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# The Valuation of Clean Spread Options: Linking Electricity, Emissions and Fuels

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Spread options are most oen used in the commodity and energy markets to encapsulate the protability of a production process by comparing the price of a rened product to the costs of production, including, but not limited to, the prices of the inputs to the production process. When the output commodity is electric power, such spread options are called spark spreads when the electricity is produced from natural gas, and dark spreads when the electricity is produced from coal. Both processes are the sources of Greenhouse Gas (GHG) emissions, in higher quantities for the latter than the former. In this paper we concentrate on the production of electricity and CO2 emissions and the resulting dependence structure between prices.
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THE VALUATION OF CLEAN SPREAD OPTIONS: LINKING ELECTRICITY, EMISSIONS
AND FUELS
REN´
E CARMONA, MICHAEL COULON, AND DANIEL SCHWARZ
Abstract. The purpose of the paper is to present a new pricing method for clean spread options, and to illustrate
its main features on a set of numerical examples produced by a dedicated computer code. The novelty of the
approach is embedded in the use of structural models as opposed to reduced-form models which fail to capture
properly the fundamental dependencies between the economic factors entering the production process.
1. Introduction
Spread options are most often used in the commodity and energy markets to encapsulate the proﬁtability of a
production process by comparing the price of a reﬁned product to the costs of production including, but not
limited to, the prices of the inputs to the production process. When the output commodity is electric power, such
spread options are called spark spreads when the electricity is produced from natural gas, and dark spreads when
the electricity is produced from coal. Both processes are the sources of GreenHouse Gas (GHG) emissions, in higher
quantities for the latter than the former. In this paper we concentrate on the production of electricity and CO2
emissions and the resulting dependence structure between prices.
Market mechanisms aimed at controlling CO2emissions have been implemented throughout the world, and whether
they are mandatory or voluntary, cap-and-trade schemes have helped to put a price on carbon in the US and in
Europe. In the academic literature, equilibrium models have been used to show what practitioners have known all
along, namely that the price put on CO2by the regulation should be included in the costs of production to set
the price of electricity. (cf. [Carmona et al., 2010]) Strings of spark spread options (options on the spread between
the price of 1MWh of electricity and the cost of the amount of natural gas needed to produce such a MWh) with
maturities covering a given period are most frequently used to value the optionality of a gas power plant which
can be run when it is proﬁtable to do so (namely when the price of electricity is greater than the cost of producing
it), and shut down otherwise. In a nutshell, if an economic agent takes control on day t, of a gas power plant
for a period [T1, T2], then for every day τ[T1, T2] of this period, he or she can decide to run the power plant
when Pτ> hgSg
τ+Kand book a proﬁt PτhgSg
τKfor each unit of power produced, and shut the plant
down if PτhgSg
τ+K. Moreover, ignoring constraints such as ramp-up rates and start-up costs, this scheduling
is automatically induced when generators bid at the level of their production costs in the day-ahead auction for
power. Here Pτdenotes the price at which one unit (1 MWh) of power can be sold on day τ,Sg
τthe price of one
unit of natural gas (typically one MMBtu), hgthe eﬃciency or heat rate of the plant (i.e. the number of units of
natural gas needed to produce one unit of electricity) and Kthe daily (ﬁxed) costs of operations and maintenance
of the plant. So in this somewhat oversimpliﬁed analysis of the optionality of the plant, the value at time tof
the control of the plant operation on day τcan be expressed as er(τt)E[(PτhgSg
τK)+|Ft] where as usual,
the exponent +stands for the positive part, i.e. x+=xwhen x0 and x+= 0 otherwise, rfor the constant
interest rate used as discount factor to compute present values of future cash ﬂows, and Ftdenotes the information
available on day twhen the conditional expectation is actually computed. So the operational control (for example
as aﬀorded by a tolling contract) of the plant over the period [T1, T2] could be valued on day tas
VP P
t=
T2
X
τ=T1
er(τt)E[(PτhgSg
τK)+|Ft].
This rather simplistic way of valuing a power generation asset in the spirit of the theory of real options, had far-
reaching implications in the developments of the energy markets, and is the main reason why spread options are
of the utmost importance. However, such a valuation procedure is ﬂawed in the presence of emission regulation
as the costs of production have to include the costs speciﬁc to the regulation. To be more speciﬁc, the day-τ
potential proﬁt (PτhgSg
τK)+of the spark spread has to be modiﬁed to (PτhgSg
τegAτK)+in order
Partially supported by NSF - DMS-0739195.
1
arXiv:1205.2302v1 [q-fin.PR] 10 May 2012
2 REN´
E CARMONA, MICHAEL COULON, AND DANIEL SCHWARZ
to accommodate the cost of the regulation. Here Aτis the price of one allowance certiﬁcate worth one ton of CO2
equivalent, and egis the emission coeﬃcient of the plant, namely the number of tons of CO2emitted by the plant
during the production of one unit of electricity. Such a spread is often called a clean spread to emphasize the fact
the externality is being paid for, and the real option approach to power plant valuation leads to the following clean
price
VCP P
t=
T2
X
τ=T1
er(τt)E[(PτhgSg
τegAτK)+|Ft]
for the control of the plant over the period [T1, T2] in the presence of the regulation.
In order to price such cross-commodity derivatives, a joint model is clearly required for fuel prices, electricity
prices and carbon allowance prices. Various studies have analyzed the strong links between these price series
(cf. [De Jong and Schneider, 2009, Koenig, 2011]). Many reduced-form price models have been proposed for elec-
tricity (cf. [Benth et al., 2008, Eydeland and Wolyniec, 2003] for examples) with a focus on capturing its styl-
ized features such as seasonality, high volatility, spikes, mean-reversion and fuel price correlation. On the other
hand, many authors have argued that these same features are better captured via a structural approach, mod-
elling the dynamics of underlying factors such as demand (load), capacity and fuel prices (early examples include
[Barlow, 2002, Cartea and Villaplana, 2008, Pirrong and Jermakyan, 2008, Coulon and Howison, 2009]).
Similarly, for carbon emission allowances, exogenously speciﬁed processes that model prices directly have been
proposed by some (cf. [Carmona and Hinz, 2011]). Others have instead treated the emissions process as the
exogenously speciﬁed underlying factor; in this case the allowance certiﬁcate becomes a derivative on cumulative
emissions (cf. [Seifert et al., 2008, Chesney and Taschini, 2012]). However, these models do not take into account
the important feedback from the allowance price to the rate at which emissions are produced in the electricity sector
— a feature, which is crucial for the justiﬁcation of any implementation of a cap-and-trade scheme. In a discrete-
time framework this feedback mechanism has been addressed, for example in [Coulon, 2009, Carmona et al., 2010].
In continuous-time the problem has been treated in [Carmona et al., 2012b] and [Howison and Schwarz, 2012],
whereby the former models the accumulation of emissions as a function of an exogenously speciﬁed electricity price
process, while the latter uses the bid-stack mechanism to infer the emissions rate.
The literature on spread options is extensive. In industry, Margrabe’s classical spread option formula (cf. [Margrabe, 1978])
is still widely used, and has been extended by various authors (see [Carmona and Durrleman, 2003] for an overview)
including to the three commodity case, as required for the pricing of clean spreads (cf. [Alos et al., 2011]).
[Carmona and Sun, 2012] analyse the pricing of two-asset spread options in a multiscale stochastic volatility model.
For electricity markets, pricing formulae for dirty spreads based on structural models have been proposed in
[Carmona et al., 2012a], in which a closed-form formula is derived in the case of K= 0, and in [A¨ıd et al., 2012],
in which semi-closed form formulae are derived for K6= 0 at the expense of a ﬁxed merit order.
The original contributions of the paper are twofold. First, we express the value of clean spread options in a formula-
tion where demand for power and fuel prices are the only factors whose stochastic dynamics are given exogenously,
and where the prices of power and emission allowances are derived from a bid-stack based structural model and
a forward backward stochastic diﬀerential system respectively. The second contribution is the development of a
numerical code for the computation of the solution of the pricing problem. First we solve a 4+1 dimensional
semilinear partial diﬀerential equation to compute the price of an emission allowance, and then we use Monte Carlo
techniques to compute the price of the spread option. These computational tools are used to produce the numerical
results of case studies presented in §6 of the paper for the purpose of illustrating the impact of a carbon regulation
on the price of spread options. In this section we ﬁrst compare the price of spark and dark spread options in two
diﬀerent markets, one with no emissions regulation in place and the other governed by an increasingly strict cap-
and-trade system. Second, we analyze the impact that diﬀerent merit order scenarios have on the option prices.
Third, we demonstrate the diﬀerence between the structural and the reduced-form approach by comparing the
option prices produced by our model with those produced by two key candidate reduced-form models. Fourth and
last, we contrast two competing policy instruments: cap-and-trade, represented by the model we propose, and a
ﬁxed carbon tax.
2. The Bid Stack: Price Setting in Electricity Markets
In order to capture the dependency of electricity price on production costs and fundamental factors in a realistic
manner, we use a structural model in the spirit of those reviewed in the recent survey of [Carmona and Coulon, 2012].
The premises of structural models for electricity prices depend upon an explicit construction of the supply curve.
THE VALUATION OF CLEAN SPREAD OPTIONS: LINKING ELECTRICITY, EMISSIONS AND FUELS 3
Since electricity is sold at its marginal cost, the electricity spot price is given by evaluation of the supply function
for the appropriate values of factors used to describe the costs of production in the model.
In practice, electricity producers submit day-ahead bids to a central market operator, whose task it is to allocate
the production of electricity amongst them. Typically, ﬁrms’ bids have the form of price-quantity pairs, with each
pair comprising the amount of electricity the ﬁrm is willing to produce, and the price at which the ﬁrm is willing
to sell this quantity. Given the large number of generators in most markets, it is common in structural models
to approximate the resulting step function of market bids by a continuous increasing curve. Firms’ bid levels are
determined by their costs of production. An important feature of our model, distinguishing it from most of the
commonly used structural models is to include, as part of the production costs, the costs incurred because of the
existence of an emissions regulation.
We assume that, when deciding which ﬁrms to call upon to produce electricity, the market operator adheres to the
merit order, a rule by which cheaper production units are called upon before more expensive ones. For simplicity,
operational and transmission constraints are not considered.
Assumption 1. The market operator arranges bids according to the merit order, in increasing order of production
costs.
The map resulting from ordering market supply in increasing order of electricity costs of production is what is
called the bid stack. As it is one of the important building blocks of our model, we deﬁne it in a formal way for
later convenience.
Deﬁnition 1. The bid stack is given by a measurable function
b: [0,¯x]×R×Rn3(x, a, s)b(x, a, s)R,
with the property that for each ﬁxed (a, s)R×Rn, the function [0,¯x]3x b(x, a, s) is strictly increasing.
In this deﬁnition, ¯xR++ represents the market capacity (measured in MWh) and the variable xthe supply of
electricity. The integer nN\ {0}gives the number of economic factors (typically the prices in eof the fuels
used in the production of electricity), and sRnthe numeric values of these factors. Here and throughout the
rest of the paper the cost of carbon emissions (measured in eper metric ton of CO2) is denoted by a. So for
a given allowance price, say a, and fuel prices, say s, the market is able to supply xunits of electricity at price
level b=b(x, a, s) (measured in eper MWh). In other words, b(x, a, s) represents the bid level of the marginal
production unit in the event that demand equals x.
The choice of a function bwhich captures the subtle dependence of the electricity price upon the level of supply
and the production costs, is far from trivial, and diﬀerent approaches have been considered in the literature, as
reviewed recently by [Carmona and Coulon, 2012]. In §5.1 we extend the model proposed in [Carmona et al., 2012a]
to include the cost of carbon as part of the variable costs driving bid levels.
3. Risk-Neutral Pricing of Allowance Certificates
As the inclusion of the cost of emission regulation in the valuation of spread options is the main thrust of
the paper, we explain how emission allowances are priced in our model. The model we introduce is close to
[Howison and Schwarz, 2012]. However we extend the results found therein to allow the equilibrium bids of gener-
ators to be stochastic and driven by fuel prices, a generalization that is vital for our purpose.
We suppose that carbon emissions in the economy are subject to cap-and-trade regulation structured as follows:
at the end of the compliance period, each registered ﬁrm needs to oﬀset its cumulative emissions with emission
allowances or incur a penalty for each excess ton of CO2not covered by a redeemed allowance certiﬁcate. Initially,
ﬁrms acquire allowance certiﬁcates through free allocation, e.g. through National Allocation Plans (NAP) like in
the initial phase of the European Union (EU) Emissions Trading Scheme (ETS), or by purchasing them at auctions
like in the Regional Greenhouse Gas Initiative (RGGI) in the North East of the US. Allowances change hands
throughout the compliance period. Typically, a ﬁrm which thinks that its initial endowment will not suﬃce to
cover its emissions will buy allowances, while ﬁrms expecting a surplus will sell them. Adding to these naturals,
speculators enter the market providing liquidity. Allowances are typically traded in the form of forward contracts
and options. In this paper, we denote by Atthe spot price of an allowance certiﬁcate maturing at the end of the
compliance period. Because their cost of carry is negligible, we treat them as ﬁnancial products liquidly traded in
a market without frictions, and in which long and short positions can be taken.
4 REN´
E CARMONA, MICHAEL COULON, AND DANIEL SCHWARZ
In a competitive equilibrium, the level of cumulative emissions relative to the cap (i.e. the number of allowance
certiﬁcates issued by the regulation authority) determines whether — at the end of the compliance period — ﬁrms
will be subjected to a penalty payment and create a demand for allowance certiﬁcates. See [Carmona et al., 2010]
for details. For this reason, allowance certiﬁcates should be regarded as derivatives on the emissions accumulated
throughout the trading period. This type of option written on a non-tradable underlying interest is rather frequent
in the energy markets: temperature options are a case in point.
3.1. The Market Emissions Rate. As evidenced by the above discussion, the rate at which CO2is emitted
in the atmosphere as a result of electricity production has to be another important building block of our model.
Clearly at any given time, this rate is a function of the amount of electricity produced and because of their impact
on the merit order, the variable costs of production, including fuel prices, and notably, the carbon allowance price
itself.
Deﬁnition 2. The market emissions rate is given by a bounded function
µe: [0,¯x]×R×Rn3(x, a, s)µe(x, a, s)R+,
which is Lipschitz continuous in its three variables, strictly increasing in xwhen aand sare held ﬁxed, and strictly
decreasing in awhen xand sare ﬁxed.
With the deﬁnition above, for a given level of electricity supply and for given allowance and fuel prices, µe=
µe(x, a, s) represents the rate at which the market emits, measured in tons of CO2per hour. Cumulative emissions
are then computed by integrating the market emissions rate over time. The monotonicity property in xmakes
sense since any increase in supply can only increase the emissions rate. Similarly, as the cost of carbon increases the
variable costs (and hence the bids) of pollution intensive generators increase by more than those of environmentally
friendlier ones. Dirtier technologies become relatively more expensive and are likely to be scheduled further down
in the merit order. As a result cleaner technologies are brought online earlier, hence the monotonicity in a.
In §5.2 we propose a speciﬁc functional form for µeconsistent with the bid stack model introduced in §5.1.
3.2. The Pricing Problem. We shall use the following notation. For a ﬁxed time horizon TR+, let (W0
t, Wt)t[0,T ]
be a (n+ 1)-dimensional standard Wiener process on a probability space (Ω,F,P), F0:= (F0
t) the ﬁltration gener-
ated by W0,FW:= (FW
t) the ﬁltration generated by W, and F:= F0∨F Wthe market ﬁltration. All relationships
between random variables are to be understood in the almost surely sense.
Consumers’ demand for electricity is given by an F0
t-adapted stochastic process (Dt). In response to this demand
producers supply electricity, and we assume that demand and supply are always in equilibrium, so that at any
time t[0, T ] an amount Dtof electricity is supplied. The prices of fuels are observed FW
processes (St)t[0,T ], where St:= (S1
t, . . . , Sn
t). If the price of an allowance certiﬁcate at time t, say At, becomes
available, as we will see in §3.3, (At)T[0,T ]will be constructed as a Ft-adapted stochastic process solving a Forward
Backward Stochastic Diﬀerential Equation (FBSDE). The rate of emission µe(Dt, At, St) can then be evaluated
and the cumulative emissions computed by integration over time, resulting in a Ft-adapted process (Et)t[0,T ].
In order to avoid the diﬃculties of estimating the market price of risk (see for example [Eydeland and Wolyniec, 2003]
for a discussion of some possible ways to approach this thorny issue), we choose to specify the dynamics of the
processes (Dt)t[0,T ]and (St)t[0,T ]under a risk neutral measure QPchosen by the market for pricing purposes.
