Electricity price modeling and asset valuation: A multi-fuel structural approach

Abstract

We introduce a new and highly tractable structural model for spot and
derivative prices in electricity markets. Using a stochastic model of the bid
stack, we translate the demand for power and the prices of generating fuels
into electricity spot prices. The stack structure allows for a range of
generator efficiencies per fuel type and for the possibility of future changes
in the merit order of the fuels. The derived spot price process captures
important stylized facts of historical electricity prices, including both
spikes and the complex dependence upon its underlying supply and demand
drivers. Furthermore, under mild and commonly used assumptions on the
distributions of the input factors, we obtain closed-form formulae for
electricity forward contracts and for spark and dark spread options. As merit
order dynamics and fuel forward prices are embedded into the model, we capture
a much richer and more realistic dependence structure than can be achieved by
classical reduced-form models. We illustrate these advantages by comparing with
Margrabe's formula and a simple cointegration model, and highlight important
implications for the valuation of power plants.

Full text

ELECTRICITY PRICE MODELING AND ASSET VALUATION: A MULTI-FUEL
STRUCTURAL APPROACH
REN ´
E CARMONA, MICHAEL COULON, AND DANIEL SCHWARZ
ABS TR ACT. We introduce a new and highly tractable structural model for spot and derivative prices
in electricity markets. Using a stochastic model of the bid stack, we translate the demand for power
and the prices of generating fuels into electricity spot prices. The stack structure allows for a range of
generator efficiencies per fuel type and for the possibility of future changes in the merit order of the
fuels. The derived spot price process captures important stylized facts of historical electricity prices,
including both spikes and the complex dependence upon its underlying supply and demand drivers.
Furthermore, under mild and commonly used assumptions on the distributions of the input factors, we
obtain closed-form formulae for electricity forward contracts and for spark and dark spread options. As
merit order dynamics and fuel forward prices are embedded into the model, we capture a much richer
and more realistic dependence structure than can be achieved by classical reduced-form models. We
illustrate these advantages by comparing with Margrabe’s formula and a simple cointegration model,
and highlight important implications for the valuation of power plants.
Electricity markets and structural model and forward prices and spread options and power plant
valuation
JEL Classification Numbers: C60, G12, G13, Q40
1. INTRODUCTION
Since the onset of electricity market deregulation in the 1990s, the modeling of prices in these
markets has become an important topic of research both in academia and industry. Energy companies
own large portfolios of generation units and require sophisticated models of price dynamics in order
to manage risk. Asset valuation is also of utmost importance in capital intensive industries and
real option theory is typically used to associate the management of a plant to a string of spread
options, spanning many years or decades. One of the main thrusts of this paper is to provide new
and versatile tools for these valuations which efficiently capture the complex dependencies upon
demand and production fuel prices.
In electricity price modeling, important challenges include prominent seasonalities and mean-
reversion at various time scales, sudden spikes and the strong link between the prices of electricity
and other energy commodities (see Figures 1b and 5 for sample daily historical spot and forward
prices from the PJM market) – features which mostly stem from the non-storability of electricity
and the resulting matching of supply and demand at all times. While every model should attempt
to capture these properties as well as possible, at the same time there is a need for fast and efficient
methods to value power plants and other derivatives on the spot price. To achieve the latter goal
of efficiency much literature has ignored or oversimplified the former goal of modeling structural
relationships. In this paper, we propose a model that realistically captures the dependency of power
prices on their primary drivers; yet we obtain closed-form expressions for spot, forward and option
prices.
Partially supported by NSF - DMS-0739195.
1
arXiv:1205.2299v1 [q-fin.PR] 10 May 2012

2 REN ´
E CARMONA, MICHAEL COULON, AND DANIEL SCHWARZ
The existing literature on electricity price modeling can be approximately divided into three cat-
egories. At one end of the spectrum are so called full production cost models. These rely on knowl-
edge of all generation units, their corresponding operational constraints and network transmission
constraints. Prices are then typically solved for by complex optimization routines (cf. [19]). Al-
though this type of model may provide market insights and forecasts in the short term, it is — due to
its complexity — unsuited to handling uncertainty, and hence to derivative pricing or the valuation
of physical assets. Other related approaches which share this weakness include models of strategic
bidding (cf. [22]) and other equilibrium approaches (cf. [6]). At the other end of the spectrum are
reduced form models. These are characterised by an exogenous specification of electricity prices,
with either the forward curve (cf. [12] and [5]) or the spot price (cf. [16, 20, 4]) representing the
starting point for the model. Reduced form models typically either ignore fuel prices or introduce
them as exogenous correlated processes; hence they are not successful at capturing the important
afore mentioned dependence structure between fuels and electricity. Further, spikes are usually only
obtained through the inclusion of jump processes or regime switches, which provide little insight
into the causes that underly these sudden price swings.
In between these two extremes is the structural approach to electricity price modeling, which
stems from the seminal work of Barlow [3]. We use the adjective structural to describe models,
which — to varying degrees of detail and complexity — explicitly approximate the supply curve in
electricity markets (commonly known as the bid stack due to the price-setting auction). The market
price is then obtained under the equilibrium assumption that demand and supply must match. In
Barlow’s work the bid stack is simply a fixed parametric function, which is evaluated at a random
demand level. Later works have refined the modeling of the bid curve and taken into account its
dependency on the available capacity (cf. [8, 7, 10]), as well as fuel prices (cf. [27, 14, 1, 2]) and
the cost of carbon emissions (cf. [23, 13]). The raison d’ˆ
etre of all structural models is very clear.
If the bid curve is chosen appropriately, then observed stylized facts of historic data can be well
matched. Moreover, because price formation is explained using fundamental variables and costs of
production, these models offer insight into the causal relationships in the market; for example, prices
in peak hours are most closely correlated with natural gas prices in markets with many gas ‘peaker’
plants; similarly, price spikes are typically observed to coincide with states of very high demand
or low capacity. As a direct consequence, this class of models also performs best at capturing the
varied dependencies between electricity, fuel prices, demand and capacity.
The model we propose falls into the category of structural models. Our work breaks from the cur-
rent status quo by providing closed-form formulae for the prices of a number of derivative products
in a market driven by two underlying fuels and featuring a continuum of efficiencies (heat rates). In
the considered multi-fuel setting, our model of the bid stack allows the merit order to be dynamic:
each fuel can become the marginal fuel and hence set the market price of electricity. Alternatively,
several fuels can set the price jointly. Despite this complexity, under only mild assumptions on the
distribution, under the pricing measure, of the terminal value of the processes representing electric-
ity demand and fuels, we obtain explicit formulae for spot prices, forwards and spread options, as
needed for power plant valuation. Moreover, our formulae capture very clearly and conveniently
the dependency of electricity derivatives upon the prices of forward contracts written on the fuels
that are used in the production process. This allows the model to easily ‘see’ additional information
contained in the fuel forward curves, such as states of contango or backwardation — another feature,
which distinguishes it from other approaches.
The parametrization of the bid stack we propose combines an exponential dependency on de-
mand, suggested several times in the literature (cf. [30, 11, 24]), with the need for a heat rate
function multiplicative in the fuel price, as stressed by Pirrong and Jermakyan [27]. Eydeland and

