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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 Pτ−hgSg

τ−Kfor each unit of power produced, and shut the plant

down if Pτ≤hgSg

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 e−r(τ−t)E[(Pτ−hgSg

τ−K)+|Ft] where as usual,

the exponent +stands for the positive part, i.e. x+=xwhen x≥0 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

e−r(τ−t)E[(Pτ−hgSg

τ−K)+|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 (Pτ−hgSg

τ−K)+of the spark spread has to be modiﬁed to (Pτ−hgSg

τ−egAτ−K)+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

e−r(τ−t)E[(Pτ−hgSg

τ−egAτ−K)+|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, ¯x∈R++ represents the market capacity (measured in MWh) and the variable xthe supply of

electricity. The integer n∈N\ {0}gives the number of economic factors (typically the prices in eof the fuels

used in the production of electricity), and s∈Rnthe 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 T∈R+, 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

t-adapted stochastic

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 Q∼Pchosen 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=s0∈Rn, 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:R→Ris bounded, measurable and non-decreasing. Usually φ(·) := πI[Γ,∞)(·), where π∈R+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(T−t)) 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=s0∈Rn,

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:Rn→Rn,σs:Rn→Rn×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

(Pτ−hvSv

τ−evAτ−K)+,

where Krepresents the value of the ﬁxed operation and maintenance costs, hv∈R++ and ev∈R++ 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(τ−t))EQh(Pτ−hvSv

τ−evAτ−K)+|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,

Qi∈Iˆeia+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

b−1

g(b(x, a, s), a, sg)−1+ˆec

mcexp mcˆ

b−1

c(b(x, a, s), a, sc)−1

for (x, a, s)∈[0,¯x]×R×R2, where for i∈ {c, g}we deﬁne

ˆ

b−1

i(p, a, si) := 0 ∨¯xi∧1

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

b−1

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(¯x−Dt)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, ˆσ∈R++ 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−¯si−ˆσ2

i

2ηiSi

tdt+ ˆσiSi

td˜

Wi

t, Si

0=si

0∈R++,

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 ≤t≤T, 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 At≈0, 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

At≈πexp(−r(T−t)), 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

spread maturity

dark spread price

0 0.2 0.4 0.6 0.8 1

0

2

4

6

8

10

12

14

Low Efficiency Coal Plant

spread maturity

dark spread price

0 0.2 0.4 0.6 0.8 1

15

20

25

30

35

40

45

High Efficiency Gas Plant

spread maturity

spark spread price

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

6

Low Efficiency Gas Plant

spread maturity

spark spread price

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

0 0.2 0.4 0.6 0.8 1

2

3

4

5

6

7

8

9

10 x 106

maturity

power market profits

1 1.2 1.4 1.6 1.8 2

x 108

1.8

2

2.2

2.4

2.6

2.8

3

3.2 x 109

cap level (Γ)

power market profits

cap−and−trade

carbon tax

cap−and−trade

carbon tax

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 ] PτDτ−ZDτ

0

b(x, Aτ, Sτ)dx!

=X

τ∈[0,T ]Z¯x

0

(Pτ−b(x, Aτ, Sτ))+dx

=X

τ∈[0,T ]Z¯xc

0

(Pτ−hc(x)Sc

τ−ec(x)Aτ)+dx+Z¯xg

0

(Pτ−hg(x)Sg

τ−eg(x)Aτ)+dx,

where the second line follows from the fact that the events {Pτ≥b(x, Aτ, Sτ)}and {Dτ≥x}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)2∂2

∂d2+1

2σc(sc)2∂2

∂s2

c

+1

2σg(sg)2∂2

∂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[Γ,∞)(e), for e∈R.

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) := µd−1

2

∂

∂d σ2

dnd+µc−1

2

∂

∂sc

σ2

c−∂

∂sc

ρσcσgnc

+µg−1

2

∂

∂sg

σ2

g−∂

∂sg

ρσcσgng+µene,on ∂UT.

At points of the boundary where f≥0 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 f≥0 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 f≥0 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 f≥0 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(T−t))π, on UT|e=¯e.

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:= k∆t, dm:= m∆d,

sci:= i∆sc, sgj:= j∆sg, en:= n∆e.

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 < . . . < nk∆t=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(τ−t)) 1

nmc

nmc

X

i=1 b(Di

nk, Sc,i

nk, Sg ,i

nk, Ai

nk)−hvSg,i

nk−evAi

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

nk−ˆ

Vτ2.

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fuel structural approach. Working paper, Princeton University, University of Oxford, Oxford-Man Institute.

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ential equations and emission derivatives. Annals of Applied Probability.