3.3. An FBSDE for the Allowance Price. We assume that at time t= 0, demand for electricity is known.
Thereafter, it evolves according to an Itˆo diﬀusion. Speciﬁcally, for t[0, T ], demand for electricity Dtis the
unique strong solution of a stochastic diﬀerential equation of the form
(1) dDt=µd(t, Dt)dt+σd(Dt)d ˜
W0
t, D0=d0(0,¯x),
where ( ˜
Wt) is an Ft-adapted Q-Brownian motion. The time dependence of the drift allows us to capture the
seasonality observed in electricity demand.
Similarly to demand, the prices of the fuels used in the production processes satisfy a system of stochastic diﬀerential
equations written in a vector form as follows:
(2) dSt=µs(St)dt+σs(St)d ˜
Wt, S0=s0Rn, t [0, T ].
Cumulative emissions are measured from the beginning of the compliance period when time t= 0, so that E0= 0.
Subsequently, they are determined by integrating over the market emissions rate µeintroduced in Deﬁnition 2. So
THE VALUATION OF CLEAN SPREAD OPTIONS: LINKING ELECTRICITY, EMISSIONS AND FUELS 5
assuming that the price Atof an allowance certiﬁcate is knwon, the cumulative emissions process is represented by
a bounded variation process; i.e. for t[0, T ],
(3) dEt=µe(Dt, At, St)dt, E0= 0.
Note that with this deﬁnition the process (Et) is non-decreasing, which makes intuitive sense considering that it
represents a cumulative quantity.
To complete the formulation of the pricing model, it remains to characterize the allowance certiﬁcate price process
(At)t[0,T ]. If our model is to apply to a one compliance period scheme, in a competitive equilibrium, at the end
of the compliance period t=T, its value is given by a deterministic function of the cumulative emissions:
(4) AT=φ(ET),
where φ:RRis bounded, measurable and non-decreasing. Usually φ(·) := πI,)(·), where πR+denotes the
penalty paid in the event of non-compliance and Γ R+the cap chosen by the regulator as the aggregate allocation
of certiﬁcates. See [Carmona et al., 2010] for details. Since the discounted allowance price is a martingale under
Q, it is equal to the conditional expectation of its terminal value, i.e.
(5) At= exp (r(Tt)) EQ[φ(ET)| Ft],for t[0, T ],
which implies in particular that the allowance price process (At) is bounded. Since the ﬁltration (Ft) is being gener-
ated by the Wiener processes, it is a consequence of the Martingale Representation Theorem (cf. [Karatzas and Shreve, 1999])
that the allowance price can be represented as an Itˆo integral with respect to the Brownian motion ( ˜
W0
t,˜
Wt). It
follows that
(6) dAt=rAtdt+Z0
td˜
W0
t+Zt·d˜
Wt,for t[0, T ]
for some Ft-adapted square integrable process (Z0
t, Zt).
Combining equations (1), (2), (3), (4) and (6), the pricing problem can be reformulated as the solution of the
FBSDE
(7)
dDt=µd(t, Dt)dt+σd(Dt)d ˜
W0
t, D0=d0(0,¯x),
dSt=µs(St)dt+σs(St)d ˜
Wt, S0=s0Rn,
dEt=µe(Dt, At, St)dt, E0= 0,
dAt=rAtdt+Z0
td˜
W0
t+Zt·d˜
Wt, AT=φ(ET).
Notice that the ﬁrst two equations are standard stochastic diﬀerential equations (in the forward direction of time)
which do not depend upon the cumulative emissions and the allowance price. We will choose their coeﬃcients so
that existence and uniqueness of solutions hold. To be more speciﬁc we make the following assumptions on the
coeﬃcients of (7):
Assumption 2. The functions µd: [0, T ]×[0,¯x]R,σd: [0,¯x]R,µs:RnRn,σs:RnRn×Rnare
such that the ﬁrst two equations in (7) have a unique strong solution.
3.4. Existence of a Solution to the Allowance Pricing Problem.
Theorem 1. We assume that Assumption 2 holds and that µeis Lipshitz with respect to the variable auniformly
in xand s, and that µe(x, 0, s)is uniformly bounded in xand s. Then if φis bounded and Lipschitz, the FBSDE
(7) has a unique square integrable solution.
Proof. Assumption 2 being satisﬁed, and the ﬁrst two equations of (7) being decoupled from the remaining ones,
there exist adapted processes (Dt) and (St) with values in [0,¯x] and Rnrespectively, unique strong solutions of
the ﬁrst two equations of (7). Once these two processes are constructed, we can plug their values into the last
two equations of (7), and treat the resulting equations as an FBSDE with random coeﬃcients. Existence and
uniqueness hold because of Theorem 7.1 of [Ma et al., 2011] 1. Strictly speaking this result is only proved for one-
dimensional processes. In the present situation, while Etand Atare indeed one-dimensional, the Wiener process
is (n+ 1)-dimensional and we cannot apply directly Theorem 7.1 of [Ma et al., 2011]. However, a close look at
the proof of this result shows that what is really needed is to prove the well-posedness of the characteristic BSDE,
and the boundedness of its solution and the solutions of the dominating Ordinary Diﬀerential Equations (ODE).
In the present situation, these equations are rather simple due to the fact that Ethas bounded variation, and as a
consequence, its volatility vanishes. The two dominating ODEs can be solved explicitly and one can check that the
1We would like to thank Francois Delarue for suggesting this strategy and the use of [Ma et al., 2011] in the present set-up.
6 REN´
E CARMONA, MICHAEL COULON, AND DANIEL SCHWARZ
solutions are bounded by inspection. Moreover, the function φfor the terminal condition being uniformly Lipschitz,
the characteristic BSDE is one-dimensional, though driven by a multi-dimensional Brownian motion, its terminal
condition is bounded, and Kobylanski’s comparison results (see the original contribution [Kobylanski, 2000]) can
be used to conclude the proof.
The above existence result is proven for a terminal condition given by a smooth function φ. As already mentioned
earlier, single compliance period equilibrium models most often require that the function φtake two values, and
the terminal condition φ(ET) equal the penalty when the regulatory cap is exceeded, i.e. when ET>Γ, and zero
when ET<Γ. [Carmona and Delarue, 2012] proved that a weaker form of existence and uniqueness of a solution
to the FBSDE still holds when φis discontinuous (in particular when φis an indicator function). Given that the
decoupling ﬁeld constructed in [Ma et al., 2011] is uniformly Lipschitz, we conjecture that a proof in the spirit of
the one given in [Carmona and Delarue, 2012] should work here and provide this weaker form of existence and
uniqueness. However, Carmona and Delarue also proved that under a strict monotonicity assumption on µe(which
should hold in our case for intuitive reasons), the aggregate emissions were equal with positive probability to the
cap at the end of the compliance period, and the terminal condition could not be prescribed for all the scenarios.
We suspect that in the present situation, the cumulative emissions equal the cap (i.e. ET= Γ) for a set of scenarios
of positive probability, and the terminal price of an allowance ATcannot be prescribed in advance on this set of
scenarios.
4. Valuing Clean Spread Options
In this section we consider the problem of spread option pricing as described in the introduction. Whether the goal
is to value a physical asset or risk manage ﬁnancial positions, one needs to compute the price of a European call
option on the diﬀerence between the price of electricity and the costs of production for a particular power plant.
The costs that we take into account are the ﬁxed operation and maintenance costs, the cost of the fuel needed to
generate one MWh of electricity and the cost of the ensuing emissions. Letting the Ft-adapted process (Pt) denote
the spot price of electricity, and recasting the informal discussion of the introduction with the notation we chose
to allow for several input fuels, a clean spread option with maturity τ[0, T ] is characterized by the payoﬀ
(PτhvSv
τevAτK)+,
where Krepresents the value of the ﬁxed operation and maintenance costs, hvR++ and evR++ denote the
speciﬁc heat and emissions rates of the power plant under consideration, and Sv∈ {S1, . . . , Sn}is the price at time
τof the fuel used in the production of electricity. In the special case when Svis the price of coal (gas) the option
is known as a clean dark (spark) spread option.
Since we are pricing by expectation, the value Vv
tof the clean spread is given by the conditional expectation under
the pricing measure of the discounted payoﬀ; i.e.
Vv
t= exp(r(τt))EQh(PτhvSv
τevAτK)+|Fti,for t[0, τ ].
5. A Concrete Two-Fuel Model
We now turn to the special case of two fuels, coal and gas.
5.1. The Bid Stack. Our bid stack model is a slight variation on the one we proposed in [Carmona et al., 2012a].
Here we extend to include the cost of emissions as part of the variable costs driving ﬁrms’ bids.
We assume that the coal and gas generators have aggregate capacities ¯xcand ¯xgrespectively, so that the market
capacity is ¯x= ¯xc+ ¯xg, and their bid levels are given by linear functions of the allowance price and the price of the
fuel used for the generation of electricity. We denote these bid functions by bcand bgrespectively. The coeﬃcients
appearing in these linear functions correspond to the marginal emissions rate (measured in ton equivalent of CO2
per MWh) and the heat rate (measured in MMBtu per MWh) of the technology in question. Speciﬁcally, for
i∈ {c, g}, we assume that
(8) bi(x, a, s) := ei(x)a+hi(x)s, for (x, a, s)[0,¯xi]×R×R,
where the marginal emissions rate eiand the heat rate hiare given by
ei(x) := ˆeiexp (mix)
hi(x) := ˆ
hiexp (mix),for x[0,¯xi].
THE VALUATION OF CLEAN SPREAD OPTIONS: LINKING ELECTRICITY, EMISSIONS AND FUELS 7
Here ˆei,ˆ
hi,miare strictly positive constants. We allow the marginal emissions rate and the heat rate of each
technology to vary to reﬂect diﬀerences in eﬃciencies within the ﬂeet of coal and gas generators. Since less eﬃcient
plants with higher heat rates have correspondingly higher emissions rates, it is a reasonable approximation to
assume that for each technology the ratio hi/eiis ﬁxed.
Proposition 1. With bcand bgas above and I={c, g}, the market bid stack bis given by
b(x, a, s) =
ˆeia+ˆ
hisiexp (mix),if bi(x, a, si)bj(0, a, sj)for i, j I , i 6=j,
ˆeia+ˆ
hisiexp (mi(x¯xj)) ,if bi(x¯xj, a, si)> bj(0, a, sj)for i, j I , i 6=j,
QiIˆeia+ˆ
hisiβiexp (γx),otherwise
for (x, a, s)[0,¯x]×R×R2, where βi=mM\{i}
mc+mgand γ=mcmg
mc+mg.
Proof. The proof is a straightforward extension of Corollary 1 in [Carmona et al., 2012a].
5.2. The Emissions Stack. In order to determine the rate at which the market emits we need to know which
generators are supplying electricity at any time. By the merit order assumption the market operator calls upon ﬁrms
in increasing order of their bid levels. Therefore, given electricity, allowance and fuel prices (p, a, s)R×R×R2,
for i∈ {c, g}, the set of active generators of fuel type iis given by {x[0,¯xi] : bi(x, a, s)p}.
Proposition 2. Assuming that the market bid stack is of the form speciﬁed in Proposition 1, the market emissions
rate µeis given by
(9) µe(x, a, s) := ˆeg
mgexp mgˆ
b1
g(b(x, a, s), a, sg)1+ˆec
mcexp mcˆ
b1
c(b(x, a, s), a, sc)1
for (x, a, s)[0,¯x]×R×R2, where for i∈ {c, g}we deﬁne
ˆ
b1
i(p, a, si) := 0 ¯xi1
mi
log p
ˆeia+ˆ
hisi,
for (p, a, s)R×R×R2.
Proof. The market emissions rate follows from integrating the marginal emissions rate eifor each technology over
the corresponding set of active generators and then summing the two. Given the monotonicity of biin xand its
range [0,¯xi], the function ˆ
b1
idescribes the quantity of electricity supplied by fuel i∈ {c, g}, and hence the required
upper limit of integration.
5.3. Specifying the Exogenous Stochastic Factors.
The Demand Process. We posit that under Q, the process (Dt) satisﬁes for t[0, T ] the stochastic diﬀerential
equation
dDt=ηDt¯
D(t)dt+p2ηˆσDt(¯xDt)d ˜
Wt, D0=d0(0,¯x),
where [0, T ]3t ¯
D(t)(0,¯x) is a deterministic function giving the level of mean reversion of the demande and
η, ˆσR++ are constants. With this deﬁnition (Dt) is a Jacobi diﬀusion process; it has a linear, mean-reverting
drift component and degenerates on the boundary. Moreover, subject to min( ¯
D(t),¯x¯
D(t)) ¯xˆσ, for t[0, T ], the
process remains within the interval (0,¯x) at all times (cf. [Forman and Sørensen, 2008]). To capture the seasonal
character of demand, we choose a function ¯
D(t) of the form:
¯
D(t) := ϕ0+ϕ1sin(2πϑt),
where the coeﬃcients will be chosen below.
The Fuel Price Processes. We assume that the prices of coal (Sc
t) and gas (Sg
t) follow correlated exponential OU
processes under the measure Q; i.e., for i∈ {c, g}and t[0, T ],
dSi
t=ηilog Si
t¯siˆσ2
i
2ηiSi
tdt+ ˆσiSi
td˜
Wi
t, Si
0=si
0R++,
where d hWc, W git=ρdt.
8 REN´
E CARMONA, MICHAEL COULON, AND DANIEL SCHWARZ
ˆ
hcˆecmc¯xcˆ
hgˆegmg¯xg¯x
3 0.9 0.00005 12000 7 0.4 0.00003 18000 30000
Table 1. Parameters relating to the bid stack and the emissions stack.
η ϕ0ϕ1ϑˆσ d0
50 21000 3000 1 0.1 21000
Table 2. Parameters relating to the demand process.
ηc¯scˆσcsc
0ηg¯sgˆσgsg
0ρ
1.5 2 0.5 exp(2) 1.5 2 0.5 exp(2) 0.3
Table 3. Parameters relating to the fuel price processes.
πΓT r
100 1.4e+08 1 0.05
Table 4. Parameters relating to the cap-and-trade scheme.
6. Numerical Analysis
We now turn to the detailed analysis of the model we propose. For this purpose we consider a number of case
studies in §6.2 to §6.5. To produce the following results we used the numerical schemes explained in Appendix A
and Appendix B.
6.1. Choice of Parameters. The tables in this section summarise the parameters used for the numerical analysis
of our model that follows below. We refer to the parameters speciﬁed in Tables 1 - 5 as the ‘base case’ and indicate
whenever we depart from this choice. Note that our parameter choices do not correspond to a particular electricity
market, but that all values are within a realistic realm.
Table 1 summarises the parameters specifying the bid curves. We consider a medium sized electricity market served
by coal and gas generators and with gas being the dominant technology. For the marginal emission rates, Table
1 implies that ec[0.9,1.64] and eg[0.4,0.69] (both measured in tCO2per MWh), so that all gas plants are
‘cleaner’ than all coal plants. For the heat rates, we observe that hc[3,5.5] and hg[7,12] (both measured in
MMBtu per MWh). Using (9) now with Dt= ¯x, for 0 tT, and the assumption that there are 8760 production
hours in the year, we ﬁnd, denoting the the maximum cumulative emissions by ¯e, that ¯e= 2.13e+ 08.
Table 2 contains the parameters for the demand process (Dt). We model periodicities on an annual and a weekly
time scale and the chosen rate of mean reversion assumes that demand reverts to its (time dependent) mean over
the course of one week.
In Table 3 we summarise the parameters that specify the behavior of the prices of coal and gas. Both are chosen
to be slowly mean-reverting, at least in comparison to demand. To ease analysis, we assume that all parameters
are identical for the two fuels, including mean price levels, both measured in MMBtu.
Table 4 deﬁnes the cap-and-trade scheme that we assume to be in place. The duration of the compliance period
Tis measured in years and we set the cap at 70% of the upper bound ¯efor the cumulative emissions, in order to
incentivise a reduction in emissions. This choice of parameters results in A0being approximately equal to π/2,
a value for which there is signiﬁcant initial overlap between gas and coal bids in the stack. Furthermore, the
parameters imply a bid stack structure such that at mean levels of coal and gas prices, At= 0 pushes all coal bids
below gas bids, while for At=πalmost all coal bids are above all gas bids.
Finally, in Table 5 we specify the four spread option contracts used in the base case scenario to represent high
and low eﬃciency coal plants, and high and low eﬃcency gas plants. (Note that low eﬃciency means dirtier and
corresponds to high hvand evand vice versa.)
THE VALUATION OF CLEAN SPREAD OPTIONS: LINKING ELECTRICITY, EMISSIONS AND FUELS 9
High Eﬀ. Coal Low Eﬀ. Coal High Eﬀ. Gas Low Eﬀ. Gas
hcechcechgeghgeg
3.5 1.05 5.0 1.5 7.5 0.43 11.5 0.66
Table 5. Parameters relating to the spread options.