ELECTRICITY PRICE MODELING AND ASSET VALUATION: A MULTI-FUEL STRUCTURAL APPROACH 3
Geman [18] propose a similar structure for forward prices and note that Black-Scholes like deriv-
ative prices are available if the power price is log-normal. However, this requires the assumption
of a single marginal fuel type and ignores capacity limits. Coulon and Howison [14] construct the
stack by approximating the distribution of the clusters of bids from each technology, but their ap-
proach relies heavily on numerical methods when it comes to derivative pricing. In the work of Aid
et al [1], the authors simplify the stack construction by allowing only one heat rate (constant heat
rate function) per fuel type, a significant oversimplification of spot price dynamics for mathematical
convenience. Aid et al [2] extend this approach to improve spot price dynamics and capture spikes,
but at the expense of a static merit order, ruling out, among other things, the possibility that coal and
gas can change positions in the stack in the future. In both cases the results obtained by the authors
only lead to semi-closed form formulae, which still have to be evaluated numerically.
The importance of the features incorporated in our model is supported by prominent develop-
ments observed in recent data. In particular, shale gas discoveries have led to a dramatic drop in US
natural gas prices in recent years, from a high of over $13 in 2008 to under $3 in January 2012. Such
a large price swing has rapidly pushed natural gas generators down the merit order, and highlights
the need to account for uncertainty in future merit order changes, particularly for longer term prob-
lems like plant valuation. In addition, studying hourly data from 2004 to 2010 on marginal fuels in
the PJM market (published by Monitoring Analytics), we observe that the electricity price was fully
set by a single technology (only one marginal fuel) in only 16.1 per cent of the hours. For the year
2010 alone, the number drops to less than 5 per cent. Substantial overlap of bids from different fuels
therefore exists, and changes in merit order occur gradually as prices move. We believe that our
model of the bid stack adheres to many of the true features of the bid stack structure, which leads to
a reliable reproduction of observed correlations and price dynamics, while retaining mathematical
tractability.
2. ST RUCT URA L AP PROAC H TO E LEC TRI CI TY PR ICI NG
In the following we work on a complete probability space (Ω,F,P). For a fixed time horizon
T∈R+, we define the (n+1)-dimensional standard Wiener process (W0
t,Wt)t∈[0,T], where W:=
(W1,...,Wn). Let F0:= (F0
t)denote the filtration generated by W0and FW:= (FW
t)the filtration
generated by W. Further, we define the market filtration F:=F0∨FW. All relationships between
random variables are to be understood in the almost surely sense.
2.1. Price Setting in Electricity Markets. We consider a market in which individual firms generate
electricity. All firms submit day-ahead bids to a central market administrator, whose task it is to
allocate the production of electricity amongst them. Each firm’s bids take the form of price-quantity
pairs representing an amount of electricity the firm is willing to produce, and the price at which the
firm is willing to sell it1. An important part is therefore played by the merit order, a rule by which
cheaper production units are called upon before more expensive ones in the electricity generation
process. This ultimately guarantees that electricity is supplied at the lowest possible price.2
Assumption 1. The market administrator arranges bids according to the merit order and hence in
increasing order of costs of production.
1Alternatively, firms may, in some markets, submit continuous bid curves, which map an amount of electricity to the
price at which it is offered. For our purposes this distinction will however not be relevant.
2This description is of course a simplification of the market administrator’s complicated unit commitment problem,
typically solved by optimization in order to satisfy various operational constraints of generators, as well as transmission
constraints. Details vary from market to market and we do not address these issues here, as our goal is to approximate the
price setting mechanism and capture the key relationships needed for derivative pricing.

4 REN ´
E CARMONA, MICHAEL COULON, AND DANIEL SCHWARZ
We refer to the resulting map from the total supply of electricity and the factors that influence
the bid levels to the price of the marginal unit as the market bid stack and assume that it can be
represented by a measurable function
b:[0,¯
ξ]×Rn3(ξ,s)→b(ξ,s)∈R,
which will be assumed to be strictly increasing in its first variable. Here, ¯
ξ∈R+represents the
combined capacity of all generators in the market, henceforth the market capacity (measured in
MW), and s∈Rnrepresents factors of production which drive firm bids (e.g. fuel prices).
Demand for electricity is assumed to be price-inelastic and given exogenously by an F0
t-adapted
process (Dt)(measured in MW). As we shall see later, the prices of the factors of production used
in the electricity generation process will be assumed to be FW
t-adapted. So under the objective
historical measure P, the demand is statistically independent of these prices. This is a reasonable
assumption as power demand is typically driven predominantly by temperature, which fluctuates at
a faster time scale and depends more on local or regional conditions than fuel prices. The market
responds to this demand by supplying an amount ξt∈[0,¯
ξ]of electricity. We assume that the market
is in equilibrium with respect to the supply of and demand for electricity; i.e.
(1) Dt=ξt,for t∈[0,T].
This implies that Dt∈[0,¯
ξ]for t∈[0,T]and (ξt)is F0
t-adapted.
The market price of electricity (P
t)is now defined as the price at which the last unit that is needed
to satisfy demand sells its electricity; i.e. using (1),
(2) P
t:=b(Dt,·),for t∈[0,T].
We emphasize the different roles played by the first variable (i.e. demand) and all subsequent
variables (i.e. factors driving bid levels) of the bid stack function b. Due to the inelasticity as-
sumption, the level of demand fully determines the quantity of electricity that is being generated; all
subsequent variables merely impact the merit order arrangement of the bids.
Remark 1.The price setting mechanism described above applies directly to day-ahead spot prices
set by uniform auctions, as in most exchanges today. However, we believe that in a competitive
market with rational agents, the day-ahead auction price also serves as the key reference point for
real-time and over-the-counter prices.
2.2. Mathematical Model of the Bid Stack. From the previous subsection, it is clear that the
price of electricity in a structural model like the one we are proposing depends critically on the
construction of the function b. Before we explain how this is done in the current setting, we make
the following assumption about the formation of firms’ bids.
Assumption 2. Bids are driven by production costs. Furthermore,
(1) costs depend on fuel prices and firm-specific characteristics only;
(2) firms’ marginal costs are strictly increasing.
To back up Assumption 2, we briefly consider it in light of historic bid and spot price data. Figure
1a plots the bid stack from the PJM market region in the US on the first day of two consecutive
months. Firstly, rapidly increasing marginal costs lead to the steep slope of the stack near the market
capacity (70,000 MW); this feature translates directly into spikes in spot prices (see Figure 1b)
when demand is high. Between the two sample dates in February and March 2003, the prices of gas
increased rapidly. In the PJM market gas fired plants have historically featured mainly in the second
half of the bid stack. Therefore, it is precisely the gas price related increase in production costs,
which explains the increase in bid levels observed beyond about 40,000MW in Figure 1a (see also
§4.2 for a discussion of recent merit order changes).

ELECTRICITY PRICE MODELING AND ASSET VALUATION: A MULTI-FUEL STRUCTURAL APPROACH 5
0 7
x 104
300
600
Supply (MW)
Price ($/MWh)
1st February 2003
1st March 2003
(A) Sample bid stacks.
0
100
200
2001 2004 2007 2010
Year
Price ($/MWh)
(B) Historical daily average prices.
FIGURE 1. Historical prices and bids from the PJM market in the North East US
Production costs are typically linked to a particular fuel price (e.g. coal, natural gas, lignite, oil,
etc.). Furthermore, within each fuel class, the cost of production may vary significantly, for example
as old generators may have a higher heat rate (lower efficiency) than new units. It is not our aim
to provide a mathematical model that explains how to aggregate individual bids or capture strategic
bidding. Instead, we group together generators of the same fuel type and assume the resulting bid
curve to be exogenously given and to satisfy Assumption 2. From this set of bid curves, the merit
order rule then determines the construction of the market bid stack.
Let I={1,...,n}denote the index set of all the fuels used in the market to generate electricity.
We assume that their prices are the only factors influencing the bid stacks. With each i∈Iwe
associate an FW
t-adapted fuel price process (Si
t)and we define the fuel bid curve for fuel ito be a
measurable function
bi:[0,¯
ξi]×R3(ξ,s)→bi(ξ,s)∈R,
where the argument ξrepresents the amount of electricity supplied by generators utilizing fuel type
i,sa possible value of the price Si
t, and ¯
ξi∈R+the aggregate capacity of all the generators utilizing
fuel type i. We assume that biis strictly increasing in its first argument, as required by Assumption
2. Further, also for i∈I, let the Ft-adapted process (ξi
t)represent the amount of electricity supplied
by generators utilizing fuel type i. It follows that
∑
i∈I
¯
ξi=¯
ξ,and Dt=∑
i∈I
ξi
t,for t∈[0,T].
In order to simplify the notation below, for i∈I, and for each s∈Rwe denote by bi(·,s)−1the
generalized (right continuous) inverse of the function ξ→bi(ξ,s). Thus
bi(·,s)−1(p):=¯
ξi∧infξ∈(0,¯
ξi]:bi(ξ,s)>p,for (p,s)∈R×R,
where we use the standard convention inf /0 = +∞. Using the notation
bi(s):=bi(0,s)and ¯
bi(s):=bi(¯
ξi,s)
and writing ˆ
b−1
i(p,s) = bi(·,s)−1(p)to ease the notation, we see that ˆ
b−1
i(p,s) = 0 if p∈(−∞,bi(s)),
ˆ
b−1
i(p,s) = ¯
ξiif p∈[¯
bi(s),∞), and ˆ
b−1
i(p,s)∈[0,¯
ξi]if p∈[bi(s),¯
bi(s)). For fuel i∈Iat price Si
t=s
and for electricity prices below bi(s)no capacity from the ith technology will be available. Similarly,
once all resources from a technology are exhausted, increases in the electricity price will not lead
to further production units being brought online. So defined, the inverse function ˆ
b−1
imaps a given

