case, because in both markets the effect on the aggregate supply of certificates during the second period is the same. Suppose the market is in compliance at time t = T 1 . Then, a number (E 1 cap − E 1 ) of certificates are banked to the second period and added to E 2 cap , independently of whether borrowing is allowed or not. Otherwise, if the market is not in compliance at time t = T 1 a number min(E 1 − E 1 cap , E 2 cap ) of certificates are withdrawn from E 2 cap (if borrowing is not allowed) or the same number are brought forward and hence also deducted from E 2 cap (if borrowing is allowed). Therefore, for i = 2 we solve the partial differential equation (14) with terminal condition (17) and obtain α 2 = α 2 (·, ·, ·; E 1 ). Subsequently, we solve (14) for i = 1 together with the terminal condition φ 1 given by (17), where A 2 T1 = α 2 (T 1 , D, 0; E). The boundary condition at E = E max is given by (23).  Figure 7. The value of the first-period allowance certificate (A 1 t ) t∈[0,T 1 ] at different times up to compliance, with two compliance periods, connected by the borrowing, banking and withdrawal mechanisms.

5.3.
The Impact of Cap-and-Trade. The raison d'être of any cap-and-trade scheme is to reduce emissions. More precisely its aim is to incentivise sufficient load shifting throughout the trading period for the cumulative emissions not to reach the cap. In our modelling setup we illustrate the effectiveness of cap-and-trade by calculating the expected cumulative emissions at the end of the compliance period for different levels of the penalty π. Recall that the penalty represents an upper bound for the allowance price; the case π = 0 corresponds to business-as-usual and increasing π to a gradually more aggressive cap-and-trade scheme.
We simulate the cumulative emissions process (E t ) using the Monte Carlo scheme explained in Appendix B, choosing D 0 = 0.7ξ max . We repeat this simulation for values of the penalty ranging from zero to 200 and calculate the mean of E T , denoted byÊ T . We note that for the present purpose of analysing the cumulative emissions, the simulation of demand should take place under the physical measure P, which is related to Q by the market price of demand risk. This measure can be accurately estimated from market data (see for example [15] for different approaches). In the absence of a detailed data analysis however, we follow the time honoured approach of letting the market price of demand risk be constant and equal to zero. Therefore, for the purposes of our simulation we work with the SDE (20).  Table 4. Monte Carlo estimate of the mean cumulative emissionsÊT and the corresponding standard errorσÊ. Figure 8 plots the results of the simulation of 10 6 paths. Under business-as-usual the cumulative emissions are expected to exceed the cap. As emissions trading is introduced the market reacts by abating but initially (at a penalty level of π = 25) the cumulative emissions still exceed the cap. Further increases in the penalty (close to π = 100) lead to sufficient load shifting in order for the market to reach a state of compliance. More aggressive regulation now only leads to small reductions in the cumulative emissions; i.e. our analysis confirms the well known stylized fact that emissions trading cannot incentivise firms to reduce cumulative emissions far below the cap.

A Call Option on Emissions.
For the numerical solution of the call option, we specify (24) v(t, D, E) = e −r(T −t) π − e r(T −τ ) K The condition (24) follows from recalling that when E = E max the value of the allowance certificate α is given by α(t, D, E) = e −r(T −t) π. For the same reasons as put forward in §3.1 boundary conditions at D = 0, ξ max and E = 0 are not necessary.
Because the partial differential equation (19) requires the allowance price as an input parameter, it is necessary to solve (19) and (11) in parallel in order to obtain the value of the option. The numerical scheme that determines the price of the call as a function of the demand for electricity and the cumulative emissions is explained in Appendix A.3 and the resulting value surface is plotted in Figure 9.