We now consider a series of case studies to investigate various features of the model’s results in turn. As the model
captures many diﬀerent factors and eﬀects, this allows us to isolate some of the most important implications. In
Case Study I, we investigate the impact on coal and gas plants of diﬀerent eﬃciencies of creating an increasingly
strict carbon emissions market. In Case Study II, we assess the impact on these plants of changes in initial fuel
prices. In Case Study III, we compare spread option prices in our model with two simple reduced-form approaches
for At, which allows us to better understand the role of key model features such as bid stack driven abatement.
Finally in Case Study IV, we consider the overall impact of cap-and-trade markets in the electricity sector, by
comparing with a well-known alternative, a ﬁxed carbon tax.
6.2. Case Study I: Impact of the Emissions Market. The ﬁrst eﬀect that we are interested in studying in
the model is the impact of the cap-and-trade market on clean spread option prices, for increasingly strict levels
of the cap Γ. At one extreme (when the cap is so generous that At0, for all t[0, T ]), results correspond to
the case of a market without a cap and trade system, while at the other extreme (when the cap is so strict that
Atπexp(r(Tt)), for all t[0, T ]), there is essentially a very high carbon tax which tends to push most
coal generators above gas generators in the stack. It is intuitively clear that higher carbon prices typically lead to
higher spark spread option prices and lower dark spread option prices, thus favouring gas plants over coal plants,
but the relationships can be more involved as they vary between low eﬃciency and high eﬃciency plants.
In Figure 1, we compare spread option prices corresponding to diﬀerent eﬃciency generators (i.e., to diﬀerent
hv, evin the spread payoﬀ) as a function of maturity τ. ‘High’ and ‘low’ eﬃciency plant indicates values of hv, ev
chosen to be near the lowest and highest respectively in the stack, as given by Table 5. Within each of the four
subplots, the ﬁve lines correspond to ﬁve diﬀerent values of the cap Γ, ranging from very lenient to very strict. We
immediately observe in Figure 1 the seasonality in spread prices caused by the seasonality in power demand. This
is most striking for the low eﬃciency cases (high hv, ev), as such plants would rarely be used in shoulder months,
particularly in the case of gas. For low eﬃciency plants, the relationship with cap level (and corresponding initial
allowance price) is as one would expect: a stricter cap greatly increases the value of gas plants and greatly decreases
the value of the dirtier coal plants. This is also true for high eﬃciency gas plants, although the price diﬀerence (in
percentage terms) for diﬀerent Γ is less, since these are eﬀectively ‘in-the-money’ options, unlike those discussed
above. However, the analysis becomes more complicated for high eﬃciency coal plants, which tend to be chosen to
run in most market conditions, irrespective of emissions markets. Interestingly, we ﬁnd that for these options the
relationship with Γ (and hence A0) can be non-monotonic under certain conditions, particularly for high levels of
demand, when the price is set near the very top of the stack. In such cases a stricter cap provides extra beneﬁt
for the cleaner coal plants via higher power prices (typically set by the dirtier coal plants on the margin) which
outweighs the disadvantage of coal plants being replaced by gas plants in the merit order.
6.3. Case Study II: Impact of Fuel Price Changes. Notice that in Table 3, the initial conditions of both gas
and coal have been set to be equal to their long term median levels. We now consider the case that gas price sg
0is
either above or below its long term level, thus inducing a change in the initial merit order. Given the record low
prices of under $2 recently witnessed in the US natural gas market (due primarily to shale gas discoveries), it is natural to ask how such fuel price variations aﬀect our spread option results. Note however that since ηc=ηg= 1.5 (implying a typical mean reversion time of 8 months), by the end of the trading period, the simulated fuel price distributions will again be centred near their mean reversion levels. Thus in this case study, we capture a temporary, not permanent, shift in fuel prices. In Figure 2, we plot the value of coal and gas power plants, as given by the sum of spread options of all maturities τ[0, T ]. In the ﬁrst plot, we consider high eﬃciency (low hvand ev) plants, while in the second we consider low eﬃciencies. The former are much more likely to operate each day and to generate proﬁts, and are hence much more valuable than the latter. However, they also show diﬀerent relationships with sg 0, as illustrated for several diﬀerent cap levels Γ (like in Case Study I above) which correspond to high, low or medium (base case) values of A0. Firstly, for low eﬃciency plants (right plot), we observe that gas plant value is typically decreasing in sg 0, as we expect, since higher gas prices tend to push the bids from gas above those from coal, meaning there is less chance 10 REN´ E CARMONA, MICHAEL COULON, AND DANIEL SCHWARZ 0 0.2 0.4 0.6 0.8 1 0 10 20 30 40 50 60 High Efficiency Coal Plant spread maturity dark spread price 0 0.2 0.4 0.6 0.8 1 0 10 20 30 40 50 Low Efficiency Coal Plant spread maturity dark spread price 0 0.2 0.4 0.6 0.8 1 0 10 20 30 40 50 60 70 80 High Efficiency Gas Plant spread maturity spark spread price 0 0.2 0.4 0.6 0.8 1 0 5 10 15 20 25 30 Low Efficiency Gas Plant spread maturity spark spread price Γ=2e+08 Γ=1.8e+08 Γ=1.6e+08 Γ=1.4e+08 Γ=1.2e+08 Γ=2e+08 Γ=1.8e+08 Γ=1.6e+08 Γ=1.4e+08 Γ=1.2e+08 Γ=2e+08 Γ=1.8e+08 Γ=1.6e+08 Γ=1.4e+08 Γ=1.2e+08 Γ=2e+08 Γ=1.8e+08 Γ=1.6e+08 Γ=1.4e+08 Γ=1.2e+08 Figure 1. Cap strictness analysis for high eﬃciency coal (top left), low eﬃciency coal (top right), high eﬃciency gas (bottom left) and low eﬃciency gas (bottom right): Spark and dark spread option values plotted against maturity, for varying levels of the cap Γ. Note that the ﬁve equally- spaced cap values from 2e+ 08 to 1.2e+ 08 tons of CO2imply initial allowance prices of$5, $28,$52, $80, and$94.
2 4 6 8 10 12
0
0.5
1
1.5
2
2.5 x 104
initial gas price
power plant value
coal − low A
coal − mid A
coal − high A
gas − low A
gas − mid A
gas − high A
2 4 6 8 10 12
0
200
400
600
800
1000
1200
1400
1600
initial gas price
power plant value
coal − mid A
coal − high A
gas − low A
gas − mid A
Figure 2. Power Plant Value (sum of spreads over τ) versus sg
0for high eﬃciency (left) and low
eﬃciency (right). ‘High A’ corresponds to Γ = 1e+ 08, ‘Mid A’ to Γ = 1.4e+ 08 (base case) and
‘Low A’ to Γ = 1.8e+ 08, with corresponding values A0= 94, A0= 52, A0= 5
THE VALUATION OF CLEAN SPREAD OPTIONS: LINKING ELECTRICITY, EMISSIONS AND FUELS 11
σaρac ρag σeρec ρeg
0.6 -0.2 0.4 0.006 -0.2 0.2
Table 6. Parameters for reduced-form comparisons, treating Atand Etas GBMs.
that the gas plant will be used for electricity generation. Similarly, coal plant values are typically increasing in sc
0,
as more coal plants will be used.2Note however, that for some cases, the curves ﬂatten out, as no more merit order
changes are possible. This is particularly true for the coal plant when A0is very high (and hence once gas drops
below a certain point, the coal plant is almost certainly going to remain more expensive than all gas plants) and
for the gas plant when A0is very low (and hence once coal increases above a certain point, the gas plant is almost
certainly going to remain more expensive than all coal plants).
We now turn our attention to the high eﬃciency case (left plot), meaning the relatively cheap and clean plants for
each technology. As expected, coal beneﬁts from low values of A0(ie, a lenient cap) and gas from high values of
A0(ie, a strict cap). On the other hand, the relationship with sg
0is now increasing for almost all six cases plotted
except that of a gas plant with high A0. While it may seem surprising that for low or medium values of A0, the gas
plant value increase with sg
0, this is quite intuitive when one considers that the range of bids from gas generators
widens as sg
0increases, implying that the eﬃcient plants can make a larger proﬁt when the ineﬃcient plants set
the power price. Indeed, as demand is quite high on average, and gas is 60 percent of the market, it is likely that
these eﬃcient gas plants will almost always be ‘in-the-money’ even if coal is lower in the stack. Only in the case
that coal is typically above gas and now marginal (i.e. the high A0case) is the value of the gas plant decreasing in
sg
0since the plant’s proﬁt margins shrink as gas and coal bids converge.
6.4. Case Study III: Comparison with Reduced-Form. The second analysis we consider is to compare the
results of our structural model for the allowance price, with two other simpler models, both of which belong
to the class of ‘reduced-form’ models. The ﬁrst of these treats the allowance price itself as a simple Geometric
Brownian Motion (with drift runder Q), and hence Aτis lognormal at spread maturity, like Sc
τand Sg
τ. The
second comparison treats the emissions process as a Geometric Brownian Motion (GBM), and retains the digital
terminal condition AT=πI{ET>Γ}. As the drift of (Et) is then simply a constant (chosen to match the initial
value A0in the full model), there is no feedback from (At) on (Et), or in other words, no abatement induced by
the allowance price. For any time t,Atis then given in closed-form by a formula resembling the Black-Scholes
digital option price. In order to fully specify the reduced-form models, we need to choose volatility parameters σa
and σefor each of the GBMs, as well as correlations ρac, ρag and ρec, ρeg with the Brownian Motions driving the
other exogenous factors, coal and gas prices. All of these parameters are chosen to approximately match the levels
of volatility and correlation produced by simulations in the full structural model, and are given in Table 6. Finally,
note that in all three models we compare, the power price is given by the same bid stack function as usual, so our
aim is to isolate and evaluate the eﬀect of our more sophisticated framework for the allowance price, in comparison
to simpler approaches. The cap throughout is Γ = 1.4e+ 08, the base case.
Figure 3 reveals that the diﬀerence between the reduced form models and the full structural model is relatively
small for high eﬃciency gas and coal plants which are typically ‘in-the-money’. In contrast a larger gap appears
for low eﬃciency cases, where the reduced form models signiﬁcantly overprice spread options relative to the stack
model. In particular, the case of lognormal emissions produces much higher prices, especially for dark spreads.
The intuition is as follows. In the full model, the bid stack structure automatically leads to lower emissions when
the allowance price is high, and higher emissions when the allowance price is low, producing a mean-reversion-like
eﬀect on the cumulative emissions, keeping the process moving roughly towards the cap, with the ﬁnal outcome
(compliance or not) in many simulations only becoming clear very close to maturity. In contrast, if Etis a GBM,
much of the uncertainty is often resolved early in the trading period, with Atthen sticking near zero or πfor much
of the period. In such cases, there is a much larger beneﬁt for deep OTM options (low eﬃciency plants), for which
the tails of the allowance price distribution provide great value either for coal (when the price is near zero) or
for gas (when the price is near the penalty). We observe that in some of the subplots (particularly low eﬃciency
coal), this extra beneﬁt is indeed realized in the full model, but only very near the end of the trading period when
the volatility of (At) spikes, and the process either rises or falls sharply. In contrast, for the other reduced-form
model with lognormal (At), the volatility of the allowance price is constant throughout and Atnever moves rapidly
towards zero or the penalty. However, the overall link with fuel and power prices is much weaker when simply
2In this plot, the cases ‘coal - low A’ and ‘gas - high A’ are not included as their values are much greater and hence cannot be shown
conveniently on the same axis.
12 REN´
E CARMONA, MICHAEL COULON, AND DANIEL SCHWARZ
0 0.2 0.4 0.6 0.8 1
10
15
20
25
30
35
40
45
High Efficiency Coal Plant
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
10
12
14
Low Efficiency Coal Plant
0 0.2 0.4 0.6 0.8 1
15
20
25
30
35
40
45
High Efficiency Gas Plant
0 0.2 0.4 0.6 0.8 1
0
1
2
3
4
5
6
Low Efficiency Gas Plant
full model (FBSDE)
lognormal allowance
lognormal emissions
full model (FBSDE)
lognormal allowance
lognormal emissions
full model (FBSDE)
lognormal allowance
lognormal emissions
full model (FBSDE)
lognormal allowance
lognormal emissions
Figure 3. Model comparison against reduced-form: Spark and dark spread option values for
varying heat rates, emissions rates and maturities.
using correlated Brownian Motions, which serves to widen the spread distribution in most cases relative to the full
structural model. This result is somewhat similar to the observation in [Carmona et al., 2012a] that a stack model
generally produces lower spread option prices than Margrabe’s formula for correlated lognormals.
6.5. Case Study IV: Cap-and-Trade vs. Carbon Tax. Finally, we wish to investigate the implications of
the model for cap-and-trade systems, as compared with ﬁxed carbon taxes. This question has been much debated
by policy makers as well as academics, and can be roughly summarized as ﬁxing quantity versus ﬁxing price. In
[Carmona et al., 2010], several diﬀerent designs for cap-and-trade systems are compared to a carbon tax, using
criteria such as cost to society and windfall proﬁts to power generators. Here we follow a related approach by
analyzing the power sector as a whole, but we build on our previous case studies by using spread option prices as
a starting point. Firstly we observe that the total expected discounted proﬁts of the power sector are equal to the
value of all the power plants implied by the bid stack structure, which in turn equals a portfolio of (or integral over)
sums of spread option prices with varying hvand ev. i.e, for each simulation over the period [0, T ], total proﬁts
THE VALUATION OF CLEAN SPREAD OPTIONS: LINKING ELECTRICITY, EMISSIONS AND FUELS 13
Figure 4. Cap-and-trade vs. carbon tax: Power sector proﬁts versus time for the ‘base case’
(left); Total proﬁts over one year for equally-spaced cap values from 1e+ 08 to 1.95e+ 08 tons of
CO2. (right)
(total revenues minus total costs) are3
Total Proﬁts = X
τ[0,T ] PτDτZDτ
0
b(x, Aτ, Sτ)dx!
=X
τ[0,T ]Z¯x
0
(Pτb(x, Aτ, Sτ))+dx
=X
τ[0,T ]Z¯xc
0
(Pτhc(x)Sc
τec(x)Aτ)+dx+Z¯xg
0
(Pτhg(x)Sg
τeg(x)Aτ)+dx,
where the second line follows from the fact that the events {Pτb(x, Aτ, Sτ)}and {Dτx}are equal.
Hence, instead of picking particular coal and gas plants with eﬃciencies speciﬁed by the parameters in Table 5,
we now integrate power plant value over all the eﬃciencies of plants in the stack, as deﬁned by the parameters in
Table 1. For the case of the carbon tax, we simply force At=A0exp(rt) for all t[0, T ], including the exponential
function in order to match the mean of the process in the cap-and-trade model. This is equivalent to setting the
volatility σaequal to zero in the GBM model for the allowance price in Case Study III.
In Figure 4, we ﬁrst plot the expected total market proﬁts in the base case as a function of time. It is interesting
to observe that two important eﬀects occur, pulling the proﬁts in opposite directions, but varying in strength over
the trading period. In particular, although the proﬁts must be equal at time zero, a gap quickly appears in the
early part of the trading period, with expected proﬁts to power generators signiﬁcantly higher under a carbon
tax than cap-and-trade. However, as maturity approaches, the gap narrows and the order reverses over the ﬁnal
days, as cap-and-trade generates higher expected proﬁts. These eﬀects can be understood with a little thought.
Firstly, as A0= 52 in the base case, the bids of coal and gas begin the period at very similar levels, a state which
generally keeps proﬁts low, since the variance of electricity prices is low and the proﬁt margins of both coal and
gas generators are quite low. As time progresses and fuel prices move, the coal and gas bids will tend to drift apart
in most simulations, for example with gas sometimes moving above coal, say. However, in our structural model for
the cap-and-trade scheme, in such a case the higher emissions will induce a higher allowance price, and in turn a
feedback eﬀect due to the coupling in (7), which acts to keep coal and gas bids closer together. A similar argument
can be made for the case of gas bids tending to move below coal bids but then being counteracted by lower allowance
prices. Again we see that the power market structure induces mean reversion on (Et), which in this scenario (of an
averagely strict cap) corresponds to keeping coal and gas bids close together. On the other hand, under a carbon
tax with ﬁxed (or deterministic) At, there is of course no feedback mechanism (price-sensitive abatement), and
3Note that we do not consider here additional issues such as whether allowances are auctioned or freely allocated to generators.
Instead, we assume that allowances are bought on the market by generators as and when they need them.