6 REN ´
E CARMONA, MICHAEL COULON, AND DANIEL SCHWARZ
price of electricity and the price of fuel ito the amount of electricity supplied by generators relying
on this fuel type.
Proposition 1. For a given vector (Dt,St), where St:= (S1
t,...,Sn
t), the market price of electricity
(P
t)is determined by
(3) P
t=min
i∈IbiSi
t∨sup(p∈R:∑
i∈I
ˆ
b−1
ip,Si
t<Dt),for t ∈[0,T].
Proof. By the definition of ˆ
b−1
i, the function ˜
b−1, defined by
˜
b−1(p,s1,·· · ,sn):=∑
i∈I
ˆ
b−1
i(p,si),
is, when the prices of all the fuels are fixed, a non-decreasing map taking the electricity price to the
corresponding amount of electricity generated by the market. Similarly to the case of one fixed fuel
price, for each fixed set of fuel prices, say s:= (s1,·· · ,sn), we define the bid stack function ξ→
b(ξ,s)as the generalized (left continuous) inverse of the function ξ→˜
b−1(p,s1,·· · ,sn)defined
above, namely
b(ξ,s):=min
i∈Ibisi∨sup(p∈R:∑
i∈I
ˆ
b−1
ip,si<ξ),for (ξ,s)∈[0,¯
ξ]×Rn,
where we use the convention sup /0 =−∞.
The desired result follows from the definition of the market price of electricity in (2). 
2.3. Defining a Pricing Measure in the Structural Setting. The results presented in this paper
do not depend on a particular model for the evolution of the demand for electricity and the prices
of fuels. In particular, the concrete bid stack model for the electricity spot price introduced in §3
is simply a deterministic function of the exogenously given factors under the real world measure
P. However, for the pricing of derivatives in §4 and §5 we need to define a pricing measure Qand
the distribution of the random factors at maturity under this new measure will be important for the
results that we obtain later.
For an Ft-adapted process θ
θ
θt, where θ
θ
θt:= (θ0
t,θ1
t,...,θn
t), a measure Q∼Pis characterized
by the Radon-Nikodym derivative
(4) dQ
dP:=exp−ZT
0
θ
θ
θu·dWu−1
2ZT
0
|θ
θ
θu|2du,
where we assume that (θ
θ
θt)satisfies the so-called Novikov condition
Eexp1
2ZT
0
|θ
θ
θu|2du<∞.
We identify (θ0
t)and (θi
t)with the market prices of risk for demand and for fuel irespectively.
We choose to avoid the difficulties of estimating the market price of risk (see for example [19] for
several possibilities) and instead make the following assumption.
Assumption 3. The market chooses a pricing measure Q∼P, such that
Q∈{Q∼P: all discounted prices of traded assets are local Q-martingales}.
Note that we make no assumption regarding market completeness here. Because of the non-
storability condition, certainly electricity cannot be considered a traded asset. Further, there are
different approaches to modeling fuel prices; they may be treated as traded assets (hence local mar-
tingales under Q) or — more realistically — assumed to exhibit mean reversion under the pricing

ELECTRICITY PRICE MODELING AND ASSET VALUATION: A MULTI-FUEL STRUCTURAL APPROACH 7
measure. Either way, demand is a fundamental factor and the noise (W0
t)associated with it means
that the joined market of fuels and electricity is bound to be incomplete. Note however, that all
derivative products that we price later in the paper (forward contracts and spread options) are clearly
traded assets and covered by Assumption 3.
3. EXPONENTIAL BI D STACK MO DEL
Equation (3) in general cannot be solved explicitly. The reason for this is that any explicit solution
essentially requires the inversion of the sum of inverses of individual fuel bid curves.
We now propose a specific form for the individual fuel bid curves, which allows us to obtain a
closed form solution for the market bid stack b. Here and throughout the rest of the paper, for i∈I,
we define bito be explicitly given by
(5) bi(ξ,s):=sexp(ki+miξ),for (ξ,s)∈[0,¯
ξi]×R+,
where kiand miare constants and miis strictly positive. Note that biclearly satisfies (1) and since it
is strictly increasing on its domain of definition it also satisfies (2).
3.1. The Case of nFuels. For observed (Dt,St), let us define the sets M,C⊆Iby
M:={i∈I: generators using fuel iare partially used}
and C:=i∈I: the entire capacity ¯
ξiof generators using fuel iis used.
A possible procedure for establishing the members of Mand Cis to order all the values of biand ¯
bi
and determine the corresponding cumulative amounts of electricity that are supplied at these prices.
Then find where demand lies in this ordering.
With the above definition of Mand Cwe arrive at the following corollary to Proposition 1.
Corollary 1. For biof exponential form, as defined in (5), the market price of electricity is given
explicitly by the left continuous version of
(6) P
t= ∏
i∈MSi
tαi!exp(β+γ Dt−∑
i∈C
¯
ξi!),for t ∈[0,T],
where
αi:=1
ζ ∏
j∈M,j6=i
mj!,β:=1
ζ ∑
l∈M
kl∏
j∈M,j6=l
mj!,
γ:=1
ζ ∏
j∈M
mj!and ζ:=∑
l∈M
∏
j∈M,j6=l
mj.
Proof. At any time t∈[0,T]the electricity price depends on the composition of the sets Mand C;
i.e. the current set of marginal and fully utilized fuel types.
For i∈M,ˆ
b−1
i=b−1
i, for i∈C,ˆ
b−1
i=¯
ξiand for i∈I\ {M∪C},ˆ
b−1
i=0. Therefore, inside
the supremum in (3), we replace Iwith Mand take ∑i∈C¯
ξito the right hand side. By Proposition
1 the electricity price is given by the left continuous inverse of the function ∑i∈Mˆ
b−1
i, which in the
exponential case under consideration, simplifies to a single log function and yields (6). 
It is clear from equation (6) that the number of possible expressions for the electricity price is
fully determined by the different configurations the sets Mand Ccan take. In fact, fluctuations in

8 REN ´
E CARMONA, MICHAEL COULON, AND DANIEL SCHWARZ
0q3q2q1
Supply (MW)
Price (Euro/MWh)
b1
b2
b3
(A) Fuel bid curves bi.
0q1+q2q
Supply (MW)
Price (Euro/MWh)
(B) Market bid stack b.
FIGURE 2. Example of fuel bid curves and resulting market bid stack for I:=
{1,2,3},q:=¯
ξ
demand and fuel prices can lead to
(7)
n
∑
i=1n
i"n−i
∑
j=0n−i
j#
distinct cases for (6). Nonetheless, the market bid stack is always a piece-wise exponential function
of demand (see Figure 2) with constantly evolving shape as fuel prices move. This captures in a very
tractable way the influence of fuel prices on the merit order and resulting power price dynamics.
3.2. The Case of Two Fuels. For the remainder of the paper, we restrict our attention to the case
of a two-fuel market, consisting of coal and natural gas generators. Our results can in principle
be extended to the general case of n>2 fuels. However, the level of complexity of the formulas
increases rapidly, as evidenced by the number of possible expressions given in (7). We also choose
to omit the analysis of the one fuel case, which leads to far simpler expressions throughout, but
cannot lead to merit order changes. From now on, we set I:={c,g}and carry over all notation
introduced in §2 and §3 so far.
From (7) we know that there are five possible expressions for the electricity spot price. We list
them in Table 1. Note that fixing Dtreduces this list to some subset of three, each of which —
depending on the state of St— can set the electricity price. (A similar reduction to three expressions
occurs by fixing St.) We exploit this property to write formula (6) in a form more amenable to
calculations, identifying all cases explicitly. To simplify the presentation in the text below, we
define
bcg (ξ,s):= (sc)αc(sg)αgexp(β+γ ξ ),for (ξ,s)∈[0,¯
ξ]×R2
+,
where αc,αg,βand γare defined in Corollary 1 and simplify for two fuels to
αc=mg
mc+mg
,αg=1−αc=mc
mc+mg
,β=kcmg+kgmc
mc+mg
,γ=mcmg
mc+mg
.
Further, we set i−:=argmin¯
ξc,¯
ξgand i+:=argmax¯
ξc,¯
ξg.
Corollary 2. With I :={c,g}, for t ∈[0,T], the electricity spot price is given by
P
t=blow (Dt,St)I[0,¯
ξi−](Dt) + bmid (Dt,St)I(¯
ξi−,¯
ξi+](Dt) + bhigh (Dt,St)I(¯
ξi+,¯
ξ](Dt),