Conclusion
Emissions trading has become one of the most popular policy instruments employed by regulators to reduce global emissions. Allowance certificates -the key financial instruments in emission markets -and derivatives on them are traded actively on exchanges nowadays despite the lack of an established theoretical pricing framework taking the subtleties of these markets into account.
The contribution of this paper is threefold. First, by appealing to the bid stack, the key price-setting mechanism in electricity markets (which we suggest is a reasonable description of more general markets as well), we introduce the idea of load-shifting and show how the cost of carbon affects which firms are supplying electricity to the market. This immediately allows us to deduce the rate at which emissions accumulate during a compliance period. Because we derive the cumulative emissions process starting with an exogenously defined stochastic process for demand, we offer an explanation for the non-compliance event, which is the main price driver of allowance certificates. Second, we embed the load-shifting mechanism in a continuous time pricing framework for allowance certificates, taking the form of a forward-backward stochastic differential equation. Within this setup the extension to price derivatives on the allowance certificate is immediate. Third, in the setting of an emission market with multiple compliance periods, we analyse the impact different connecting mechanisms have on the allowance price.
Appendix A. Numerical Solution of the Allowance and Option Pricing Problem We comment on the numerical scheme employed to solve the allowance and option pricing PDEs.
A.1. One Compliance Period. We discretize the domainŪ T by choosing mesh widths ∆D, ∆E and a time step ∆t. The discrete mesh points (D i , E j , t k ) are then defined by D i := i∆D, E j := j∆E, t k := k∆t.
The finite difference scheme we employ produces approximations α k i,j , which are assumed to converge to the true solution α as the mesh width tends to zero.
Since the partial differential equation (11) is posed backwards in time with a terminal condition, we choose a backward finite difference for the time derivative in order to work with an explicit scheme.
In the E-direction we are approximating a conservation law PDE with discontinuous terminal condition. (For an in depth discussion of numerical schemes for these type of equations see [23]) The first derivative in the E-direction, relating to the nonlinear part of the partial differential equation, is discretised against the drift direction using a one-sided upwind difference. Because characteristic information is propagating in the direction of decreasing E, this one-sided difference is also used to calculate the value of the approximation on the part of the boundary corresponding to E = 0.
In the D-direction the equation is parabolic everywhere except on the boundary, where it degenerates. Hence we use central differences to discretize the first and second order derivative. At the boundaries corresponding to D = 0 and D = ξ max , where the second derivative vanishes and no boundary conditions need to be specified, we again use a one-sided difference in our numerical scheme.
With smooth boundary data, on a smooth domain and with a strictly decreasing (in α) coefficient µ e , the scheme described above can be expected to exhibit first order convergence. In our setting, we expect the discontinuous terminal condition and the fact that µ e is merely decreasing in α, to have adverse effects on the convergence rate.
We analyse the convergence of our numerical scheme in the supremum norm and the 1-norm at t = 0; i.e. we calculate where α h l represents the approximate solution given the vector of mesh parameters h l at refinement level l and α k i,j 1 := i,j α k i,j ∆D∆E.
The parameters that define the different mesh widths are displayed in Table 5; here we paid careful attention that our choice honours the Courant-Friedrichs-Lewy condition for the convergence of explicit schemes (cf. [23]). As mentioned above we expect the main contributions to the error to stem from the hyperbolic part of the equation. Therefore, we choose a very fine grid in the E-direction for our analysis.  Table 5. Parameters for the convergence analysis of the numerical scheme. Table 6 displays the results from our convergence study. Note that the error decays much faster in the 1-norm. This is not at all surprising, as we expect the error from the discontinuous terminal condition to propagate in the direction of decreasing E. This leads to a significant error on a small part of the gird, which is picked up by the infinity norm, whereas the approximation converges much faster everywhere else, as shown by our analysis of the error in the 1-norm.  Table 6. Numerical error in the supremum and the 1-norm. Figure 10 plots the error Err ∞ l as a function of the mesh width. From the slope of the line of best fit through the error points, we estimate the convergence rate of our scheme to be 0.9131 in the infinity norm. A.2. Multiple Compliance Periods. To deal with the path-dependency that enters the pricing problem in an emissions market with multiple compliance periods -in the case of banking and withdrawal -through the terminal condition (16) and (17), we have introduced the extra variable E 1 .
The problem is then solved backwards beginning with the second compliance period, for which we solve the partial differential equation (14) with corresponding boundary conditions and the terminal condition (17), introducing an extra dimension for the variable E 1 . We then store α 2 (T 1 , D, 0; E 1 ).
Subsequently, we solve the partial differential equation corresponding to the first compliance period, using the stored values of α 2 for the evaluation of the terminal condition.
We compute the numerical approximation to each α i using the scheme described in A.1. For a market in which banking, borrowing and withdrawal are implemented, the terminal condition at the end of the first period is modified in the obvious way.
We note that compared with the single-period problem, the multi-period problem has an extra dimension due to the variable E 1 . When solving the problem beginning with the second compliance period, this increases the complexity in two ways. First, the partial differential equation (14) must be solved for a sufficiently large number of values of E 1 . Second, we need to store α 2 (T 2 , D, 0) since it is needed for the terminal condition of α 1 .
A.3. Option Pricing Problem. We use the obvious modification of the earlier scheme described in Appendix A.1 to solve the option pricing partial differential equation (19).

Appendix B. Monte Carlo Simulation of Cumulative Emissions
Let (D k , E k , A k ) denote the discrete time approximation to the FBSDE solution (D t , E t , A t ) on the time grid 0 < ∆t < 2∆t < . . . < n k ∆t = T . At each time step we calculate A k by interpolating the discrete approximation α k i,j at D k , E k , beginning with the initial values D 0 = d, E 0 = 0. The approximations (D k , E k ) are obtained using a simple Euler scheme (cf. [17]). The discretized version of (D t ) is forced to be instantaneously reflecting at the boundaries D k = 0 and D k = ξ max .
Using this discretization we simulate n c paths and, as usual, calculate the mean cumulative emissionsÊ T , given bŷ where E i n k denotes the outcome of the simulation of the ith path. The corresponding standard errorσÊ is obtained bŷ Oxford-Man Institute, Oxford, UK E-mail address: schwarz@maths.ox.ac.uk