14 REN´
E CARMONA, MICHAEL COULON, AND DANIEL SCHWARZ
bids tend to wander apart. However, as the end of the trading period approaches, in the cap-and-trade system the
allowance price eventually gets pulled to either zero or π, which will separate the bids in one way or the other,
either leading to very large proﬁts for coal plants (if AT= 0) or for gas plants (if AT=π). This is a similar eﬀect
to that discussed when comparing with a lognormal allowance price in Case Study III, as neither a carbon tax nor
a lognormal allowance price model sees the extra volatility near maturity caused by the terminal condition.
Finally, in the second plot of Figure 4, we consider how these conclusions change if the cap is made stricter or more
lenient. Instead of plotting against maturity, we consider the total proﬁts of the power sector over the entire period
[0, T ]. Firstly, we observe that under both forms of emissions regulation, power sector proﬁts are lowest if the cap
is chosen close to base case, under which the bids from coal and gas generators are more tightly clustered together.
Secondly, it is important to notice that the conclusion in the previous discussion that a carbon tax provides more
proﬁts to the power sector does not hold for all scenarios of the cap. In particular, for either very high or very
low values of the cap, the cap-and-trade scheme provides more proﬁts than a tax. The explanation here is that for
the automatic abatement mechanism in the stack to have its largest impact (keeping bids together, and emissions
heading towards the cap), there needs to be signiﬁcant uncertainty at time zero as to whether the cap will be
reached. The feedback mechanism of a cap-and-trade system then allows this uncertainty to be prolonged through
the period. On the other hand, for an overly strict or overly lenient cap (or similarly for a merit order which does
not allow for much abatement), the second eﬀect discussed above dominates over the ﬁrst. In other words, the
terminal condition which guarantees large proﬁts to either coal or gas at maturity begins to take precedence earlier
in the trading period, instead of just before maturity as in the base case. Although in practice there are many other
details to consider when comparing diﬀerent forms of emissions legislation, our stylized single-period model sheds
some light on the diﬀerences between cap-and-trade and carbon tax, as well as the clear importance of choosing an
appropriate cap level.
7. Conclusion
As policy makers debate the future of global carbon emissions legislation, the existing cap-and-trade schemes around
the world have already signiﬁcantly impacted the dynamics of electricity prices and the valuation of real assets,
such as power plants, particularly under the well-known European Union Emissions Trading Scheme. Together
with the recent volatile behaviour of all energy prices (e.g., gas, coal, oil), the introduction of carbon markets has
increased the risk of changes in the merit order of fuel types, known to be a crucial factor in the price setting
mechanism of electricity markets. In the US, the recent sharp drop in natural gas prices is already causing changes
in the merit order, which would be further magniﬁed by any new emissions regulation. Such considerations are
vital for describing the complex dependence structure between electricity, its input fuels, and emissions allowances,
and thus highly relevant for both market participants and policy makers designing emissions trading schemes. In
this paper, we derived the equilibrium carbon allowance price as the solution of an FBSDE, in which feedback from
allowance price on market emission rates is linked to the electricity stack structure. The resulting model speciﬁes
simultaneously both electricity and allowance price dynamics as a function of fuel prices, demand and accumulated
emissions; in this way, it captures consistently the highly state-dependent correlations between all the energy
prices, which would not be achievable in a typical reduced-form approach. We used a PDE representation for the
solution of the pricing FBSDE and implemented a ﬁnite diﬀerence scheme to solve for the price of carbon allowances.
Finally we compared our model for allowance prices with other reduced-form approaches and analysed its important
implications on price behaviour, spread option pricing and the valuation of physical assets in electricity markets
covered by emissions regulation. The four case studies illustrated the many important considerations needed to
understand the complex joint dynamics of electricity, emissions and fuels, as well as the additional insight that can
be provided by our structural approach.
Appendix A. Numerical Solution of the FBSDE
A.1. Candidate Pricing PDE. The construction of a solution to the FBSDE 7 was done in Theorem 1 by means
of a decoupling random ﬁeld urepresenting the solution in the form At=u(t, Et). The existence of this random ﬁeld
was derived from the results of [Ma et al., 2011], and given its uniqueness and the Markov nature of FBSDE 7, it is
possible to show that uis in fact a function of Dtand St, so that Atis in fact of the form At=α(t, Dt, Et, Sc
t, Sg
t)
for some deterministic function α: [0, T ]×[0,¯x]×R++ ×[0,¯e][0, π]. Standard arguments in the theory of
THE VALUATION OF CLEAN SPREAD OPTIONS: LINKING ELECTRICITY, EMISSIONS AND FUELS 15
FBSDEs show that this αis a viscosity solution of the semilinear PDE:
Lα+Nα= 0,on UT
(10)
α=φ(e),on {t=T} × U,(11)
where U:= (0,¯x)×R++ ×R++ ×(0,¯e) and UT:= [0, T )×U; the operators Land Nare deﬁned by
L:=
∂t +1
2σd(d)22
∂d2+1
2σc(sc)22
∂s2
c
+1
2σg(sg)22
∂s2
g
+µd(t, d)
∂d +µc(sc)
∂sc
+µg(sg)
∂sg
r·
and N:= µe(d, ·,(sc, sg))
∂e . As previously, we specify for our purposes that φ(e) = πI,)(e), for eR.
With regards to the problem (10) the question arises at which parts of the boundary we need to specify boundary
conditions and, given the original stochastic problem (7), of what form these conditions should be. To answer the
former question we consider the Fichera function fat points of the boundary where one or more of the diﬀusion
coeﬃcients disappear (cf. [Oleinik and Radkevic, 1973]). Deﬁning n:= (nd, nc, ng, ne) to be the inward normal
vector to the boundary, Fichera’s function for the operator (N+L) reads
(12) f(t, d, sc, sg, e) := µd1
2
∂d σ2
dnd+µc1
2
∂sc
σ2
c
∂sc
ρσcσgnc
+µg1
2
∂sg
σ2
g
∂sg
ρσcσgng+µene,on ∂UT.
At points of the boundary where f0 the direction of information propagation is outward and we do not need
to specify any boundary conditions; at points where f < 0 information is inward ﬂowing and boundary conditions
have to be speciﬁed. We evaluate (12) for the choice of coeﬃcients presented in §5.3.
Considering the parts of the boundary corresponding to d= 0 and d= ¯x, we ﬁnd that f0 if and only if
min( ¯
D(t),¯x¯
D(t)) ¯xˆσ, which is the same condition prescribed in §5.3 to guarantee that the Jacobi diﬀusion
stays within the interval (0,¯x). At points of the boundary corresponding to e= 0, we ﬁnd that f0 always. On
the part of the boundary on which e= ¯e,f < 0 except at the point (d, ·,·, e) = (0,·,·,¯e), where f= 0, an ambiguity
which could be resolved by smoothing the domain. Similarly, we ﬁnd that f0 on parts of the boundary where
sc= 0 or sg= 0. Therefore, no boundary conditions are necessary except when e= ¯e, where we prescribe
(13) α= exp(r(Tt))π, on UT|ee.
In addition we need to specify an asymptotic condition for large values of scand sg. We choose to consider solutions
that, for i∈ {c, g}, satisfy
(14) ∂α
∂si
0,on UT|si→∞.
A.2. An Implicit - Explicit Finite Diﬀerence Scheme. We approximate the domain ¯
UTby a ﬁnite grid
spanning [0, T ]×[0,¯x]×[0,¯sc]×[0,¯sg]×[0,¯e]. For the discretization we choose mesh widths ∆d, ∆sc, ∆sg, ∆e
and a time step ∆t. The discrete mesh points (tk, dm, sci, sgj, en) are then deﬁned by
tk:= kt, dm:= md,
sci:= isc, sgj:= jsg, en:= ne.
The ﬁnite diﬀerence scheme we employ produces approximations αk
m,i,j,n, which are assumed to converge to the
true solution αas the mesh width tends to zero.
Since the partial diﬀerential equation (10) is posed backwards in time with a terminal condition, we choose a
backward ﬁnite diﬀerence for the time derivative. In order to achieve better stability properties we make the part
of the scheme relating to the linear operator Limplicit; the part relating to the operator Nis made explicit in
order to handle the nonlinearity.
In the e-direction we are approximating a conservation law PDE with discontinuous terminal condition. (For an
in depth discussion of numerical schemes for this type of equation see [LeVeque, 1990]) The ﬁrst derivative in
the s-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
16 REN´
E CARMONA, MICHAEL COULON, AND DANIEL SCHWARZ
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. At the part of the boundary corresponding to e= ¯ewe apply the condition (13).
In the d-direction the equation is elliptic everywhere except on the boundary, where it degenerates. Therefore,
we expect the convection coeﬃcient to be much larger than the diﬀusion coeﬃcient near the boundaries. In order
to keep the discrete maximum principle we again use a one-sided upwind diﬀerence for the ﬁrst order derivative.
Thereby we have to pay attention that due to the mean-reverting nature of (Dt) the direction of information
propagation and therefore the upwind direction changes as the sign of µdchanges. The same upwind diﬀerence is
also used to calulate the value of the approximation at the boundaries d= 0 and d= ¯x. To discretize the second
order derivative we use central diﬀerences.
The scand sg-direction are treated similarly to the d-direction. We use one-sided upwind diﬀerences for the ﬁrst
order derivatives, thereby taking care of the boundaries corresponding to sc= 0 and sg= 0. The second order
derivatives are discretized using central diﬀerences. At the boundary corresponding to sc= ¯scand sg= ¯sgwe
apply the asymptotic condition (14) as a boundary condition.
With smooth boundary data, on a smooth domain, the scheme described above can be expected to exhibit ﬁrst
order convergence. In our setting, we expect the discontinuous terminal condition to have adverse eﬀects on the
convergence rate.
Appendix B. Numerical Calculation of Spread Prices
B.1. Time Discretisation of SDEs. Let (Dk, Sc
k, Sg
k, Ek, Ak) denote the discrete time approximation to the
FBSDE solution (Dt, Sc
t, Sg
t, Et, At) on the time grid 0 <t < 2∆t < . . . < nkt=τ. At each time step we
calculate Akby interpolating the discrete approximation αk
m,i,j,n at (Dk, Sc
k, Sg
k, Ek), beginning with the initial
values D0=d0, Sc
0=sc
0, Sg
0=sg
0, E0= 0. The approximations (Dk, Sc
k, Sg
k, Ek) are obtained using a simple Euler
scheme (cf. [Glasserman, 2004]). The discretized version of (Dt) is forced to be instantaneously reﬂecting at the
boundaries Dk= 0 and Dk= ¯x; similarly, the discretized versions of (Sc
t) and (Sg
t) are made instantaneously
reﬂecting at Sc
k= 0 and Sg
k= 0.
B.2. Monte Carlo Calculation of Option Prices. Using this discretization we simulate nmc paths and, as
usual, for t[0, τ ), calculate the mean spark spread price ˆ
Vt, given by
ˆ
Vt:= exp(r(τt)) 1
nmc
nmc
X
i=1 b(Di
nk, Sc,i
nk, Sg ,i
nk, Ai
nk)hvSg,i
nkevAi
nk+,
where the index irefers to the simulation scenario. The corresponding standard error ˆσvis obtained by
ˆσv:= v
u
u
t
1
nmc (nmc 1)
nmc
X
i=1 Vi
nkˆ
Vτ2.
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Bendheim Center for Finance, Dept. ORFE, University of Princeton, Princeton NJ 08544, USA
ORFE, University of Princeton, Princeton NJ 08544, USA
Oxford-Man Institute, University of Oxford, Oxford, UK
THE VALUATION OF CLEAN SPREAD OPTIONS: LINKING ELECTRICITY, EMISSIONS
AND FUELS
REN´
E CARMONA, MICHAEL COULON, AND DANIEL SCHWARZ
Abstract. The purpose of the paper is to present a new pricing method for clean spread options, and to illustrate
its main features on a set of numerical examples produced by a dedicated computer code. The novelty of the
approach is embedded in the use of structural models as opposed to reduced-form models which fail to capture
properly the fundamental dependencies between the economic factors entering the production process.
1. Introduction
Spread options are most often used in the commodity and energy markets to encapsulate the proﬁtability of a
production process by comparing the price of a reﬁned product to the costs of production including, but not
limited to, the prices of the inputs to the production process. When the output commodity is electric power, such
spread options are called spark spreads when the electricity is produced from natural gas, and dark spreads when
the electricity is produced from coal. Both processes are the sources of GreenHouse Gas (GHG) emissions, in higher
quantities for the latter than the former. In this paper we concentrate on the production of electricity and CO2
emissions and the resulting dependence structure between prices.
Market mechanisms aimed at controlling CO2emissions have been implemented throughout the world, and whether
they are mandatory or voluntary, cap-and-trade schemes have helped to put a price on carbon in the US and in
Europe. In the academic literature, equilibrium models have been used to show what practitioners have known all
along, namely that the price put on CO2by the regulation should be included in the costs of production to set
the price of electricity. (cf. [Carmona et al., 2010]) Strings of spark spread options (options on the spread between
the price of 1MWh of electricity and the cost of the amount of natural gas needed to produce such a MWh) with
maturities covering a given period are most frequently used to value the optionality of a gas power plant which
can be run when it is proﬁtable to do so (namely when the price of electricity is greater than the cost of producing
it), and shut down otherwise. In a nutshell, if an economic agent takes control on day t, of a gas power plant
for a period [T1, T2], then for every day τ[T1, T2] of this period, he or she can decide to run the power plant
when Pτ> hgSg
τ+Kand book a proﬁt PτhgSg
τKfor each unit of power produced, and shut the plant
down if PτhgSg
τ+K. Moreover, ignoring constraints such as ramp-up rates and start-up costs, this scheduling
is automatically induced when generators bid at the level of their production costs in the day-ahead auction for
power. Here Pτdenotes the price at which one unit (1 MWh) of power can be sold on day τ,Sg
τthe price of one
unit of natural gas (typically one MMBtu), hgthe eﬃciency or heat rate of the plant (i.e. the number of units of
natural gas needed to produce one unit of electricity) and Kthe daily (ﬁxed) costs of operations and maintenance
of the plant. So in this somewhat oversimpliﬁed analysis of the optionality of the plant, the value at time tof
the control of the plant operation on day τcan be expressed as er(τt)E[(PτhgSg
τK)+|Ft] where as usual,
the exponent +stands for the positive part, i.e. x+=xwhen x0 and x+= 0 otherwise, rfor the constant
interest rate used as discount factor to compute present values of future cash ﬂows, and Ftdenotes the information
available on day twhen the conditional expectation is actually computed. So the operational control (for example
as aﬀorded by a tolling contract) of the plant over the period [T1, T2] could be valued on day tas
VP P
t=
T2
X
τ=T1
er(τt)E[(PτhgSg
τK)+|Ft].
This rather simplistic way of valuing a power generation asset in the spirit of the theory of real options, had far-
reaching implications in the developments of the energy markets, and is the main reason why spread options are
of the utmost importance. However, such a valuation procedure is ﬂawed in the presence of emission regulation
as the costs of production have to include the costs speciﬁc to the regulation. To be more speciﬁc, the day-τ
potential proﬁt (PτhgSg
τK)+of the spark spread has to be modiﬁed to (PτhgSg
τegAτK)+in order
Partially supported by NSF - DMS-0739195.
1
arXiv:1205.2302v1 [q-fin.PR] 10 May 2012
2 REN´
E CARMONA, MICHAEL COULON, AND DANIEL SCHWARZ
to accommodate the cost of the regulation. Here Aτis the price of one allowance certiﬁcate worth one ton of CO2
equivalent, and egis the emission coeﬃcient of the plant, namely the number of tons of CO2emitted by the plant
during the production of one unit of electricity. Such a spread is often called a clean spread to emphasize the fact
the externality is being paid for, and the real option approach to power plant valuation leads to the following clean
price
VCP P
t=
T2
X
τ=T1
er(τt)E[(PτhgSg
τegAτK)+|Ft]
for the control of the plant over the period [T1, T2] in the presence of the regulation.
In order to price such cross-commodity derivatives, a joint model is clearly required for fuel prices, electricity
prices and carbon allowance prices. Various studies have analyzed the strong links between these price series
(cf. [De Jong and Schneider, 2009, Koenig, 2011]). Many reduced-form price models have been proposed for elec-
tricity (cf. [Benth et al., 2008, Eydeland and Wolyniec, 2003] for examples) with a focus on capturing its styl-
ized features such as seasonality, high volatility, spikes, mean-reversion and fuel price correlation. On the other
hand, many authors have argued that these same features are better captured via a structural approach, mod-
elling the dynamics of underlying factors such as demand (load), capacity and fuel prices (early examples include
[Barlow, 2002, Cartea and Villaplana, 2008, Pirrong and Jermakyan, 2008, Coulon and Howison, 2009]).