ELECTRICITY PRICE MODELING AND ASSET VALUATION: A MULTI-FUEL STRUCTURAL APPROACH 9
P
t, for t∈[0,T]Criterion Composition of
M C
Sc
texp(kc+mcDt)bc(Dt,Sc
t)≤bgSg
t{c} {∅}
Sg
texp(kg+mgDt)bgDt,Sg
t≤bc(Sc
t){g} {∅}
Sc
texpkc+mcDt−¯
ξg bcDt−¯
ξg,Sc
t>¯
bgSg
t{c} {g}
Sg
texpkg+mgDt−¯
ξc bgDt−¯
ξc,Sg
t>¯
bc(Sc
t){g} {c}
(Sc
t)αcSg
tαgexp(β+γDt)otherwise {c,g} {∅}
TABL E 1. Distinct cases for the electricity price (6) in the two fuel case
where, for (ξ,s)∈[0,¯
ξ]×R2
+,
blow (ξ,s):=bc(ξ,sc)I{bc(ξ,sc)<bg(sg)}+bg(ξ,sg)I{bg(ξ,sg)<bc(sc)}
+bcg (ξ,s)I{bc(ξ,sc)≥bg(sg),bg(ξ,sg)≥bc(sc)},
bmid (ξ,s):=bi+ξ,si+Inbi+(ξ,si+)<bi−(si−)o+bi+ξ−¯
ξi−,si+I{bi+(ξ−¯
ξi+,si+)>¯
bi−(si−)}
+bcg (ξ,s)Inbi+(ξ,si+)≥bi−(si−),bi+(ξ−¯
ξi−,si+)≤¯
bi−(si−)o,
bhigh (ξ,s):=bcξ−¯
ξg,scI{bc(ξ−¯
ξg,sc)>¯
bg(sg)}+bgξ−¯
ξc,sgI{bg(ξ−¯
ξc,sg)>¯
bc(sc)}
+bcg (ξ,s)I{bc(ξ−¯
ξg,sc)≤¯
bg(sg),bg(ξ−¯
ξc,sg)≤¯
bc(sc)}.
Proof. The expressions for blow ,bmid,bhigh are obtained from (6) by fixing Dtin the intervals
(0,¯
ξi−],(¯
ξi−,¯
ξi+],(¯
ξi+,¯
ξ]respectively and considering the different scenarios for Mand C.
Sc
blow
Sg
P
Sc
bmid (c=i+)
Sg
P
Sc
bhigh
Sg
P
Sc
bmid (c=i−)
Sg
P
(A) Surface plots of blow,bmid and bhigh
0
P1
P2
P3
P4
P5
Gas Price
Power Price
Low Demand
Medium Demand
High Demand
(B)blow,bmid ,bhigh for fixed Sc(¯
ξc>¯
ξg).
FIGURE 3. Illustration of the dependence of power spot price on fuel prices and demand
Figure 3 illustrates the dependence of the power price on the input factors, namely the prices of
coal and gas and the demand for electricity. For the purpose of these plots we assumed throughout
that ¯
ξc>¯
ξg. The surfaces in Figure 3a represent the functions blow ,bmid and bhigh for demand
fixed in the corresponding intervals. In Figure 3b we aggregate the information contained in Figure

10 REN ´
E CARMONA, MICHAEL COULON, AND DANIEL SCHWARZ
3a for a fixed coal price; i.e. we compare the dependency of the electricity price on the gas price
for each of the three relevant demand levels. In all three cases electricity is non-decreasing in fuel
price and is constant against Si
tif fuel iis not at the margin (i.e., i/∈M). Furthermore, in each case
P
tis linear in the sole marginal fuel for sufficiently low or high gas price, and non-linear in both
fuels in the quadrilateral in the middle. This characterises the region of coal-gas overlap, where both
technologies jointly set the price. Finally, note that labels P1 to P5 indicate regions corresponding
to rows one to five of Table 1.
3.3. Extension to Capture Spikes and Negative Prices. In this section we suggest a simple exten-
sion of the bid stack model in order to more accurately capture the spot price density in markets that
are prone to dramatic price spikes during peak hours or sudden negative prices at off-peak times.
Importantly, this modification does not impact the availability of closed-form solutions for forwards
or spread options, which we introduce in §4 and §5.
Let (Xt)be a stochastic process adpated to the filtration F0
t, as is Dt. Further, we assume that the
relationship between the two processes satisfies
{Xt≤0}={ω∈Ω:Dt=0}and {Xt≥0}=Dt=¯
ξ.
The difference between (Dt)and (Xt)is that (Dt)is restricted to take values in [0,¯
ξ]only, whereas
the process (Xt)can potentially take values on the entire real line.
In the event that demand hits zero or ¯
ξ, we say that the market is in a negative price regime
(for which negative prices are possible, but not guaranteed) or a spike regime and we redefine the
electricity price at these points to be given respectively by
bn(x,s):=b(0,s)−exp(−mnx) + 1,for (x,s)∈(−∞,0]×R2
+,
and bs(x,s):=b¯
ξ,s+expmsx−¯
ξ−1,for (x,s)∈¯
ξ,∞×R2
+.
Under this extension, the power price expression from Corollary 2 is therefore replaced by
ˆ
P
t:=bn(Xt,St)I{0}(Dt) + blow (Dt,St)I(0,¯
ξi](Dt)
+bmid (Dt,St)I(¯
ξi,¯
ξj](Dt) + bhigh (Dt,St)I(¯
ξj,¯
ξ)(Dt) + bs(Xt,St)I{¯
ξ}(Dt),
where blow,bmid,bhigh are defined in Corollary 2.
Note that the constants mn,ms>0 determine how volatile prices are in these two regimes. In such
cases, the price of electricity may now be interpreted as being set by a thin tail of miscellaneous bids,
which correspond to no particular technology. Therefore, the difference between the electricity price
implied by the bid stack and that defined by the negative price or spike regime is independent of fuel
prices.
Remark 2.It is possible to generate realistic spikes even in the base model (without the inclusion of
the spike regime), simply by choosing one of the exponential fuel bid curves to be very steep (large
mi). However, this would come at the expense of realistically capturing changes in the merit order,
as it artificially stretches the bids associated with that technology.
Figure 4 displays the electricity price through time as generated by the stack model for three
different choices of parameters, for the same scenario. In Figure 4b we show a typical price path in
the case that mc,mg,msand mnare very small. This corresponds to a step function bid stack and
has been suggested by Aid et al [1]. Clearly, the prices do not exhibit enough variation to match
observed time series. The solid line in Figure 4a corresponds to more realistic values of mcand mg;
the dashed line illustrates the modification of this path due to the choice of larger values for msand
mn. Both paths capture the stylized facts of electricity price time series reasonably well. For the

ELECTRICITY PRICE MODELING AND ASSET VALUATION: A MULTI-FUEL STRUCTURAL APPROACH 11
100
200
0 1
Time (Years)
Power Price (Euro/MWh)
Pt
Pt (with spikes)
(A)P
tand ˆ
P
twith mc,mg>0.
100
200
0 1
Time (Years)
Power Price (Euro/MWh)
(B)P
twith mc,mgsmall.
FIGURE 4. Simulation of the power price for typical parameters
purpose of this simulation the prices of coal and gas have been modeled as exponential Ornstein-
Uhlenbeck (OU) processes, and demand as an OU process with seasonality truncated at zero and ¯
ξ
(with (Xt)its untruncated version). However, the choice of model for these factors is secondary at
this stage, as we are emphasizing the consequences of our choice for the bid stack itself.
4. FO RWARD CONTRACTS
We now turn to the analysis of forward contracts in our structural framework. For the sake of
simplicity, we ignore delivery periods and suggest that Tbe considered as a representative date in a
typical monthly delivery period (see [4] for more on handling delivery periods). For the purpose of
the present discussion, a forward contract with maturity Tis defined by the payoff
P
T−Fp
t,
where Fp
tis the delivery price agreed at the initial date t, and paid by the holder of the long position
at T. Simple arbitrage arguments (cf. [26]) imply that
Fp
t=EQ[P
T|Ft].
The result of Corollary 2 shows that the payoff of the forward is a function of demand and fuels, so
that the electricity forward contract becomes a derivative on fuel prices and demand.
4.1. Closed Form Forward Prices. For the explicit calculation of forward prices the following
property of Gaussian densities will be useful (see [21] for an application of this result to pricing
compound options). Let ϕ1denote the density of the standard univariate Gaussian distribution,
and Φ1(·)and Φ2(·,·;ρ)the cumulative distribution functions (cdfs) of the univariate and bivariate
(correlation ρ) standard Gaussian distributions respectively.
Lemma 1. The following relationship holds between ϕ1,Φ1and Φ2:
(8) Za
−∞
exp(l1+q1x)ϕ1(x)Φ1(l2+q2x)dx=expl1+q2
1
2Φ2
a−q1,l2+q1q2
q1+q2
2
;−q2
q1+q2
2
,
for all l1,l2,q1,q2∈Rand a ∈R∪ {∞}.