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[Carmona et al., 2010] Carmona, R., Fehr, F., Hinz, J., and Porchet, A. (2010). Market designs for emissions trading schemes. SIAM

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Bendheim Center for Finance, Dept. ORFE, University of Princeton, Princeton NJ 08544, USA

E-mail address:rcarmona@princeton.edu

ORFE, University of Princeton, Princeton NJ 08544, USA

E-mail address:mcoulon@princeton.edu

Oxford-Man Institute, University of Oxford, Oxford, UK

E-mail address:schwarz@maths.ox.ac.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 Pτ−hgSg

τ−Kfor each unit of power produced, and shut the plant

down if Pτ≤hgSg

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 e−r(τ−t)E[(Pτ−hgSg

τ−K)+|Ft] where as usual,

the exponent +stands for the positive part, i.e. x+=xwhen x≥0 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

e−r(τ−t)E[(Pτ−hgSg

τ−K)+|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 (Pτ−hgSg

τ−K)+of the spark spread has to be modiﬁed to (Pτ−hgSg

τ−egAτ−K)+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

e−r(τ−t)E[(Pτ−hgSg

τ−egAτ−K)+|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, ¯x∈R++ represents the market capacity (measured in MWh) and the variable xthe supply of

electricity. The integer n∈N\ {0}gives the number of economic factors (typically the prices in eof the fuels

used in the production of electricity), and s∈Rnthe 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 T∈R+, 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

t-adapted stochastic

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 Q∼Pchosen 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=s0∈Rn, 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:R→Ris bounded, measurable and non-decreasing. Usually φ(·) := πI[Γ,∞)(·), where π∈R+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(T−t)) 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=s0∈Rn,

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:Rn→Rn,σs:Rn→Rn×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

(Pτ−hvSv

τ−evAτ−K)+,

where Krepresents the value of the ﬁxed operation and maintenance costs, hv∈R++ and ev∈R++ 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(τ−t))EQh(Pτ−hvSv

τ−evAτ−K)+|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,

Qi∈Iˆeia+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

b−1

g(b(x, a, s), a, sg)−1+ˆec

mcexp mcˆ

b−1

c(b(x, a, s), a, sc)−1

for (x, a, s)∈[0,¯x]×R×R2, where for i∈ {c, g}we deﬁne

ˆ

b−1

i(p, a, si) := 0 ∨¯xi∧1

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

b−1

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(¯x−Dt)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, ˆσ∈R++ 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−¯si−ˆσ2

i

2ηiSi

tdt+ ˆσiSi

td˜

Wi

t, Si

0=si

0∈R++,

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 ≤t≤T, 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 At≈0, 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

At≈πexp(−r(T−t)), 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

spread maturity

dark spread price

0 0.2 0.4 0.6 0.8 1

0

2

4

6

8

10

12

14

Low Efficiency Coal Plant

spread maturity

dark spread price

0 0.2 0.4 0.6 0.8 1

15

20

25

30

35

40

45

High Efficiency Gas Plant

spread maturity

spark spread price

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

6

Low Efficiency Gas Plant

spread maturity

spark spread price

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

0 0.2 0.4 0.6 0.8 1

2

3

4

5

6

7

8

9

10 x 106

maturity

power market profits

1 1.2 1.4 1.6 1.8 2

x 108

1.8

2

2.2

2.4

2.6

2.8

3

3.2 x 109

cap level (Γ)

power market profits

cap−and−trade

carbon tax

cap−and−trade

carbon tax

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 ] PτDτ−ZDτ

0

b(x, Aτ, Sτ)dx!

=X

τ∈[0,T ]Z¯x

0

(Pτ−b(x, Aτ, Sτ))+dx

=X

τ∈[0,T ]Z¯xc

0

(Pτ−hc(x)Sc

τ−ec(x)Aτ)+dx+Z¯xg

0

(Pτ−hg(x)Sg

τ−eg(x)Aτ)+dx,

where the second line follows from the fact that the events {Pτ≥b(x, Aτ, Sτ)}and {Dτ≥x}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)2∂2

∂d2+1

2σc(sc)2∂2

∂s2

c

+1

2σg(sg)2∂2

∂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[Γ,∞)(e), for e∈R.

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) := µd−1

2

∂

∂d σ2

dnd+µc−1

2

∂

∂sc

σ2

c−∂

∂sc

ρσcσgnc

+µg−1

2

∂

∂sg

σ2

g−∂

∂sg

ρσcσgng+µene,on ∂UT.

At points of the boundary where f≥0 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 f≥0 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 f≥0 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 f≥0 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(T−t))π, on UT|e=¯e.

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:= k∆t, dm:= m∆d,

sci:= i∆sc, sgj:= j∆sg, en:= n∆e.

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 < . . . < nk∆t=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(τ−t)) 1

nmc

nmc

X

i=1 b(Di

nk, Sc,i

nk, Sg ,i

nk, Ai

nk)−hvSg,i

nk−evAi

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

nk−ˆ

Vτ2.

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Bendheim Center for Finance, Dept. ORFE, University of Princeton, Princeton NJ 08544, USA

E-mail address:rcarmona@princeton.edu

ORFE, University of Princeton, Princeton NJ 08544, USA

E-mail address:mcoulon@princeton.edu

Oxford-Man Institute, University of Oxford, Oxford, UK

E-mail address:schwarz@maths.ox.ac.uk