Similarly, for carbon emission allowances, exogenously speciﬁed processes that model prices directly have been
proposed by some (cf. [Carmona and Hinz, 2011]). Others have instead treated the emissions process as the
exogenously speciﬁed underlying factor; in this case the allowance certiﬁcate becomes a derivative on cumulative
emissions (cf. [Seifert et al., 2008, Chesney and Taschini, 2012]). However, these models do not take into account
the important feedback from the allowance price to the rate at which emissions are produced in the electricity sector
— a feature, which is crucial for the justiﬁcation of any implementation of a cap-and-trade scheme. In a discrete-
time framework this feedback mechanism has been addressed, for example in [Coulon, 2009, Carmona et al., 2010].
In continuous-time the problem has been treated in [Carmona et al., 2012b] and [Howison and Schwarz, 2012],
whereby the former models the accumulation of emissions as a function of an exogenously speciﬁed electricity price
process, while the latter uses the bid-stack mechanism to infer the emissions rate.
The literature on spread options is extensive. In industry, Margrabe’s classical spread option formula (cf. [Margrabe, 1978])
is still widely used, and has been extended by various authors (see [Carmona and Durrleman, 2003] for an overview)
including to the three commodity case, as required for the pricing of clean spreads (cf. [Alos et al., 2011]).
[Carmona and Sun, 2012] analyse the pricing of two-asset spread options in a multiscale stochastic volatility model.
For electricity markets, pricing formulae for dirty spreads based on structural models have been proposed in
[Carmona et al., 2012a], in which a closed-form formula is derived in the case of K= 0, and in [A¨ıd et al., 2012],
in which semi-closed form formulae are derived for K6= 0 at the expense of a ﬁxed merit order.
The original contributions of the paper are twofold. First, we express the value of clean spread options in a formula-
tion where demand for power and fuel prices are the only factors whose stochastic dynamics are given exogenously,
and where the prices of power and emission allowances are derived from a bid-stack based structural model and
a forward backward stochastic diﬀerential system respectively. The second contribution is the development of a
numerical code for the computation of the solution of the pricing problem. First we solve a 4+1 dimensional
semilinear partial diﬀerential equation to compute the price of an emission allowance, and then we use Monte Carlo
techniques to compute the price of the spread option. These computational tools are used to produce the numerical
results of case studies presented in §6 of the paper for the purpose of illustrating the impact of a carbon regulation
on the price of spread options. In this section we ﬁrst compare the price of spark and dark spread options in two
diﬀerent markets, one with no emissions regulation in place and the other governed by an increasingly strict cap-
and-trade system. Second, we analyze the impact that diﬀerent merit order scenarios have on the option prices.
Third, we demonstrate the diﬀerence between the structural and the reduced-form approach by comparing the
option prices produced by our model with those produced by two key candidate reduced-form models. Fourth and
last, we contrast two competing policy instruments: cap-and-trade, represented by the model we propose, and a
ﬁxed carbon tax.
2. The Bid Stack: Price Setting in Electricity Markets
In order to capture the dependency of electricity price on production costs and fundamental factors in a realistic
manner, we use a structural model in the spirit of those reviewed in the recent survey of [Carmona and Coulon, 2012].
The premises of structural models for electricity prices depend upon an explicit construction of the supply curve.
THE VALUATION OF CLEAN SPREAD OPTIONS: LINKING ELECTRICITY, EMISSIONS AND FUELS 3
Since electricity is sold at its marginal cost, the electricity spot price is given by evaluation of the supply function
for the appropriate values of factors used to describe the costs of production in the model.
In practice, electricity producers submit day-ahead bids to a central market operator, whose task it is to allocate
the production of electricity amongst them. Typically, ﬁrms’ bids have the form of price-quantity pairs, with each
pair comprising the amount of electricity the ﬁrm is willing to produce, and the price at which the ﬁrm is willing
to sell this quantity. Given the large number of generators in most markets, it is common in structural models
to approximate the resulting step function of market bids by a continuous increasing curve. Firms’ bid levels are
determined by their costs of production. An important feature of our model, distinguishing it from most of the
commonly used structural models is to include, as part of the production costs, the costs incurred because of the
existence of an emissions regulation.
We assume that, when deciding which ﬁrms to call upon to produce electricity, the market operator adheres to the
merit order, a rule by which cheaper production units are called upon before more expensive ones. For simplicity,
operational and transmission constraints are not considered.
Assumption 1. The market operator arranges bids according to the merit order, in increasing order of production
costs.
The map resulting from ordering market supply in increasing order of electricity costs of production is what is
called the bid stack. As it is one of the important building blocks of our model, we deﬁne it in a formal way for
later convenience.
Deﬁnition 1. The bid stack is given by a measurable function
b: [0,¯x]×R×Rn3(x, a, s)b(x, a, s)R,
with the property that for each ﬁxed (a, s)R×Rn, the function [0,¯x]3x b(x, a, s) is strictly increasing.
In this deﬁnition, ¯xR++ represents the market capacity (measured in MWh) and the variable xthe supply of
electricity. The integer nN\ {0}gives the number of economic factors (typically the prices in eof the fuels
used in the production of electricity), and sRnthe numeric values of these factors. Here and throughout the
rest of the paper the cost of carbon emissions (measured in eper metric ton of CO2) is denoted by a. So for
a given allowance price, say a, and fuel prices, say s, the market is able to supply xunits of electricity at price
level b=b(x, a, s) (measured in eper MWh). In other words, b(x, a, s) represents the bid level of the marginal
production unit in the event that demand equals x.
The choice of a function bwhich captures the subtle dependence of the electricity price upon the level of supply
and the production costs, is far from trivial, and diﬀerent approaches have been considered in the literature, as
reviewed recently by [Carmona and Coulon, 2012]. In §5.1 we extend the model proposed in [Carmona et al., 2012a]
to include the cost of carbon as part of the variable costs driving bid levels.
3. Risk-Neutral Pricing of Allowance Certificates
As the inclusion of the cost of emission regulation in the valuation of spread options is the main thrust of
the paper, we explain how emission allowances are priced in our model. The model we introduce is close to
[Howison and Schwarz, 2012]. However we extend the results found therein to allow the equilibrium bids of gener-
ators to be stochastic and driven by fuel prices, a generalization that is vital for our purpose.
We suppose that carbon emissions in the economy are subject to cap-and-trade regulation structured as follows:
at the end of the compliance period, each registered ﬁrm needs to oﬀset its cumulative emissions with emission
allowances or incur a penalty for each excess ton of CO2not covered by a redeemed allowance certiﬁcate. Initially,
ﬁrms acquire allowance certiﬁcates through free allocation, e.g. through National Allocation Plans (NAP) like in
the initial phase of the European Union (EU) Emissions Trading Scheme (ETS), or by purchasing them at auctions
like in the Regional Greenhouse Gas Initiative (RGGI) in the North East of the US. Allowances change hands
throughout the compliance period. Typically, a ﬁrm which thinks that its initial endowment will not suﬃce to
cover its emissions will buy allowances, while ﬁrms expecting a surplus will sell them. Adding to these naturals,
speculators enter the market providing liquidity. Allowances are typically traded in the form of forward contracts
and options. In this paper, we denote by Atthe spot price of an allowance certiﬁcate maturing at the end of the
compliance period. Because their cost of carry is negligible, we treat them as ﬁnancial products liquidly traded in
a market without frictions, and in which long and short positions can be taken.
4 REN´
E CARMONA, MICHAEL COULON, AND DANIEL SCHWARZ
In a competitive equilibrium, the level of cumulative emissions relative to the cap (i.e. the number of allowance
certiﬁcates issued by the regulation authority) determines whether — at the end of the compliance period — ﬁrms
will be subjected to a penalty payment and create a demand for allowance certiﬁcates. See [Carmona et al., 2010]
for details. For this reason, allowance certiﬁcates should be regarded as derivatives on the emissions accumulated
throughout the trading period. This type of option written on a non-tradable underlying interest is rather frequent
in the energy markets: temperature options are a case in point.
3.1. The Market Emissions Rate. As evidenced by the above discussion, the rate at which CO2is emitted
in the atmosphere as a result of electricity production has to be another important building block of our model.
Clearly at any given time, this rate is a function of the amount of electricity produced and because of their impact
on the merit order, the variable costs of production, including fuel prices, and notably, the carbon allowance price
itself.
Deﬁnition 2. The market emissions rate is given by a bounded function
µe: [0,¯x]×R×Rn3(x, a, s)µe(x, a, s)R+,
which is Lipschitz continuous in its three variables, strictly increasing in xwhen aand sare held ﬁxed, and strictly
decreasing in awhen xand sare ﬁxed.
With the deﬁnition above, for a given level of electricity supply and for given allowance and fuel prices, µe=
µe(x, a, s) represents the rate at which the market emits, measured in tons of CO2per hour. Cumulative emissions
are then computed by integrating the market emissions rate over time. The monotonicity property in xmakes
sense since any increase in supply can only increase the emissions rate. Similarly, as the cost of carbon increases the
variable costs (and hence the bids) of pollution intensive generators increase by more than those of environmentally
friendlier ones. Dirtier technologies become relatively more expensive and are likely to be scheduled further down
in the merit order. As a result cleaner technologies are brought online earlier, hence the monotonicity in a.
In §5.2 we propose a speciﬁc functional form for µeconsistent with the bid stack model introduced in §5.1.
3.2. The Pricing Problem. We shall use the following notation. For a ﬁxed time horizon TR+, let (W0
t, Wt)t[0,T ]
be a (n+ 1)-dimensional standard Wiener process on a probability space (Ω,F,P), F0:= (F0
t) the ﬁltration gener-
ated by W0,FW:= (FW
t) the ﬁltration generated by W, and F:= F0∨F Wthe market ﬁltration. All relationships
between random variables are to be understood in the almost surely sense.
Consumers’ demand for electricity is given by an F0
t-adapted stochastic process (Dt). In response to this demand
producers supply electricity, and we assume that demand and supply are always in equilibrium, so that at any
time t[0, T ] an amount Dtof electricity is supplied. The prices of fuels are observed FW
processes (St)t[0,T ], where St:= (S1
t, . . . , Sn
t). If the price of an allowance certiﬁcate at time t, say At, becomes
available, as we will see in §3.3, (At)T[0,T ]will be constructed as a Ft-adapted stochastic process solving a Forward
Backward Stochastic Diﬀerential Equation (FBSDE). The rate of emission µe(Dt, At, St) can then be evaluated
and the cumulative emissions computed by integration over time, resulting in a Ft-adapted process (Et)t[0,T ].
In order to avoid the diﬃculties of estimating the market price of risk (see for example [Eydeland and Wolyniec, 2003]
for a discussion of some possible ways to approach this thorny issue), we choose to specify the dynamics of the
processes (Dt)t[0,T ]and (St)t[0,T ]under a risk neutral measure QPchosen by the market for pricing purposes.
3.3. An FBSDE for the Allowance Price. We assume that at time t= 0, demand for electricity is known.
Thereafter, it evolves according to an Itˆo diﬀusion. Speciﬁcally, for t[0, T ], demand for electricity Dtis the
unique strong solution of a stochastic diﬀerential equation of the form
(1) dDt=µd(t, Dt)dt+σd(Dt)d ˜
W0
t, D0=d0(0,¯x),
where ( ˜
Wt) is an Ft-adapted Q-Brownian motion. The time dependence of the drift allows us to capture the
seasonality observed in electricity demand.
Similarly to demand, the prices of the fuels used in the production processes satisfy a system of stochastic diﬀerential
equations written in a vector form as follows:
(2) dSt=µs(St)dt+σs(St)d ˜
Wt, S0=s0Rn, t [0, T ].
Cumulative emissions are measured from the beginning of the compliance period when time t= 0, so that E0= 0.
Subsequently, they are determined by integrating over the market emissions rate µeintroduced in Deﬁnition 2. So
THE VALUATION OF CLEAN SPREAD OPTIONS: LINKING ELECTRICITY, EMISSIONS AND FUELS 5
assuming that the price Atof an allowance certiﬁcate is knwon, the cumulative emissions process is represented by
a bounded variation process; i.e. for t[0, T ],
(3) dEt=µe(Dt, At, St)dt, E0= 0.
Note that with this deﬁnition the process (Et) is non-decreasing, which makes intuitive sense considering that it
represents a cumulative quantity.
To complete the formulation of the pricing model, it remains to characterize the allowance certiﬁcate price process
(At)t[0,T ]. If our model is to apply to a one compliance period scheme, in a competitive equilibrium, at the end
of the compliance period t=T, its value is given by a deterministic function of the cumulative emissions:
(4) AT=φ(ET),
where φ:RRis bounded, measurable and non-decreasing. Usually φ(·) := πI,)(·), where πR+denotes the
penalty paid in the event of non-compliance and Γ R+the cap chosen by the regulator as the aggregate allocation
of certiﬁcates. See [Carmona et al., 2010] for details. Since the discounted allowance price is a martingale under
Q, it is equal to the conditional expectation of its terminal value, i.e.
(5) At= exp (r(Tt)) EQ[φ(ET)| Ft],for t[0, T ],
which implies in particular that the allowance price process (At) is bounded. Since the ﬁltration (Ft) is being gener-
ated by the Wiener processes, it is a consequence of the Martingale Representation Theorem (cf. [Karatzas and Shreve, 1999])
that the allowance price can be represented as an Itˆo integral with respect to the Brownian motion ( ˜
W0
t,˜
Wt). It
follows that
(6) dAt=rAtdt+Z0
td˜
W0
t+Zt·d˜
Wt,for t[0, T ]
for some Ft-adapted square integrable process (Z0
t, Zt).
Combining equations (1), (2), (3), (4) and (6), the pricing problem can be reformulated as the solution of the
FBSDE
(7)
dDt=µd(t, Dt)dt+σd(Dt)d ˜
W0
t, D0=d0(0,¯x),
dSt=µs(St)dt+σs(St)d ˜
Wt, S0=s0Rn,
dEt=µe(Dt, At, St)dt, E0= 0,
dAt=rAtdt+Z0
td˜
W0
t+Zt·d˜
Wt, AT=φ(ET).
Notice that the ﬁrst two equations are standard stochastic diﬀerential equations (in the forward direction of time)
which do not depend upon the cumulative emissions and the allowance price. We will choose their coeﬃcients so
that existence and uniqueness of solutions hold. To be more speciﬁc we make the following assumptions on the
coeﬃcients of (7):
Assumption 2. The functions µd: [0, T ]×[0,¯x]R,σd: [0,¯x]R,µs:RnRn,σs:RnRn×Rnare
such that the ﬁrst two equations in (7) have a unique strong solution.
3.4. Existence of a Solution to the Allowance Pricing Problem.
Theorem 1. We assume that Assumption 2 holds and that µeis Lipshitz with respect to the variable auniformly
in xand s, and that µe(x, 0, s)is uniformly bounded in xand s. Then if φis bounded and Lipschitz, the FBSDE
(7) has a unique square integrable solution.
Proof. Assumption 2 being satisﬁed, and the ﬁrst two equations of (7) being decoupled from the remaining ones,
there exist adapted processes (Dt) and (St) with values in [0,¯x] and Rnrespectively, unique strong solutions of
the ﬁrst two equations of (7). Once these two processes are constructed, we can plug their values into the last
two equations of (7), and treat the resulting equations as an FBSDE with random coeﬃcients. Existence and
uniqueness hold because of Theorem 7.1 of [Ma et al., 2011] 1. Strictly speaking this result is only proved for one-
dimensional processes. In the present situation, while Etand Atare indeed one-dimensional, the Wiener process
is (n+ 1)-dimensional and we cannot apply directly Theorem 7.1 of [Ma et al., 2011]. However, a close look at
the proof of this result shows that what is really needed is to prove the well-posedness of the characteristic BSDE,
and the boundedness of its solution and the solutions of the dominating Ordinary Diﬀerential Equations (ODE).
In the present situation, these equations are rather simple due to the fact that Ethas bounded variation, and as a
consequence, its volatility vanishes. The two dominating ODEs can be solved explicitly and one can check that the
1We would like to thank Francois Delarue for suggesting this strategy and the use of [Ma et al., 2011] in the present set-up.
6 REN´
E CARMONA, MICHAEL COULON, AND DANIEL SCHWARZ
solutions are bounded by inspection. Moreover, the function φfor the terminal condition being uniformly Lipschitz,
the characteristic BSDE is one-dimensional, though driven by a multi-dimensional Brownian motion, its terminal
condition is bounded, and Kobylanski’s comparison results (see the original contribution [Kobylanski, 2000]) can
be used to conclude the proof.
The above existence result is proven for a terminal condition given by a smooth function φ. As already mentioned
earlier, single compliance period equilibrium models most often require that the function φtake two values, and
the terminal condition φ(ET) equal the penalty when the regulatory cap is exceeded, i.e. when ET>Γ, and zero
when ET<Γ. [Carmona and Delarue, 2012] proved that a weaker form of existence and uniqueness of a solution
to the FBSDE still holds when φis discontinuous (in particular when φis an indicator function). Given that the
decoupling ﬁeld constructed in [Ma et al., 2011] is uniformly Lipschitz, we conjecture that a proof in the spirit of
the one given in [Carmona and Delarue, 2012] should work here and provide this weaker form of existence and
uniqueness. However, Carmona and Delarue also proved that under a strict monotonicity assumption on µe(which
should hold in our case for intuitive reasons), the aggregate emissions were equal with positive probability to the
cap at the end of the compliance period, and the terminal condition could not be prescribed for all the scenarios.