12 REN ´
E CARMONA, MICHAEL COULON, AND DANIEL SCHWARZ
Proof. In equation (8) combine the explicit exponential term with the one contained in ϕ1and com-
plete the square. Then, define the change of variable (x,y)→(z,w)by
x=z+q1,y=wq1+q2
2+q2(x−q1).
The determinant of the Jacobian matrix Jassociated with this transformation is |J|=q1+q2
2.
Performing the change of variable leads to the right hand side of (8). 
For the main result in this section we denote by Fi
t,i∈I, the delivery price of a forward contract
on fuel iwith maturity Tand write Ft:= (Fc
t,Fc
t).
Proposition 2. Given I ={c,g}, if under Q, the random variables log(Sc
T)and log(Sg
T)are jointly
Gaussian with means µcand µg, variances σ2
cand σ2
gand correlation ρ, and if the demand DT
at maturity is independent of FW
T, then for t ∈[0,T], the delivery price of a forward contract on
electricity is given by:
(9) Fp
t=Z¯
ξi−
0
flow (D,Ft)φd(D)dD+Z¯
ξi+
¯
ξi−fmid (D,Ft)φd(D)dD+Z¯
ξ
¯
ξi+fhigh (D,Ft)φd(D)dD,
where φddenotes the density of the random variable DTand, for (ξ,x)∈[0,¯
ξ]×R2
+:
flow (ξ,x) = ∑
i∈I
biξ,xiΦ1(Ri(ξ,0)/σ)
+bcg(ξ,x)exp −αcαgσ2/2"1−∑
i∈I
Φ1(Ri(ξ,0)/σ+αjσ)#,
fmid (ξ,x) = bi+ξ−¯
ξi−,xi+Φ1−Ri+ξ−¯
ξi−,¯
ξi−/σ+bi+ξ,xi+Φ1Ri+(ξ,0)/σ
+bcg(ξ,x)exp −αcαgσ2/2Φ1Ri+ξ−¯
ξi−,¯
ξi−/σ+αi−σ−Φ1Ri+(ξ,0)/σ+αi−σ,
fhigh (ξ,x) = ∑
i∈I
biξ−¯
ξj,xiΦ1−Riξ−¯
ξj,¯
ξj/σ
+bcg(ξ,x)exp −αcαgσ2/2"−1+∑
i∈I
Φ1Riξ−¯
ξj,¯
ξj/σ+αjσ#,
where j =I\ {i}, the constants αc,αg,β,γare as defined in Corollary 2, and
σ2:=σ2
c−2ρσcσg+σ2
g,
Ri(ξi,ξj):=kj+mjξj−ki−miξi+logFj
t−logFi
t−1
2σ2.
Proof. By iterated conditioning, for t∈[0,T], the price of the electricity forward Fp
tis given by
(10) Fp
t:=EQ[P
T|Ft] = EQhEQ[b(DT,ST)|F0
T∨FW
tiFti.
The outer expectation can be written as the sum of three integrals corresponding to the cases DT∈
[0,¯
ξi−],DT∈[¯
ξi−,¯
ξi+]and DT∈[¯
ξi+,0]respectively. We consider the first case and derive the flow
term. The other cases corresponding to fmid and fhigh are proven similarly.
From Corollary 2 we know that b=blow for DT∈[0,¯
ξi−]. This expression for P
Tis easily written
in terms of independent standard Gaussian variables Z:= (Z1,Z2)by using the identity
log(Sc
T)
logSg
T=µc
µg+σ2
cρσcσg
ρσcσgσ2
g Z1
Z2.

ELECTRICITY PRICE MODELING AND ASSET VALUATION: A MULTI-FUEL STRUCTURAL APPROACH 13
Defining ˆ
d:=EQ[DT|F0
T], the inner expectation can now be written in integral form as
E˜
blow ˆ
d,Z=Ic+Ig+Icg,
where ˜
blow(ξ,Z):=blow(ξ,S)and the expectation is computed with respect to the law of Z. For
example, after completing the square in z1,
Ic=Z∞
−∞
exp(l1+q1z2)φ1(z2)Φ1(l2+q2z2)dz2,
with
l1:=µc+kc+mcˆ
d+σ4
c
2,l2:=−σ2
c−µc+kc+mcˆ
d+µg
σc(σc−ρσg),
q1:=ρσcσg,q2:=σg(σg−ρσc)
σc(σc−ρσg).
Lemma 1 now applies with a=∞.Igand Icg are computed similarly. In all terms, we substitute for
µiusing the following standard result. For i∈I,
(11) Fi
t=EQSi
TFt=exp µi+1
2σ2
i,for t∈[0,T].
Substituting the resulting expression for the inner expectation into the outer expectation in (10)
yields the first term in the proposition. 
Remark 3.The assumption of lognormal fuel prices in Proposition 2 is a very common and natural
choice for modeling energy (non-power) prices. Geometric Brownian Motion (GBM) with constant
convenience yield, the classical exponential OU model of Schwartz [28], and the two-factor version
of Schwartz and Smith [29] all satisfy the lognormality assumption.
The above result does not depend upon any assumption on the distribution of the demand at
maturity, and as a result, it can easily be computed numerically for any distribution. In markets where
reasonably reliable load forecasts exist, one may consider demand to be a deterministic function, in
which case the integrals in (9) are not needed and the forward price becomes explicit. For cases
when load forecasts are not reliable, we introduce another convenient special case below, where
demand at maturity has a Gaussian distribution truncated at zero and ¯
ξ.
To simplify and shorten the notation we introduce the following shorthand notation:
Φ2×1
2 x1
x2,y;ρ:=Φ2(x1,y;ρ)−Φ2(x2,y;ρ).
Corollary 3. In addition to the assumptions in Proposition 2 let demand at maturity satisfy
DT=max0,min ¯
ξ,XT,

14 REN ´
E CARMONA, MICHAEL COULON, AND DANIEL SCHWARZ
where XT∼N(µd,σ2
d)is independent of STunder Q. Then for t ∈[0,T], the delivery price of a
forward contract is given explicitly by
Fp
t=∑
i∈I
expm2
iσ2
d
2(biµd,Fi
tΦ2×1
2 " ¯
ξi−µd
σd−miσd
−µd
σd−miσd#,Ri(µd,0)−m2
iσ2
d
σi,d
;miσd
σi,d!
+biµd−¯
ξj,Fi
tΦ2×1
2 " ¯
ξ−µd
σd−miσd
¯
ξj−µd
σd−miσd#,−Riµd−¯
ξj,¯
ξj+m2
iσ2
d
σi,d
;−miσd
σi,d!)
+∑
i∈I
δiexp(η)bcg(µd,Ft)(−Φ2×1
2 " ¯
ξi−µd
σd−γσd
−µd
σd−γσd#,Ri(µd,0) + αjσ2−γmiσ2
d
δiσi,d
;miσd
δiσi,d!
+Φ2×1
2 " ¯
ξ−µd
σd−γσd
¯
ξj−µd
σd−γσd#,Riµd−¯
ξj,¯
ξj+αjσ2−γmiσ2
d
δiσi,d
;miσd
δiσi,d!)
+Φ1−µd
σd∑
i∈I
bi0,Fi
tΦ1Ri(0,0)
σ+Φ1 µd−¯
ξ
σd!∑
i∈I
bi¯
ξi,Fi
tΦ1 −Ri¯
ξi,¯
ξj
σ!,
where j =I\ {i},δi= (−1)I{i=i+}and
σ2
i,d:=m2
iσ2
d+σ2and η:=γ2σ2
d−αcαgσ2
2.
Proof. We use Lemma 1 with a<∞. After integrating over demand, each of the terms in flow ,fmid
and fhigh turns into the difference between two bivariate Gaussian distribution functions. Simplifying
the resulting terms leads to the result. 
Although the expression in Corollary 3 may appear quite involved, each of the terms can be
readily identified with one of the five cases listed in Table 1, along with four terms (the last line of
Fp
t) corresponding to the endpoints of the stack. Furthermore, it is noteworthy that as compared to
Corollary 2, the fuel forward prices now replace the fuel spot prices in the bid stack curves bi, while
µdreplaces demand. The Gaussian cdfs essentially weight these terms according to the probability
of the various bid stack permutations. Thus Fp
tcan become asymptotically linear in Fc
tor Fg
twhen
the probability of a single fuel being marginal goes to one. Finally, we note that a very similar
closed-form expression is available for higher moments of P
Tand given in Appendix A. Convenient
expressions can also be found for covariances with fuels and for the Greeks (sensitivities with respect
to underlying factors or parameters), but are not included.
Remark 4.Under the extended model introduced in §3.3 to capture spikes, forward prices are given
by the same expression as in Proposition 3 plus the following simple terms
expms(µd−¯
ξ) + 1
2m2
sσ2
dΦ1 µd−¯
ξ
σd
+msσd!−Φ1 µd−¯
ξ
σd!
−exp−mnµd+1
2m2
nσ2
dΦ1mnσd−µd
σd+Φ1−µd
σd
Remark 5.In most electricity markets, available capacity is often uncertain, due for example to the
risk of generator outages. Since ¯
ξienters linearly in the exponential function in (6) like demand,
an extension to stochastic capacity levels should be feasible, though rather involved. However, if
capacity shocks are similar for both fuel types, this additional randomness could more easily be
accounted for by adjusting demand parameters µdand σd. More generally, we note that these