We suspect that in the present situation, the cumulative emissions equal the cap (i.e. ET= Γ) for a set of scenarios
of positive probability, and the terminal price of an allowance ATcannot be prescribed in advance on this set of
scenarios.
4. Valuing Clean Spread Options
In this section we consider the problem of spread option pricing as described in the introduction. Whether the goal
is to value a physical asset or risk manage ﬁnancial positions, one needs to compute the price of a European call
option on the diﬀerence between the price of electricity and the costs of production for a particular power plant.
The costs that we take into account are the ﬁxed operation and maintenance costs, the cost of the fuel needed to
generate one MWh of electricity and the cost of the ensuing emissions. Letting the Ft-adapted process (Pt) denote
the spot price of electricity, and recasting the informal discussion of the introduction with the notation we chose
to allow for several input fuels, a clean spread option with maturity τ[0, T ] is characterized by the payoﬀ
(PτhvSv
τevAτK)+,
where Krepresents the value of the ﬁxed operation and maintenance costs, hvR++ and evR++ denote the
speciﬁc heat and emissions rates of the power plant under consideration, and Sv∈ {S1, . . . , Sn}is the price at time
τof the fuel used in the production of electricity. In the special case when Svis the price of coal (gas) the option
is known as a clean dark (spark) spread option.
Since we are pricing by expectation, the value Vv
tof the clean spread is given by the conditional expectation under
the pricing measure of the discounted payoﬀ; i.e.
Vv
t= exp(r(τt))EQh(PτhvSv
τevAτK)+|Fti,for t[0, τ ].
5. A Concrete Two-Fuel Model
We now turn to the special case of two fuels, coal and gas.
5.1. The Bid Stack. Our bid stack model is a slight variation on the one we proposed in [Carmona et al., 2012a].
Here we extend to include the cost of emissions as part of the variable costs driving ﬁrms’ bids.
We assume that the coal and gas generators have aggregate capacities ¯xcand ¯xgrespectively, so that the market
capacity is ¯x= ¯xc+ ¯xg, and their bid levels are given by linear functions of the allowance price and the price of the
fuel used for the generation of electricity. We denote these bid functions by bcand bgrespectively. The coeﬃcients
appearing in these linear functions correspond to the marginal emissions rate (measured in ton equivalent of CO2
per MWh) and the heat rate (measured in MMBtu per MWh) of the technology in question. Speciﬁcally, for
i∈ {c, g}, we assume that
(8) bi(x, a, s) := ei(x)a+hi(x)s, for (x, a, s)[0,¯xi]×R×R,
where the marginal emissions rate eiand the heat rate hiare given by
ei(x) := ˆeiexp (mix)
hi(x) := ˆ
hiexp (mix),for x[0,¯xi].
THE VALUATION OF CLEAN SPREAD OPTIONS: LINKING ELECTRICITY, EMISSIONS AND FUELS 7
Here ˆei,ˆ
hi,miare strictly positive constants. We allow the marginal emissions rate and the heat rate of each
technology to vary to reﬂect diﬀerences in eﬃciencies within the ﬂeet of coal and gas generators. Since less eﬃcient
plants with higher heat rates have correspondingly higher emissions rates, it is a reasonable approximation to
assume that for each technology the ratio hi/eiis ﬁxed.
Proposition 1. With bcand bgas above and I={c, g}, the market bid stack bis given by
b(x, a, s) =
ˆeia+ˆ
hisiexp (mix),if bi(x, a, si)bj(0, a, sj)for i, j I , i 6=j,
ˆeia+ˆ
hisiexp (mi(x¯xj)) ,if bi(x¯xj, a, si)> bj(0, a, sj)for i, j I , i 6=j,
QiIˆeia+ˆ
hisiβiexp (γx),otherwise
for (x, a, s)[0,¯x]×R×R2, where βi=mM\{i}
mc+mgand γ=mcmg
mc+mg.
Proof. The proof is a straightforward extension of Corollary 1 in [Carmona et al., 2012a].
5.2. The Emissions Stack. In order to determine the rate at which the market emits we need to know which
generators are supplying electricity at any time. By the merit order assumption the market operator calls upon ﬁrms
in increasing order of their bid levels. Therefore, given electricity, allowance and fuel prices (p, a, s)R×R×R2,
for i∈ {c, g}, the set of active generators of fuel type iis given by {x[0,¯xi] : bi(x, a, s)p}.
Proposition 2. Assuming that the market bid stack is of the form speciﬁed in Proposition 1, the market emissions
rate µeis given by
(9) µe(x, a, s) := ˆeg
mgexp mgˆ
b1
g(b(x, a, s), a, sg)1+ˆec
mcexp mcˆ
b1
c(b(x, a, s), a, sc)1
for (x, a, s)[0,¯x]×R×R2, where for i∈ {c, g}we deﬁne
ˆ
b1
i(p, a, si) := 0 ¯xi1
mi
log p
ˆeia+ˆ
hisi,
for (p, a, s)R×R×R2.
Proof. The market emissions rate follows from integrating the marginal emissions rate eifor each technology over
the corresponding set of active generators and then summing the two. Given the monotonicity of biin xand its
range [0,¯xi], the function ˆ
b1
idescribes the quantity of electricity supplied by fuel i∈ {c, g}, and hence the required
upper limit of integration.
5.3. Specifying the Exogenous Stochastic Factors.
The Demand Process. We posit that under Q, the process (Dt) satisﬁes for t[0, T ] the stochastic diﬀerential
equation
dDt=ηDt¯
D(t)dt+p2ηˆσDt(¯xDt)d ˜
Wt, D0=d0(0,¯x),
where [0, T ]3t ¯
D(t)(0,¯x) is a deterministic function giving the level of mean reversion of the demande and
η, ˆσR++ are constants. With this deﬁnition (Dt) is a Jacobi diﬀusion process; it has a linear, mean-reverting
drift component and degenerates on the boundary. Moreover, subject to min( ¯
D(t),¯x¯
D(t)) ¯xˆσ, for t[0, T ], the
process remains within the interval (0,¯x) at all times (cf. [Forman and Sørensen, 2008]). To capture the seasonal
character of demand, we choose a function ¯
D(t) of the form:
¯
D(t) := ϕ0+ϕ1sin(2πϑt),
where the coeﬃcients will be chosen below.
The Fuel Price Processes. We assume that the prices of coal (Sc
t) and gas (Sg
t) follow correlated exponential OU
processes under the measure Q; i.e., for i∈ {c, g}and t[0, T ],
dSi
t=ηilog Si
t¯siˆσ2
i
2ηiSi
tdt+ ˆσiSi
td˜
Wi
t, Si
0=si
0R++,
where d hWc, W git=ρdt.
8 REN´
E CARMONA, MICHAEL COULON, AND DANIEL SCHWARZ
ˆ
hcˆecmc¯xcˆ
hgˆegmg¯xg¯x
3 0.9 0.00005 12000 7 0.4 0.00003 18000 30000
Table 1. Parameters relating to the bid stack and the emissions stack.
η ϕ0ϕ1ϑˆσ d0
50 21000 3000 1 0.1 21000
Table 2. Parameters relating to the demand process.
ηc¯scˆσcsc
0ηg¯sgˆσgsg
0ρ
1.5 2 0.5 exp(2) 1.5 2 0.5 exp(2) 0.3
Table 3. Parameters relating to the fuel price processes.
πΓT r
100 1.4e+08 1 0.05
Table 4. Parameters relating to the cap-and-trade scheme.
6. Numerical Analysis
We now turn to the detailed analysis of the model we propose. For this purpose we consider a number of case
studies in §6.2 to §6.5. To produce the following results we used the numerical schemes explained in Appendix A
and Appendix B.
6.1. Choice of Parameters. The tables in this section summarise the parameters used for the numerical analysis
of our model that follows below. We refer to the parameters speciﬁed in Tables 1 - 5 as the ‘base case’ and indicate
whenever we depart from this choice. Note that our parameter choices do not correspond to a particular electricity
market, but that all values are within a realistic realm.
Table 1 summarises the parameters specifying the bid curves. We consider a medium sized electricity market served
by coal and gas generators and with gas being the dominant technology. For the marginal emission rates, Table
1 implies that ec[0.9,1.64] and eg[0.4,0.69] (both measured in tCO2per MWh), so that all gas plants are
‘cleaner’ than all coal plants. For the heat rates, we observe that hc[3,5.5] and hg[7,12] (both measured in
MMBtu per MWh). Using (9) now with Dt= ¯x, for 0 tT, and the assumption that there are 8760 production
hours in the year, we ﬁnd, denoting the the maximum cumulative emissions by ¯e, that ¯e= 2.13e+ 08.
Table 2 contains the parameters for the demand process (Dt). We model periodicities on an annual and a weekly
time scale and the chosen rate of mean reversion assumes that demand reverts to its (time dependent) mean over
the course of one week.
In Table 3 we summarise the parameters that specify the behavior of the prices of coal and gas. Both are chosen
to be slowly mean-reverting, at least in comparison to demand. To ease analysis, we assume that all parameters
are identical for the two fuels, including mean price levels, both measured in MMBtu.
Table 4 deﬁnes the cap-and-trade scheme that we assume to be in place. The duration of the compliance period
Tis measured in years and we set the cap at 70% of the upper bound ¯efor the cumulative emissions, in order to
incentivise a reduction in emissions. This choice of parameters results in A0being approximately equal to π/2,
a value for which there is signiﬁcant initial overlap between gas and coal bids in the stack. Furthermore, the
parameters imply a bid stack structure such that at mean levels of coal and gas prices, At= 0 pushes all coal bids
below gas bids, while for At=πalmost all coal bids are above all gas bids.
Finally, in Table 5 we specify the four spread option contracts used in the base case scenario to represent high
and low eﬃciency coal plants, and high and low eﬃcency gas plants. (Note that low eﬃciency means dirtier and
corresponds to high hvand evand vice versa.)
THE VALUATION OF CLEAN SPREAD OPTIONS: LINKING ELECTRICITY, EMISSIONS AND FUELS 9
High Eﬀ. Coal Low Eﬀ. Coal High Eﬀ. Gas Low Eﬀ. Gas
hcechcechgeghgeg
3.5 1.05 5.0 1.5 7.5 0.43 11.5 0.66
Table 5. Parameters relating to the spread options.
We now consider a series of case studies to investigate various features of the model’s results in turn. As the model
captures many diﬀerent factors and eﬀects, this allows us to isolate some of the most important implications. In
Case Study I, we investigate the impact on coal and gas plants of diﬀerent eﬃciencies of creating an increasingly
strict carbon emissions market. In Case Study II, we assess the impact on these plants of changes in initial fuel
prices. In Case Study III, we compare spread option prices in our model with two simple reduced-form approaches
for At, which allows us to better understand the role of key model features such as bid stack driven abatement.
Finally in Case Study IV, we consider the overall impact of cap-and-trade markets in the electricity sector, by
comparing with a well-known alternative, a ﬁxed carbon tax.
6.2. Case Study I: Impact of the Emissions Market. The ﬁrst eﬀect that we are interested in studying in
the model is the impact of the cap-and-trade market on clean spread option prices, for increasingly strict levels
of the cap Γ. At one extreme (when the cap is so generous that At0, for all t[0, T ]), results correspond to
the case of a market without a cap and trade system, while at the other extreme (when the cap is so strict that
Atπexp(r(Tt)), for all t[0, T ]), there is essentially a very high carbon tax which tends to push most
coal generators above gas generators in the stack. It is intuitively clear that higher carbon prices typically lead to
higher spark spread option prices and lower dark spread option prices, thus favouring gas plants over coal plants,
but the relationships can be more involved as they vary between low eﬃciency and high eﬃciency plants.
In Figure 1, we compare spread option prices corresponding to diﬀerent eﬃciency generators (i.e., to diﬀerent
hv, evin the spread payoﬀ) as a function of maturity τ. ‘High’ and ‘low’ eﬃciency plant indicates values of hv, ev
chosen to be near the lowest and highest respectively in the stack, as given by Table 5. Within each of the four
subplots, the ﬁve lines correspond to ﬁve diﬀerent values of the cap Γ, ranging from very lenient to very strict. We
immediately observe in Figure 1 the seasonality in spread prices caused by the seasonality in power demand. This
is most striking for the low eﬃciency cases (high hv, ev), as such plants would rarely be used in shoulder months,
particularly in the case of gas. For low eﬃciency plants, the relationship with cap level (and corresponding initial
allowance price) is as one would expect: a stricter cap greatly increases the value of gas plants and greatly decreases
the value of the dirtier coal plants. This is also true for high eﬃciency gas plants, although the price diﬀerence (in
percentage terms) for diﬀerent Γ is less, since these are eﬀectively ‘in-the-money’ options, unlike those discussed
above. However, the analysis becomes more complicated for high eﬃciency coal plants, which tend to be chosen to
run in most market conditions, irrespective of emissions markets. Interestingly, we ﬁnd that for these options the
relationship with Γ (and hence A0) can be non-monotonic under certain conditions, particularly for high levels of
demand, when the price is set near the very top of the stack. In such cases a stricter cap provides extra beneﬁt
for the cleaner coal plants via higher power prices (typically set by the dirtier coal plants on the margin) which
outweighs the disadvantage of coal plants being replaced by gas plants in the merit order.
6.3. Case Study II: Impact of Fuel Price Changes. Notice that in Table 3, the initial conditions of both gas
and coal have been set to be equal to their long term median levels. We now consider the case that gas price sg
0is
either above or below its long term level, thus inducing a change in the initial merit order. Given the record low
prices of under $2 recently witnessed in the US natural gas market (due primarily to shale gas discoveries), it is natural to ask how such fuel price variations aﬀect our spread option results. Note however that since ηc=ηg= 1.5 (implying a typical mean reversion time of 8 months), by the end of the trading period, the simulated fuel price distributions will again be centred near their mean reversion levels. Thus in this case study, we capture a temporary, not permanent, shift in fuel prices. In Figure 2, we plot the value of coal and gas power plants, as given by the sum of spread options of all maturities τ[0, T ]. In the ﬁrst plot, we consider high eﬃciency (low hvand ev) plants, while in the second we consider low eﬃciencies. The former are much more likely to operate each day and to generate proﬁts, and are hence much more valuable than the latter. However, they also show diﬀerent relationships with sg 0, as illustrated for several diﬀerent cap levels Γ (like in Case Study I above) which correspond to high, low or medium (base case) values of A0. Firstly, for low eﬃciency plants (right plot), we observe that gas plant value is typically decreasing in sg 0, as we expect, since higher gas prices tend to push the bids from gas above those from coal, meaning there is less chance 10 REN´ E CARMONA, MICHAEL COULON, AND DANIEL SCHWARZ 0 0.2 0.4 0.6 0.8 1 0 10 20 30 40 50 60 High Efficiency Coal Plant spread maturity dark spread price 0 0.2 0.4 0.6 0.8 1 0 10 20 30 40 50 Low Efficiency Coal Plant spread maturity dark spread price 0 0.2 0.4 0.6 0.8 1 0 10 20 30 40 50 60 70 80 High Efficiency Gas Plant spread maturity spark spread price 0 0.2 0.4 0.6 0.8 1 0 5 10 15 20 25 30 Low Efficiency Gas Plant spread maturity spark spread price Γ=2e+08 Γ=1.8e+08 Γ=1.6e+08 Γ=1.4e+08 Γ=1.2e+08 Γ=2e+08 Γ=1.8e+08 Γ=1.6e+08 Γ=1.4e+08 Γ=1.2e+08 Γ=2e+08 Γ=1.8e+08 Γ=1.6e+08 Γ=1.4e+08 Γ=1.2e+08 Γ=2e+08 Γ=1.8e+08 Γ=1.6e+08 Γ=1.4e+08 Γ=1.2e+08 Figure 1. Cap strictness analysis for high eﬃciency coal (top left), low eﬃciency coal (top right), high eﬃciency gas (bottom left) and low eﬃciency gas (bottom right): Spark and dark spread option values plotted against maturity, for varying levels of the cap Γ. Note that the ﬁve equally- spaced cap values from 2e+ 08 to 1.2e+ 08 tons of CO2imply initial allowance prices of$5, $28,$52, $80, and$94.
2 4 6 8 10 12
0
0.5
1
1.5
2
2.5 x 104
initial gas price
power plant value
coal − low A
coal − mid A
coal − high A
gas − low A
gas − mid A
gas − high A
2 4 6 8 10 12
0
200
400
600
800
1000
1200
1400
1600
initial gas price
power plant value
coal − mid A
coal − high A
gas − low A
gas − mid A
Figure 2. Power Plant Value (sum of spreads over τ) versus sg
0for high eﬃciency (left) and low
eﬃciency (right). ‘High A’ corresponds to Γ = 1e+ 08, ‘Mid A’ to Γ = 1.4e+ 08 (base case) and
‘Low A’ to Γ = 1.8e+ 08, with corresponding values A0= 94, A0= 52, A0= 5
THE VALUATION OF CLEAN SPREAD OPTIONS: LINKING ELECTRICITY, EMISSIONS AND FUELS 11
σaρac ρag σeρec ρeg
0.6 -0.2 0.4 0.006 -0.2 0.2
Table 6. Parameters for reduced-form comparisons, treating Atand Etas GBMs.
that the gas plant will be used for electricity generation. Similarly, coal plant values are typically increasing in sc
0,
as more coal plants will be used.2Note however, that for some cases, the curves ﬂatten out, as no more merit order
changes are possible. This is particularly true for the coal plant when A0is very high (and hence once gas drops
below a certain point, the coal plant is almost certainly going to remain more expensive than all gas plants) and
for the gas plant when A0is very low (and hence once coal increases above a certain point, the gas plant is almost
certainly going to remain more expensive than all coal plants).
We now turn our attention to the high eﬃciency case (left plot), meaning the relatively cheap and clean plants for
each technology. As expected, coal beneﬁts from low values of A0(ie, a lenient cap) and gas from high values of
A0(ie, a strict cap). On the other hand, the relationship with sg
0is now increasing for almost all six cases plotted
except that of a gas plant with high A0. While it may seem surprising that for low or medium values of A0, the gas
plant value increase with sg
0, this is quite intuitive when one considers that the range of bids from gas generators
widens as sg
0increases, implying that the eﬃcient plants can make a larger proﬁt when the ineﬃcient plants set
the power price. Indeed, as demand is quite high on average, and gas is 60 percent of the market, it is likely that
these eﬃcient gas plants will almost always be ‘in-the-money’ even if coal is lower in the stack. Only in the case
that coal is typically above gas and now marginal (i.e. the high A0case) is the value of the gas plant decreasing in
sg
0since the plant’s proﬁt margins shrink as gas and coal bids converge.
6.4. Case Study III: Comparison with Reduced-Form. The second analysis we consider is to compare the
results of our structural model for the allowance price, with two other simpler models, both of which belong
to the class of ‘reduced-form’ models. The ﬁrst of these treats the allowance price itself as a simple Geometric
Brownian Motion (with drift runder Q), and hence Aτis lognormal at spread maturity, like Sc
τand Sg
τ. The
second comparison treats the emissions process as a Geometric Brownian Motion (GBM), and retains the digital
terminal condition AT=πI{ET>Γ}. As the drift of (Et) is then simply a constant (chosen to match the initial
value A0in the full model), there is no feedback from (At) on (Et), or in other words, no abatement induced by
the allowance price. For any time t,Atis then given in closed-form by a formula resembling the Black-Scholes
digital option price. In order to fully specify the reduced-form models, we need to choose volatility parameters σa
and σefor each of the GBMs, as well as correlations ρac, ρag and ρec, ρeg with the Brownian Motions driving the
other exogenous factors, coal and gas prices. All of these parameters are chosen to approximately match the levels
of volatility and correlation produced by simulations in the full structural model, and are given in Table 6. Finally,
note that in all three models we compare, the power price is given by the same bid stack function as usual, so our
aim is to isolate and evaluate the eﬀect of our more sophisticated framework for the allowance price, in comparison
to simpler approaches. The cap throughout is Γ = 1.4e+ 08, the base case.
Figure 3 reveals that the diﬀerence between the reduced form models and the full structural model is relatively
small for high eﬃciency gas and coal plants which are typically ‘in-the-money’. In contrast a larger gap appears
for low eﬃciency cases, where the reduced form models signiﬁcantly overprice spread options relative to the stack
model. In particular, the case of lognormal emissions produces much higher prices, especially for dark spreads.
The intuition is as follows. In the full model, the bid stack structure automatically leads to lower emissions when
the allowance price is high, and higher emissions when the allowance price is low, producing a mean-reversion-like
eﬀect on the cumulative emissions, keeping the process moving roughly towards the cap, with the ﬁnal outcome
(compliance or not) in many simulations only becoming clear very close to maturity. In contrast, if Etis a GBM,
much of the uncertainty is often resolved early in the trading period, with Atthen sticking near zero or πfor much
of the period. In such cases, there is a much larger beneﬁt for deep OTM options (low eﬃciency plants), for which
the tails of the allowance price distribution provide great value either for coal (when the price is near zero) or
for gas (when the price is near the penalty). We observe that in some of the subplots (particularly low eﬃciency
coal), this extra beneﬁt is indeed realized in the full model, but only very near the end of the trading period when
the volatility of (At) spikes, and the process either rises or falls sharply. In contrast, for the other reduced-form
model with lognormal (At), the volatility of the allowance price is constant throughout and Atnever moves rapidly
towards zero or the penalty. However, the overall link with fuel and power prices is much weaker when simply
2In this plot, the cases ‘coal - low A’ and ‘gas - high A’ are not included as their values are much greater and hence cannot be shown
conveniently on the same axis.
12 REN´
E CARMONA, MICHAEL COULON, AND DANIEL SCHWARZ
0 0.2 0.4 0.6 0.8 1
10
15
20
25
30
35
40
45
High Efficiency Coal Plant
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
10
12
14
Low Efficiency Coal Plant
0 0.2 0.4 0.6 0.8 1
15
20
25
30
35
40
45
High Efficiency Gas Plant
0 0.2 0.4 0.6 0.8 1
0
1
2
3
4
5
6
Low Efficiency Gas Plant
full model (FBSDE)
lognormal allowance
lognormal emissions
full model (FBSDE)
lognormal allowance
lognormal emissions
full model (FBSDE)
lognormal allowance
lognormal emissions
full model (FBSDE)
lognormal allowance
lognormal emissions
Figure 3. Model comparison against reduced-form: Spark and dark spread option values for
varying heat rates, emissions rates and maturities.
using correlated Brownian Motions, which serves to widen the spread distribution in most cases relative to the full
structural model. This result is somewhat similar to the observation in [Carmona et al., 2012a] that a stack model
generally produces lower spread option prices than Margrabe’s formula for correlated lognormals.
6.5. Case Study IV: Cap-and-Trade vs. Carbon Tax. Finally, we wish to investigate the implications of
the model for cap-and-trade systems, as compared with ﬁxed carbon taxes. This question has been much debated
by policy makers as well as academics, and can be roughly summarized as ﬁxing quantity versus ﬁxing price. In
[Carmona et al., 2010], several diﬀerent designs for cap-and-trade systems are compared to a carbon tax, using
criteria such as cost to society and windfall proﬁts to power generators. Here we follow a related approach by
analyzing the power sector as a whole, but we build on our previous case studies by using spread option prices as
a starting point. Firstly we observe that the total expected discounted proﬁts of the power sector are equal to the
value of all the power plants implied by the bid stack structure, which in turn equals a portfolio of (or integral over)
sums of spread option prices with varying hvand ev. i.e, for each simulation over the period [0, T ], total proﬁts
THE VALUATION OF CLEAN SPREAD OPTIONS: LINKING ELECTRICITY, EMISSIONS AND FUELS 13
Figure 4. Cap-and-trade vs. carbon tax: Power sector proﬁts versus time for the ‘base case’
(left); Total proﬁts over one year for equally-spaced cap values from 1e+ 08 to 1.95e+ 08 tons of
CO2. (right)
(total revenues minus total costs) are3
Total Proﬁts = X
τ[0,T ] PτDτZDτ
0
b(x, Aτ, Sτ)dx!
=X
τ[0,T ]Z¯x
0
(Pτb(x, Aτ, Sτ))+dx
=X
τ[0,T ]Z¯xc
0
(Pτhc(x)Sc
τec(x)Aτ)+dx+Z¯xg
0
(Pτhg(x)Sg
τeg(x)Aτ)+dx,
where the second line follows from the fact that the events {Pτb(x, Aτ, Sτ)}and {Dτx}are equal.
Hence, instead of picking particular coal and gas plants with eﬃciencies speciﬁed by the parameters in Table 5,
we now integrate power plant value over all the eﬃciencies of plants in the stack, as deﬁned by the parameters in
Table 1. For the case of the carbon tax, we simply force At=A0exp(rt) for all t[0, T ], including the exponential
function in order to match the mean of the process in the cap-and-trade model. This is equivalent to setting the
volatility σaequal to zero in the GBM model for the allowance price in Case Study III.
In Figure 4, we ﬁrst plot the expected total market proﬁts in the base case as a function of time. It is interesting
to observe that two important eﬀects occur, pulling the proﬁts in opposite directions, but varying in strength over
the trading period. In particular, although the proﬁts must be equal at time zero, a gap quickly appears in the
early part of the trading period, with expected proﬁts to power generators signiﬁcantly higher under a carbon
tax than cap-and-trade. However, as maturity approaches, the gap narrows and the order reverses over the ﬁnal
days, as cap-and-trade generates higher expected proﬁts. These eﬀects can be understood with a little thought.
Firstly, as A0= 52 in the base case, the bids of coal and gas begin the period at very similar levels, a state which
generally keeps proﬁts low, since the variance of electricity prices is low and the proﬁt margins of both coal and
gas generators are quite low. As time progresses and fuel prices move, the coal and gas bids will tend to drift apart
in most simulations, for example with gas sometimes moving above coal, say. However, in our structural model for
the cap-and-trade scheme, in such a case the higher emissions will induce a higher allowance price, and in turn a
feedback eﬀect due to the coupling in (7), which acts to keep coal and gas bids closer together. A similar argument
can be made for the case of gas bids tending to move below coal bids but then being counteracted by lower allowance
prices. Again we see that the power market structure induces mean reversion on (Et), which in this scenario (of an
averagely strict cap) corresponds to keeping coal and gas bids close together. On the other hand, under a carbon
tax with ﬁxed (or deterministic) At, there is of course no feedback mechanism (price-sensitive abatement), and
3Note that we do not consider here additional issues such as whether allowances are auctioned or freely allocated to generators.
Instead, we assume that allowances are bought on the market by generators as and when they need them.
14 REN´
E CARMONA, MICHAEL COULON, AND DANIEL SCHWARZ
bids tend to wander apart. However, as the end of the trading period approaches, in the cap-and-trade system the
allowance price eventually gets pulled to either zero or π, which will separate the bids in one way or the other,
either leading to very large proﬁts for coal plants (if AT= 0) or for gas plants (if AT=π). This is a similar eﬀect
to that discussed when comparing with a lognormal allowance price in Case Study III, as neither a carbon tax nor
a lognormal allowance price model sees the extra volatility near maturity caused by the terminal condition.
Finally, in the second plot of Figure 4, we consider how these conclusions change if the cap is made stricter or more
lenient. Instead of plotting against maturity, we consider the total proﬁts of the power sector over the entire period
[0, T ]. Firstly, we observe that under both forms of emissions regulation, power sector proﬁts are lowest if the cap
is chosen close to base case, under which the bids from coal and gas generators are more tightly clustered together.
Secondly, it is important to notice that the conclusion in the previous discussion that a carbon tax provides more
proﬁts to the power sector does not hold for all scenarios of the cap. In particular, for either very high or very
low values of the cap, the cap-and-trade scheme provides more proﬁts than a tax. The explanation here is that for
the automatic abatement mechanism in the stack to have its largest impact (keeping bids together, and emissions
heading towards the cap), there needs to be signiﬁcant uncertainty at time zero as to whether the cap will be
reached. The feedback mechanism of a cap-and-trade system then allows this uncertainty to be prolonged through
the period. On the other hand, for an overly strict or overly lenient cap (or similarly for a merit order which does
not allow for much abatement), the second eﬀect discussed above dominates over the ﬁrst. In other words, the
terminal condition which guarantees large proﬁts to either coal or gas at maturity begins to take precedence earlier
in the trading period, instead of just before maturity as in the base case. Although in practice there are many other
details to consider when comparing diﬀerent forms of emissions legislation, our stylized single-period model sheds
some light on the diﬀerences between cap-and-trade and carbon tax, as well as the clear importance of choosing an
appropriate cap level.
7. Conclusion
As policy makers debate the future of global carbon emissions legislation, the existing cap-and-trade schemes around
the world have already signiﬁcantly impacted the dynamics of electricity prices and the valuation of real assets,
such as power plants, particularly under the well-known European Union Emissions Trading Scheme. Together
with the recent volatile behaviour of all energy prices (e.g., gas, coal, oil), the introduction of carbon markets has
increased the risk of changes in the merit order of fuel types, known to be a crucial factor in the price setting
mechanism of electricity markets. In the US, the recent sharp drop in natural gas prices is already causing changes
in the merit order, which would be further magniﬁed by any new emissions regulation. Such considerations are
vital for describing the complex dependence structure between electricity, its input fuels, and emissions allowances,
and thus highly relevant for both market participants and policy makers designing emissions trading schemes. In
this paper, we derived the equilibrium carbon allowance price as the solution of an FBSDE, in which feedback from
allowance price on market emission rates is linked to the electricity stack structure. The resulting model speciﬁes
simultaneously both electricity and allowance price dynamics as a function of fuel prices, demand and accumulated
emissions; in this way, it captures consistently the highly state-dependent correlations between all the energy
prices, which would not be achievable in a typical reduced-form approach. We used a PDE representation for the
solution of the pricing FBSDE and implemented a ﬁnite diﬀerence scheme to solve for the price of carbon allowances.
Finally we compared our model for allowance prices with other reduced-form approaches and analysed its important
implications on price behaviour, spread option pricing and the valuation of physical assets in electricity markets
covered by emissions regulation. The four case studies illustrated the many important considerations needed to
understand the complex joint dynamics of electricity, emissions and fuels, as well as the additional insight that can
be provided by our structural approach.
Appendix A. Numerical Solution of the FBSDE
A.1. Candidate Pricing PDE. The construction of a solution to the FBSDE 7 was done in Theorem 1 by means
of a decoupling random ﬁeld urepresenting the solution in the form At=u(t, Et). The existence of this random ﬁeld
was derived from the results of [Ma et al., 2011], and given its uniqueness and the Markov nature of FBSDE 7, it is
possible to show that uis in fact a function of Dtand St, so that Atis in fact of the form At=α(t, Dt, Et, Sc
t, Sg
t)
for some deterministic function α: [0, T ]×[0,¯x]×R++ ×[0,¯e][0, π]. Standard arguments in the theory of
THE VALUATION OF CLEAN SPREAD OPTIONS: LINKING ELECTRICITY, EMISSIONS AND FUELS 15
FBSDEs show that this αis a viscosity solution of the semilinear PDE:
Lα+Nα= 0,on UT
(10)
α=φ(e),on {t=T} × U,(11)
where U:= (0,¯x)×R++ ×R++ ×(0,¯e) and UT:= [0, T )×U; the operators Land Nare deﬁned by
L:=
∂t +1
2σd(d)22
∂d2+1
2σc(sc)22
∂s2
c
+1
2σg(sg)22
∂s2
g
+µd(t, d)
∂d +µc(sc)
∂sc
+µg(sg)
∂sg
r·
and N:= µe(d, ·,(sc, sg))
∂e . As previously, we specify for our purposes that φ(e) = πI,)(e), for eR.
With regards to the problem (10) the question arises at which parts of the boundary we need to specify boundary
conditions and, given the original stochastic problem (7), of what form these conditions should be. To answer the
former question we consider the Fichera function fat points of the boundary where one or more of the diﬀusion
coeﬃcients disappear (cf. [Oleinik and Radkevic, 1973]). Deﬁning n:= (nd, nc, ng, ne) to be the inward normal
vector to the boundary, Fichera’s function for the operator (N+L) reads
(12) f(t, d, sc, sg, e) := µd1
2
∂d σ2
dnd+µc1
2
∂sc
σ2
c
∂sc
ρσcσgnc
+µg1
2
∂sg
σ2
g
∂sg
ρσcσgng+µene,on ∂UT.
At points of the boundary where f0 the direction of information propagation is outward and we do not need
to specify any boundary conditions; at points where f < 0 information is inward ﬂowing and boundary conditions
have to be speciﬁed. We evaluate (12) for the choice of coeﬃcients presented in §5.3.