ELECTRICITY PRICE MODELING AND ASSET VALUATION: A MULTI-FUEL STRUCTURAL APPROACH 15
parameters could in practice be chosen to calibrate the model to observed power forward (or option)
prices, thus using the random variable DTas a proxy for demand, capacity, and all other non fuel-
related risk, along with corresponding risk premia.
4.2. Correlation Between Electricity and Fuel Forwards. In the American PJM market, coal
and gas are the fuel types most likely to be at the margin, with coal historically below gas in the
merit order. Therefore PJM provides a suitable case study for analyzing the dependence structure
suggested by our model. In Figure 5 we observe the historical co-movement of forward (futures)
prices for PJM electricity (both peak and off-peak), Henry Hub natural gas (scaled up by a factor of
ten) and Central Appalachian coal. We pick maturities December 2009 and 2011, and plot futures
prices over the two years just prior to maturity. Figure 5a covers the period 2007-09, characterized
by a peak during the summer of 2008, when most commodities set new record highs. Gas, coal
and power all moved fairly similarly during this two-year period, although the correlation between
power and gas forward prices is most striking.
0
100
2007 2008 2009
Year
Price ($)
Natural Gas (x10)
Coal
Peak Power
(A) Dec 2009 forward price dynamics.
2009 2010 2011
0
50
Year
Price ($)
Natural Gas (x10)
Coal
Peak Power
Off Peak Power
(B) Dec 2011 forward price dynamics.
FIGURE 5. Comparison of power, gas and coal futures prices for two delivery dates
Figure 5b depicts the period 2009-2011, during which, due primarily to shale gas discoveries, gas
prices declined steadily through 2010 and 2011, while coal prices held steady and even increased a
little. As a result, this period is more revealing, as it corresponds to a time of gradual change in the
merit order. Our bid stack model implies that the level of power prices should have been impacted
both by the strengthening coal price and the falling gas price, leading to a relatively flat power price
trajectory. This is precisely what Figure 5b reveals, with very stable forward power prices during
2010-2011. The close correlation with gas is still visible, but power prices did not fall as much as
gas, as they were supported by the price of coal. Finally, we can also see that the spread between
peak and off-peak forwards for Dec 2011 delivery has narrowed significantly, as we would also
expect when there is more overlap between coal and gas bids in the stack. This subtle change in
price dynamics is crucial for many companies exposed to multi-commodity risk, and is one which
is very difficult to capture in a typical reduced-form approach, or indeed in a stack model without a
flexible merit order and overlapping fuel types.
5. SPREAD OP TI ONS
This section deals with the pricing of spread options in the structural setting presented above. We
are concerned with spread options whose payoff is defined to be the positive part of the difference

16 REN ´
E CARMONA, MICHAEL COULON, AND DANIEL SCHWARZ
between the market spot price of electricity and the cost of the amount of fuel needed by a particular
power plant to generate one MWh. If coal (gas) is the fuel featured in the payoff then the option is
known as a dark (spark) spread. Denoting by hc,hg>0 the heat rate of coal or gas, dark and spark
spread options with maturity Thave payoffs
(12) (P
T−hcSc
T)+and P
T−hgSg
T+,
respectively. We only consider the dark spread but point out that all results in this section apply to
spark spreads if one interchanges cand g. Further, since spread options are typically traded to hedge
physical assets (generating units) the heat rates that feature in the option payoff are usually in line
with the efficiency of power plants in the market. Based on the range of market heat rates implied
by our stack model, we require3
(13) exp(kc)≤hc≤expkc+mc¯
ξc.
Then, as usually, the value (Vt)of a dark spread is given by the conditional expectation under the
pricing measure of the discounted payoff; i.e.
Vt=exp (−r(T−t))EQ(P
T−hcSc
T)+Ft,
which thanks to Corollary 2 is understood to be a derivative written on demand and fuels.
Remark 6.While spread option contracts are often written on forwards, we consider spread options
on spot prices, as required for our goal of power plant valuation. In addition, as we are interested
in closed-form expressions, we limit our attention to the payoffs with strike zero, corresponding to
a plant for which fixed operating costs are negligible or relatively small. Including a positive strike,
one requires approximation techniques to price a spread option explicitly (such as perturbation of
the strike zero case), analogously to approaches proposed for when both commodities are lognormal
(cf. [9]).
5.1. Closed Form Spread Option Prices. The results derived in this section mirror those in §4.1
derived for the forward contract. Firstly, conditioning on demand, we obtain an explicit formula for
the price of the spread. Secondly, we extend this result to give a closed form formula in the case of
truncated Gaussian demand.
We keep our earlier notation for the dominant and subordinate technology i+and i−, and define
ξh:=loghc−kc
mc
,
where 0 ≤ξh≤¯
ξc. By its definition, ξhrepresents the amount of electricity that can be supplied in
the market from coal generators whose heat rate is smaller than or equal to hc.
Proposition 3. Given I ={c,g}, if, under Q, the random variables log(Sc
T)and log(Sg
T)are jointly
Gaussian distributed with means µcand µg, variances σ2
cand σ2
gand correlation ρ, then, for
3Explicit formulae for cases of hcoutside of this range are also available, but not included here.

ELECTRICITY PRICE MODELING AND ASSET VALUATION: A MULTI-FUEL STRUCTURAL APPROACH 17
t∈[0,T], the price of a dark spread option with maturity T is given by
(14) Vt=exp (−r(T−t))(Zmin(¯
ξg,ξh)
0
vlow,2(D,Ft)φd(D)dD
+Z¯
ξi−
min(¯
ξg,ξh)vlow,1(D,Ft)φd(D)dD+Zmax(¯
ξg,ξh)
¯
ξi−vmid,3(D,Ft)φd(D)dD
+Zmin(¯
ξg+ξh,¯
ξc)
max(¯
ξg,ξh)vmid,2,i+(D,Ft)φd(D)dD+Z¯
ξi+
min(¯
ξg+ξh,¯
ξc)vmid,1(D,Ft)φd(D)dD
+Zmax(¯
ξc,¯
ξg+ξh)
¯
ξi+vhigh,2(D,Ft)φd(D)dD+Z¯
ξ
max(¯
ξc,¯
ξg+ξh)vhigh,1(D,Ft)φd(D)dD),
where the integrands are given in Appendix B and discussed in some detail.
Proof. As in the proof of Proposition 2, by iterated conditioning, for t∈[0,T], the price of the dark
spread Vtis given by
Vt:=exp (−r(T−t))EQ(P
T−hcSc
T)+Ft
=exp(−r(T−t))EQhEQ(b(DT,ST)−hcSc
T)+F0
T∨FW
tFti.
Again we write the outer expectation as the sum of integrals corresponding to the different forms
the payoff can take, since the functional form of bis different for DTlying in the intervals [0,¯
ξi−],
[¯
ξi−,¯
ξi+], and [¯
ξi+,¯
ξ]. In addition the functional form of the payoff now depends on whether
DT≤ξhor DT≥ξhand on the magnitude of ξhrelative to ¯
ξcand ¯
ξg. Therefore, the first case is
subdivided into the intervals [0,min(¯
ξg,ξh)], and [min(¯
ξg,ξh),¯
ξi−]; the second case is subdivided
into [¯
ξi−,max(¯
ξg,ξh)],[max(¯
ξg,ξh),min(¯
ξg+ξh,¯
ξc)], and [min(¯
ξg+ξh,¯
ξc),¯
ξi+]; the third case
is subdivided into [¯
ξi+,max(¯
ξc,¯
ξg+ξh)], and [max(¯
ξc,¯
ξg+ξh),¯
ξ].
The integrands vin (14) are obtained by calculating the inner expectation for each demand inter-
val listed above, in a similar fashion as in Proposition 2. 
Note that (14) requires seven terms in order to cover all possible values of hcwithin the range
given by (13), as well as the two cases c=i+and c=i−. However, only five of the seven terms
appear at once, with only the second or third appearing (depending on hc≶exp(kc+mc¯
ξg)) and only
the fifth or sixth (depending on hc≶exp(kc+mc(¯
ξc−¯
ξg))). These conditions can equivalently be
written as ξh≶¯
ξgand ξh≶¯
ξc−¯
ξg. Notice that if c=i−, we deduce that ξh<¯
ξgand ξh>¯
ξc−¯
ξg
irrespective of hc, while for c=i+several cases are possible.
Similar to the analysis of the forward contract earlier, if demand is assumed to be deterministic,
then the spread option price is given explicitly by choosing the appropriate integrand from Proposi-
tion 2. To now obtain a convenient closed-form result for unknown demand, we extend our earlier
notational tool for combining Gaussian distribution functions. For any integer n, let
Φ2×n
2 x11 x12 ·· · x1n
x21 x22 ·· · x2n,y;ρ=
n
∑
i=1
[Φ2(x1i,y;ρ)−Φ2(x2i,y;ρ)].
In addition, we introduce the following notation to capture all the relevant limits of integration.
Define the vector a:= (a1,...,a8)by
a:=1
σd 0,¯
ξg∧ξh,¯
ξi−,¯
ξg∨ξh,¯
ξc∧(¯
ξg+ξh),¯
ξi+,¯
ξc∨(¯
ξg+ξh),¯
ξ−µd.

18 REN ´
E CARMONA, MICHAEL COULON, AND DANIEL SCHWARZ
Notice that the components of aare in increasing order and correspond to the limits of integration
in equation (14). In the case that c=i+, all of these values are needed, while the case c=i−is
somewhat simpler because a3=a4and a5=a6(since by (13), ξh<¯
ξc). However, the result below
is valid in both cases as various terms simply drop out in the latter case.
Corollary 4. In addition to the assumptions in Proposition 3 let demand at maturity satisfy
DT=max0,min ¯
ξ,XT,
where XT∼N(µd,σ2
d)is independent of ST. Then for t ∈[0,T], the price of a dark spread is given
explicitly by
Vt=exp (−r(T−t))bc(µd,Fc
t)expm2
cσ2
d
2Φ2×2
2 ¯
ξca3
a4a2,Rc(µd,0)−m2
cσ2
d
σc,d
;mcσd
σc,d
+bcµd−¯
ξg,Fc
texpm2
cσ2
d
2Φ2×2
2 a8a6
a7a5,−Rcµd−¯
ξg,¯
ξg+m2
cσ2
d
σc,d
;−mcσd
σc,d!
+bgµd−¯
ξc,Fg
texp m2
gσ2
d
2!Φ2×1
2 a8
¯
ξc,−Rgµd−¯
ξc,¯
ξc+m2
gσ2
d
σg,d
;−mgσd
σg,d!
−hcFc
tΦ2×3
2 a7a5a3
a6a4a2,˜
Rc((loghc−β−γ µd)/αg)
σg,γ
;−γσd
αgσg,γ
+bcg (µd,Ft)exp(η)Φ2×2
2 ¯
ξca3
a4a2,−Rc(µd,0)−αgσ2+γmcσ2
d
σc,d
;−mcσd
σc,d
−Φ2×2
2 a8a6
a7a5,−Rcµd−¯
ξg,¯
ξg−αgσ2+γmcσ2
d
σc,d
;−mcσd
σc,d!
+Φ2×1
2 a8
¯
ξc,Rgµd−¯
ξc,¯
ξc+αcσ2−γmgσ2
d
σg,d
;mgσd
σg,d!
−Φ2×3
2 a7a5a3
a6a4a2,−˜
Rc((logH−β−γ µd)/αg)−αgσ2−γ2σ2
d/αg
σg,γ
;γσd
αgσg,γ
+Φ1(−a8)∑
i∈I
bi¯
ξi,Fi
tΦ1 −Ri¯
ξi,¯
ξj
σ!−hcFc
t(1−Φ1(a7) + Φ1(a6)−Φ1(a5))),
where
˜
Ri(z):=z+logFj
t−logFi
t−1
2σ2and σ2
i,γ:=γ2σ2
d/α2
i+σ2.
Proof. All terms in (14) have the same form as those in Proposition 2 for forwards: demand appears
linearly inside each Gaussian distribution function and in the exponential function multiplying it.
Hence, applying Lemma 1 and simplifying lead to the result in the corollary. 
Remark 7.Under the extended model introduced in §3.3 to capture spikes, spread prices are given
by the same expression as in Proposition 4 plus the following simple terms4
expms(µd−¯
ξ) + 1
2m2
sσ2
dΦ1 µd−¯
ξ
σd
+msσd!−Φ1 µd−¯
ξ
σd!.
4We require only two of the four extra terms in Remark 4 due to our assumption on hcin (13), which guarantees that for
the spike regime, the option is always in the money, while for the negative price regime, it never is. Hence, if we were to
consider put spread options instead of calls, the other two terms would be needed instead.

ELECTRICITY PRICE MODELING AND ASSET VALUATION: A MULTI-FUEL STRUCTURAL APPROACH 19
Bid curves Fuel price processes Demand Rate
kimi¯
ξiκiνiλisiµdσdr
2 1 0.5 1 0.5 log(10)10 0.5 0.2 0
TABL E 2. Parameters used throughout §6 (for i∈I)
6. NUMERICAL ANA LYS IS OF SP RE AD OPTION PRICES
In this section, we investigate the implications of the two-fuel exponential stack model of §3.2 on
spread option prices and power plant valuation, as compared to two common alternative approaches.
We analyse prices for various parameter choices and option characteristics, as well as fuel forward
curve scenarios.
Recall that the closed-form spread option prices given by Corollary 4 required no specification
of a stochastic model for fuel prices, but instead imposed only a lognormality condition at maturity
T. However, for the purpose of comparing prices across maturities and across modeling approaches,
we select a simple example of fuel price dynamics consistent with Corollary 4. We assume coal (Sc
t)
and gas (Sg
t)follow correlated exponential OU processes under the measure Q. i.e., for i∈Iand
t∈[0,T],
d(logSi
t) = κiλi−(logSi
t)dt+νidWi
t,Si
0=si
(15)
where dhWc,Wgit=ρdt. As gas and coal are treated identically in the bid stack model, we consider
the symmetric case in which all coal and gas parameters are equal,5including both the parameters
in (15) and in the exponential fuel bid curves, defined in (5). All are listed in Table 2, with the
exception of ρ, which we vary throughout our analysis. All prices are calculated for time t=0.
Note that for a given maturity T, the parameters µc,µg,σc,σg, and ρappearing in §4 and §5 are
related to those in (15) by the following standard results (for i∈I):
µi=siexp(−κiT) + λi(1−exp(−κiT)) ,
σ2
i=ν2
i
2κi
(1−exp(−2κiT)),
ρσcσg=ρνcνg
κc+κg
(1−exp(−κcT−κgT)).
As for fuel prices, recall that no particular process is required for electricity demand in our model.
Typically driven by temperature, demand is often modelled as rapidly mean-reverting to a seasonal
level. Hence, in our examples, we assume (Xt), as introduced in Corollary 4, to be an independent
OU process, with a high value for mean-reversion speed (e.g. 100 or more). For our aim of pricing
options with maturities of several months or even years, the values of these parameters are insignifi-
cant, as DTis always well approximated by its stationary distribution. Hence, in Table 2 we list only
the values µdand σd. We also assume the interest rate r=0 throughout. This completes the base
parameter set to be used throughout this section unless otherwise stated.
Next, we introduce three scenarios designed to assess the role of observed fuel forward curves
and corresponding implications for bid stack structure. Fuel forward curves reveal crucial informa-
tion about the probability of future merit order changes. For example, if bids from coal and gas
are currently at similar levels (as in our parameter set) but one fuel is in backwardation (forwards
5While it is of course not realistic for gas and coal to have identical prices (typically differ by a magnitude of about ten
and trade in different units), it is plausible that after adjusting for different heat rates, the coal and gas bids could indeed
coincide and have similar volatility, make our symmetric case less hypothetical.

20 REN ´
E CARMONA, MICHAEL COULON, AND DANIEL SCHWARZ
decreasing in T), while the other is in contango (increasing in T), then the future dynamics of power
prices (under Q) should reflect the high chance of the coal and gas bids separating. We compare the
following scenarios:
(I) No fuel forward inputs (forward prices implied from (15))
(II) Gas in contango; coal in backwardation (linear with stepsize of 0.2 per month)
(III) No fuel forwards, but gas bids above coal (λc=log(sc) = log(7),λg=log(sg) = log(13))
Note that in Scenario II, fuel forward curves are inputs assumed to be observed from the market, in-
stead of being generated by a model. The standard approach to resolving the inconsistency between
the market and the model in (15) is to calibrate to fuel forwards for each Tvia a shift in the mean
level µi(or more formally via a time dependent long-term mean λi).
6.1. Spread Option Price Comparison. To test our model’s prices for spark and dark spread op-
tions, we compare with two other typical approaches to spread option pricing: Margrabe’s formula
(cf. [25]) and a simple cointegration model (cf. [17], [15] for discussions of cointegration between
electricity and fuels).
6.1.1. Margrabe’s Formula. Assume that under the measure Q, the electricity price P
Tand fuel
price Si
T,i∈I, are jointly lognormal, with correlation ρp,i. Writing µpand σ2
pfor the mean and
variance of log(P
T), then for t∈[0,T], the price of a spread option with payoff (12) is given by
Vm
t=exp(−r(T−t))"Fp
tΦ1 logFp
t/hiFi
t+σ2
p,i/2
σp,i!−hiFi
tΦ1 logFp
t/hiFi
t−σ2
p,i/2
σp,i!#,
where σ2
p,i=σ2
p−2ρp,iσpσi+σ2
i, and all other notation is as before.
6.1.2. Cointegration Model. Let YTbe an independent Gaussian random variable under Q, with
mean µyand variance σ2
y. Then for constant weights wc,wg>0 (the cointegrating vector), we
define P
Tby
P
T:=wcSc
T+wgSg
T+YT.
No closed form results are available for spread options, so prices are determined by simulation.
6.1.3. Comments on Comparison Methodology. In order to achieve a sensible comparison between
the stack model and either Margrabe or the cointegration model, the mean and variance of P
Tshould
be chosen appropriately, for each maturity. For a single fixed Tthis simply requires choosing pa-
rameters µpand σp(in the case of Margrabe) or µyand σy(in the case of the cointegration model)
to exactly match the mean and the variance of P
Tproduced by the stack model. As we shall often
compare models simultaneously across many maturities, we use a variation of this idea, finding the
best fit of an OU process for log(P
t)or (Yt). For Margrabe,
d(logP
t) = κp(λp−(logP
t))dt+νpdWp
t,P
0=p0,
where dWp,Wit=ρp,idtfor i∈I, while for the cointegration case (Yt)is an independent OU
process. In all cases we utilize the closed form expression for the variance of P
Tin the stack model,
given in Appendix A.
Finally, to reflect the symmetry between coal and gas in our parameter set, in the cointegration
model we set wc=wg=1/2exp(kc+mc(µd¯
ξc)), such that the power price has equal dependence on
each fuel, with a price level linked to the most likely marginal bid levels of coal and gas. Note that
since both underlying fuels appear in the cointegration model like in the stack model, a comparison
across values of fuel price correlation ρis quite natural. On the other hand, the correlation parameter
which we vary in the Margrabe approach is ρp,i, which correlates electricity and fuel. Hence the
choices of ρor ρp,iin the following plots are not perfect comparisons, but rather illustrate the role