Considering the parts of the boundary corresponding to d= 0 and d= ¯x, we ﬁnd that f0 if and only if
min( ¯
D(t),¯x¯
D(t)) ¯xˆσ, which is the same condition prescribed in §5.3 to guarantee that the Jacobi diﬀusion
stays within the interval (0,¯x). At points of the boundary corresponding to e= 0, we ﬁnd that f0 always. On
the part of the boundary on which e= ¯e,f < 0 except at the point (d, ·,·, e) = (0,·,·,¯e), where f= 0, an ambiguity
which could be resolved by smoothing the domain. Similarly, we ﬁnd that f0 on parts of the boundary where
sc= 0 or sg= 0. Therefore, no boundary conditions are necessary except when e= ¯e, where we prescribe
(13) α= exp(r(Tt))π, on UT|ee.
In addition we need to specify an asymptotic condition for large values of scand sg. We choose to consider solutions
that, for i∈ {c, g}, satisfy
(14) ∂α
∂si
0,on UT|si→∞.
A.2. An Implicit - Explicit Finite Diﬀerence Scheme. We approximate the domain ¯
UTby a ﬁnite grid
spanning [0, T ]×[0,¯x]×[0,¯sc]×[0,¯sg]×[0,¯e]. For the discretization we choose mesh widths ∆d, ∆sc, ∆sg, ∆e
and a time step ∆t. The discrete mesh points (tk, dm, sci, sgj, en) are then deﬁned by
tk:= kt, dm:= md,
sci:= isc, sgj:= jsg, en:= ne.
The ﬁnite diﬀerence scheme we employ produces approximations αk
m,i,j,n, which are assumed to converge to the
true solution αas the mesh width tends to zero.
Since the partial diﬀerential equation (10) is posed backwards in time with a terminal condition, we choose a
backward ﬁnite diﬀerence for the time derivative. In order to achieve better stability properties we make the part
of the scheme relating to the linear operator Limplicit; the part relating to the operator Nis made explicit in
order to handle the nonlinearity.
In the e-direction we are approximating a conservation law PDE with discontinuous terminal condition. (For an
in depth discussion of numerical schemes for this type of equation see [LeVeque, 1990]) The ﬁrst derivative in
the s-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
16 REN´
E CARMONA, MICHAEL COULON, AND DANIEL SCHWARZ
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. At the part of the boundary corresponding to e= ¯ewe apply the condition (13).
In the d-direction the equation is elliptic everywhere except on the boundary, where it degenerates. Therefore,
we expect the convection coeﬃcient to be much larger than the diﬀusion coeﬃcient near the boundaries. In order
to keep the discrete maximum principle we again use a one-sided upwind diﬀerence for the ﬁrst order derivative.
Thereby we have to pay attention that due to the mean-reverting nature of (Dt) the direction of information
propagation and therefore the upwind direction changes as the sign of µdchanges. The same upwind diﬀerence is
also used to calulate the value of the approximation at the boundaries d= 0 and d= ¯x. To discretize the second
order derivative we use central diﬀerences.
The scand sg-direction are treated similarly to the d-direction. We use one-sided upwind diﬀerences for the ﬁrst
order derivatives, thereby taking care of the boundaries corresponding to sc= 0 and sg= 0. The second order
derivatives are discretized using central diﬀerences. At the boundary corresponding to sc= ¯scand sg= ¯sgwe
apply the asymptotic condition (14) as a boundary condition.
With smooth boundary data, on a smooth domain, the scheme described above can be expected to exhibit ﬁrst
order convergence. In our setting, we expect the discontinuous terminal condition to have adverse eﬀects on the
convergence rate.
Appendix B. Numerical Calculation of Spread Prices
B.1. Time Discretisation of SDEs. Let (Dk, Sc
k, Sg
k, Ek, Ak) denote the discrete time approximation to the
FBSDE solution (Dt, Sc
t, Sg
t, Et, At) on the time grid 0 <t < 2∆t < . . . < nkt=τ. At each time step we
calculate Akby interpolating the discrete approximation αk
m,i,j,n at (Dk, Sc
k, Sg
k, Ek), beginning with the initial
values D0=d0, Sc
0=sc
0, Sg
0=sg
0, E0= 0. The approximations (Dk, Sc
k, Sg
k, Ek) are obtained using a simple Euler
scheme (cf. [Glasserman, 2004]). The discretized version of (Dt) is forced to be instantaneously reﬂecting at the
boundaries Dk= 0 and Dk= ¯x; similarly, the discretized versions of (Sc
t) and (Sg
t) are made instantaneously
reﬂecting at Sc
k= 0 and Sg
k= 0.
B.2. Monte Carlo Calculation of Option Prices. Using this discretization we simulate nmc paths and, as
usual, for t[0, τ ), calculate the mean spark spread price ˆ
Vt, given by
ˆ
Vt:= exp(r(τt)) 1
nmc
nmc
X
i=1 b(Di
nk, Sc,i
nk, Sg ,i
nk, Ai
nk)hvSg,i
nkevAi
nk+,
where the index irefers to the simulation scenario. The corresponding standard error ˆσvis obtained by
ˆσv:= v
u
u
t
1
nmc (nmc 1)
nmc
X
i=1 Vi
nkˆ
Vτ2.
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Bendheim Center for Finance, Dept. ORFE, University of Princeton, Princeton NJ 08544, USA
ORFE, University of Princeton, Princeton NJ 08544, USA
Oxford-Man Institute, University of Oxford, Oxford, UK
... In particular, the equation satisfied by the cumulative emissions process has no volatility term. Such equations have already been used to model the evolution of the cumulative emissions and price of an emissions allowance in a carbon market such as the EU ETS; see [5,7,13,29]. ...
... At each time T k1 the spot price will depend on the value of the cumulative emissions E T k1 but also on the cap imposed by the regulator, which in turn, will depend on E T k . This model is more realistic than the single-period model introduced in previous work [5,7,13]. In the single trading period model, any unused allowances become worthless at the end of the trading period. ...
... We assume that this process is purely autonomous and independent of the level of cumulative emissions and the allowance price. For example, in the presentation of [5] for an electricity market with emissions regulation, pP t q 0ďtďT could be a vector consisting of fuel prices and an inelastic demand curve for electricity. Setting P 0 " p, a deterministic constant, we assume that its dynamics are given by dP t " bpP t q dtσpP t q dW t , t P r0, T s, ...
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We introduce a model for the evolution of emissions and the price of emissions allowances in a carbon market, such as the European Union Emissions Trading System (EU ETS). The model accounts for multiple trading periods, or phases, with multiple times at which compliance can occur. At the end of each trading period, the participating firms must surrender allowances for their emissions made during that period, and additional allowances can be used for compliance in the following periods. We show that the multiperiod allowance pricing problem is well-posed for various mechanisms (such as banking, borrowing, and withdrawal of allowances) linking the trading periods. The results are based on the analysis of a forward-backward stochastic differential equation with coupled forward and backward components, a discontinuous terminal condition, and a forward component that is degenerate. We also introduce an infinite-period model for a carbon market with a sequence of compliance times and with no end date. We show that, under appropriate conditions, the value function for the multiperiod pricing problem converges, as the number of periods increases, to a value function for this infinite-period model and that such functions are unique. Finally, we present a numerical example that demonstrates empirically the convergence of the multiperiod pricing problem. Funding: The Engineering and Physical Sciences Research Council and Climate-KIC funded all parts of H. Chotai’s PhD programme in the centre for doctoral training and provided many opportunities for travel to conferences and events during the programme. D. Crisan received financial support from École Polytechnique and Université Paris Diderot through visitor grants.
... What we aim to do is to build off the work studying how firms, individually, or collectively optimise. A non-exhaustive list of works which study this either in SREC or C&T markets is [12,17,43,64,65]. ...
... In each of these works, the authors argue for the equivalence between the solution to the optimization problem for the central planner and an equilibrium solution whereby each individual agent optimizes their profit. There are also notable works on structural models for financial instruments in emissions markets, such as [45] and [12]. Finally, the problem of dynamic allocation of carbon allowances in a C&T market is considered in [3]. ...
... It is well-known that (given µ) V (k) t satisfies the following BSDE (see Lemma 4.47 of [15]): 12) and g , Γ are the controls that optimize the Hamiltonian function. This assumes that the value function exists in a classical sense, and is once differentiable in time and twice differentiable in state. ...
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Principal agent games are a growing area of research which focuses on the optimal behaviour of a principal and an agent, with the former contracting work from the latter, in return for providing a monetary award. While this field canonically considers a single agent, the situation where multiple agents, or even an infinite amount of agents are contracted by a principal are growing in prominence and pose interesting and realistic problems. Here, agents form a Nash equilibrium among themselves, and a Stackelberg equilibrium between themselves as a collective and the principal. We apply this framework to the problem of implementing emissions markets. We do so while incorporating market clearing as well as agent heterogeneity, and distinguish ourselves from extant literature by incorporating the probabilistic approach to MFGs as opposed to the analytic approach, with the former lending itself more naturally for our problem. For a given market design, we find the Nash equilibrium among agents using techniques from mean field games. We then provide preliminary results for the optimal market design from the perspective of the regulator, who aims to maximize revenue and overall environmental benefit.
... In particular, the equation satisfied by the cumulative emissions process has no volatility term. Such equations have already been used to model the evolution of the cumulative emissions and price of an emissions allowance in a carbon market such as the EU ETS; see [5,7,13,29]. ...
... At each time T k1 the spot price will depend on the value of the cumulative emissions E T k1 but also on the cap imposed by the regulator, which in turn, will depend on E T k . This model is more realistic than the single-period model introduced in previous work [5,7,13]. In the single trading period model, any unused allowances become worthless at the end of the trading period. ...
... We assume that this process is purely autonomous and independent of the level of cumulative emissions and the allowance price. For example, in the presentation of [5] for an electricity market with emissions regulation, pP t q 0ďtďT could be a vector consisting of fuel prices and an inelastic demand curve for electricity. Setting P 0 " p, a deterministic constant, we assume that its dynamics are given by dP t " bpP t q dtσpP t q dW t , t P r0, T s, ...
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We introduce a model for the evolution of emissions and the price of emissions allowances in a carbon market such as the European Union Emissions Trading System (EU ETS). The model accounts for multiple trading periods, or phases, with multiple times at which compliance can occur. At the end of each trading period, the participating firms must surrender allowances for their emissions made during that period, and additional allowances can be used for compliance in the following periods. We show that the multi-period allowance pricing problem is well-posed for various mechanisms (such as banking, borrowing and withdrawal of allowances) linking the trading periods. The results are based on the analysis of a forward-backward stochastic differential equation with coupled forward and backward components, a discontinuous terminal condition and a forward component that is degenerate. We also introduce an infinite period model, for a carbon market with a sequence of compliance times and with no end date. We show that, under appropriate conditions, the value function for the multi-period pricing problem converges, as the number of periods increases, to a value function for this infinite period model, and that such functions are unique.
... CSSs are by design cross-commodity products consisting of fuel, electricity and carbon allowance prices, which are typically analysed by separate pricing models. Nonetheless, to uncover the price formation process of such contracts, a joint model for all commodities is required as their respective dynamic, when considered separately, differs from that of the cross-commodity product (Carmona, Coulon, & Schwarz, 2012). ...
... In a structural-form pricing model, Carmona et al. (2012) model directly the dynamics of the underlying factors of CSS options and focus on the dependence structure between electricity and CO2 prices. Reduced-form pricing models of CSS options focus on capturing the main features such as mean-reversion, spikes and high volatility, see Benth et al. (2008) for a review. ...
... Price of the substitute fuel is an important driver of future (Carmona, Coulon, & Schwarz, 2012) and ...
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The profitability of electricity market participants providing flexibility and dispatchability is affected by different determinants. We study how variable renewable energy sources (vRES) and other factors impact both electricity futures prices and hedgeable profit margins of gas generators, proxied by clean spark spreads (CSSs) in futures markets. We focus on futures markets in three European countries (Germany, UK, Nordic) over the time period 2009–2016. We use a statistical univariate ARX TGARCH model that addresses persistency and volatility clustering. We find that the growth in vRES capacity reduces both electricity prices and CSSs more strongly than the changes in the futures fuel and carbon markets. Specifically, an additional 1 GW of installed wind capacity is associated with a decline by 6 cents (1.2%) and 37 cents (0.8%) per MWh in the German CSS and peak load electricity futures, respectively. An additional 1 GW of installed PV capacity is associated with a decline by 4 cents (0.8%) and 28 cents (0.6%) per MWh in the German CSS and peak load electricity futures, respectively. The results suggest that in systems dominated by vRES new markets or instruments may be needed allowing for an adequate risk management for flexible and dispatchable power generators.
... Additionally, they focus on the implications of the solution to the optimization problem posed on the certificate price, as opposed to the nature of the optimal behaviour of the regulated agents themselves, which is the primary focus of this work. There are also notable works on structural models for financial instruments in emissions markets, such as [38] and [10]. ...
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Solar renewable energy certificate (SREC) markets are a market‐based system that incentivizes solar energy generation. A regulatory body overseeing load serving entities imposes a lower bound on the amount of energy each regulated firm must generate via solar means, providing them with a tradeable certificate for each MWh generated. Firms seek to navigate the market optimally by modulating their SREC generation and trading rates. As such, the SREC market can be viewed as a stochastic game, where agents interact through the SREC price. We study this stochastic game by solving the mean‐field game (MFG) limit with subpopulations of heterogeneous agents. Market participants optimize costs accounting for trading frictions, cost of generation, nonlinear noncompliance costs, and generation uncertainty. Moreover, we endogenize SREC price through market clearing. We characterize firms' optimal controls as the solution of McKean–Vlasov (MV) forward‐backward stochastic differential equations (FBSDEs) and determine the equilibrium SREC price. We establish the existence and uniqueness of a solution to this MV‐FBSDE, and prove that the MFG strategies form an ε$\epsilon$‐Nash equilibrium for the finite player game. Finally, we develop a numerical scheme for solving the MV‐FBSDEs and conduct a simulation study.
... The relationship between electricity and natural gas prices in Spanish wholesale markets is analysed by Furi o and Poblaci on (2018), and it is concluded that natural gas and electricity prices are not only cointegrated but also share common long-term dynamics. Carmona et al. (2012) present a pricing method for clean spread options with a set of numerical examples. Elias et al. (2018) use a real-options approach to assess the value of retrofitting carbon capture and storage technology to an existing natural gas-fired base-load power plant. ...
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The Covid-19 crisis has had a major impact on electricity markets, affecting power plant input and output prices. In this paper Spanish electricity and natural gas prices and international carbon prices are used to calculate the variable margin of natural gas combined cycles (NGCC), i.e. the Clean Spark Spread (CSS). The stochastic behavior of the CSS is modeled using an Ornstein-Uhlenbeck (OU) process because of its properties. The expected first semester 2020 CSS results based on the fitted model with daily 2016-2019 data, taking the end of 2019 as a starting point, are compared with the actual figures for the same period. In the first half of 2020 electricity and natural gas prices are significantly lower than expected at the end of 2019, but carbon allowance prices have decreased less in percentage terms. The monthly CSS values in the first half of 2020 are significantly lower than expected for March-May 2020, with the April value being -€4.15/WWh lower figure than the €7.16/MWh expected. This work calculates distributions of daily and monthly CSS values.
... Additionally, they focus on the implications of the solution to the optimization problem posed on the certificate price, as opposed to the nature of the optimal behaviour of the regulated agents themselves, which is the primary focus of this work. There is also notable work on structural models for financial instruments in emissions markets, such as [36] and [11]. ...
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SREC markets are a market-based system designed to incentivize solar energy generation. A regulatory body imposes a lower bound on the amount of energy each regulated firm must generate via solar means, providing them with a certificate for each MWh generated. Regulated firms seek to navigate the market to minimize the cost imposed on them, by modulating their SREC generation and trading activities. As such, the SREC market can be viewed through the lens of a large stochastic game with heterogeneous agents, where agents interact through the market price of the certificates. We study this stochastic game by solving the mean-field game (MFG) limit with sub-populations of heterogeneous agents. Our market participants optimize costs accounting for trading frictions, cost of generation, SREC penalty, and generation uncertainty. Using techniques from variational analysis, we characterize firms' optimal controls as the solution of a new class of McKean-Vlasov FBSDE and determine the equilibrium SREC price. We numerically solve the MV-FBSDEs and conclude by demonstrating how firms behave in equilibrium using simulated examples.
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The goal of this survey is to review the major idiosyncrasies of the commodity markets and the methods which have been proposed to handle them in spot and forward price models. We devote special attention to the most idiosyncratic of all: electricity mar-kets. Following a discussion of traded instruments, market features, historical perspectives, recent developments and various modeling approaches, we focus on the important role of other energy prices and fundamental factors in setting the power price. In doing so, we present a detailed analysis of the structural approach for electricity, arguing for its merits over traditional reduced-form models. Building on several recent articles, we advocate a broad and flexible structural framework for spot prices, incorporating demand, capacity and fuel prices in several ways, while calculating closed-form forward prices throughout.
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