ELECTRICITY PRICE MODELING AND ASSET VALUATION: A MULTI-FUEL STRUCTURAL APPROACH 21
of correlation in generating a range of prices for each model. Due to variations across maturities,
there is no direct link between a chosen value ρand an appropriate ρp,ishort of estimating these
from data.
7.5 9 10.5 12
10
20
30
hc
Spread price
stack (ρ=−0.8)
stack (ρ=0.8)
Marg. (ρ=−0.8)
Marg. (ρ=0.8)
Coint. (ρ=−0.8)
Coint. (ρ=0.8)
(A) All models (Scenario I)
7.5 9 10.5 12
0
5
10
15
hc
Dark spread price
Scenario I (ρ=−0.8)
Scenario I (ρ=0)
Scenario I (ρ=0.8)
Scenario III (ρ=−0.8)
Scenario III (ρ=0)
Scenario III (ρ=0.8)
(B) Stack model (µd=0.3,σd=0.12)
FIGURE 6. Spread option prices against hifor different correlations and demand levels
6.2. Spread Option Parameter Analysis.
6.2.1. Spread prices versus heat rate hi.For a fixed maturity T=1, we plot dark spread option
prices as a function of heat rate hc(over a range corresponding to (13)). Figure 6a illustrates the
Scenario I results for each of the three models considered, for two different correlation parameters,
ρ=±0.8. In all three models, negative correlation logically raises the option price, as it increases
the volatility of the spread. The stack model generally predicts lower spread option prices than Mar-
grabe, and a smaller gap between different correlation levels. This is because the strong structural
link keeps long-term levels of power and gas close together, thus narrowing the spread distribution
relative to the weaker case of correlated Brownian Motions. The gap between Margrabe and the
stack model widens with negative correlation, while for extreme positive correlation (not plotted),
Margrabe can underprice the stack model. Another way of understanding this phenomenon is to no-
tice that the bid stack automatically imposes a positive dependence structure between electricity and
its underlying fuels, which can only be somewhat weakened or strengthened by varying fuel price
correlation through ρ. The cointegration approach shares this characteristic, and therefore prices
much closer to the stack model than Margrabe, but still somewhat higher for ρ=0.8.
Unlike for Margrabe, Figure 6b reveals interestingly that the dependence on correlation ρdoes
not hold strictly for the stack model, as exceptions can be found. Considering a market with very
low demand (µd=0.3,σd=0.12), we observe that the relationship with ρis reversed for low values
of hc. Since demand is so low, the price is typically set by the cheaper fuel and hence only in the
money for coal bids generally below gas. Hence for negative ρ, we do not receive the typical high
payoffs from high gas and low coal states, and get zero payoff when coal moves above gas in the
stack. Figure 6b also confirms that this reversal is most pronounced if coal bids are shifted to be
below gas bids, as described by Scenario III.
6.2.2. Spread prices versus maturity T. We next investigate spread option prices against maturity T,
and for this purpose fix hi=exp(ki+mi¯
ξi/2)in option payoffs, matching the median heat rate in the
market. We again compare several correlation levels for all three models, and now include Scenario
II to test the impact of fuel forward curves. In Figure 7a (Scenario I), spread options are typically
increasing in maturity as expected, and flatten out as the price processes approach their stationary
distributions, with an ordering of the three models resembling Figure 6a. More interestingly, in

22 REN ´
E CARMONA, MICHAEL COULON, AND DANIEL SCHWARZ
0 1 2
10
20
maturity
Spread price
stack (ρ=−0.5)
stack (ρ=0.5)
Marg (ρ=−0.5)
Marg (ρ=0.5)
Coint (ρ=−0.5)
Coint (ρ=0.5)
(A) Scenario I: Spark or dark spread
0 1 2
5
10
15
maturity
Spark spread price
stack (ρ=−0.5)
stack (ρ=0.5)
Marg (ρ=−0.5)
Marg (ρ=0.5)
Coint (ρ=−0.5)
Coint (ρ=0.5)
(B) Scenario II: Spark spread
FIGURE 7. Option prices against Tfor different correlations and fuel forward scenarios
Scenario II (Figure 7b), longer term spark spread options drop significantly in value in stack model,
are thus greatly overpriced by Margrabe, and significantly overpriced by the cointegration model as
well. As the gas forward curve is now in contango, while coal is in backwardation, coal will almost
always be below gas in the future bid stack, especially for very long maturities. Hence, a spark
spread option has relatively little chance of being in the money, as this would require unusually high
demand. This is a good example of a dependency which cannot be captured by Margrabe or other
reduced-form models, but is automatically captured by the merit order built into the stack model.
Moreover, fuel forward prices are direct inputs into our expressions for spread options, avoiding the
need for an additional calibration step to first match observed fuel forwards, as is the case for the
other approaches.
0 1 2
5
10
15
maturity
Spark spread price
stack
M (i)
M (ii)
M (iii)
C (i)
C (ii)
C (iii)
(A) Scenario II: Spark Spread Comparison
0 1 2
3000
maturity
variance of PT
stack model variance (Scenario I)
stack model variance (Scenario II)
(B) Comparison of variances
FIGURE 8. Analysis of impact of matching mean and variance of P
Tfor Scenario II
So far all plots have assumed that both the mean and variance of the power price distributions
are matched in all three models, via the procedure described in Section 6.1.3. One might question
whether this is realistic. In practice, we only have history (and possibly observed forward curves)
to calibrate each model, and thus should not be borrowing extra information about the future from
the stack model’s structure when calibrating the other approaches. While matching the mean is
reasonable as it is analogous to matching observed power forwards, matching the variance is less
justifiable. In Figure 8a, we compare spark spread option prices for Margrabe and the cointegration
model in Scenario II (as in Figure 7b but now ρ=0) for three different calibration assumptions:
full matching as earlier; matching means but not variances; matching neither means nor variances.
Here ‘not matched’ implies that means and/or variances are instead fitted to Scenario I levels (a
proxy for history). We note significant differences between all cases. Failing to match the mean

ELECTRICITY PRICE MODELING AND ASSET VALUATION: A MULTI-FUEL STRUCTURAL APPROACH 23
implies a greater overpricing of spark spreads in this case, while failing to match the variances acts
in the opposite direction here, lowering the price since the forward looking variance (implied by the
stack in Scenario II) is higher than the variance in Scenario I (see Figure 8b). While other scenarios
could lead to different patterns, it is clear that significant price differences can occur due to the
likely changes in the merit order. In Margrabe, no information is transmitted from fuel forward
prices to the distribution of P
T, while in the cointegration model limited information is transmitted,
since the relative dependence on coal and gas is fixed initially by wc,wg, instead of dynamically
adapting to fuel price movements (and demand). In contrast, the stack model produces highly state-
dependent power price volatility and correlations reflecting known information about the future
market structure.
7.5 9 10.5 12
0.5
1
hc
implied correlation
ξg=0.1 ξg=0.3 ξg=0.5 ξg=0.7 ξg=0.9
(A) Implied correlation varying ¯
ξg
7.5 9 10.5 12
0.6
0.7
0.8
hc
implied correlation
µd=0.3 µd=0.5 µd=0.7
(B) Implied correlation varying µd
7.5 910.5 12 0
1
2
0.5
0.6
0.7
T
hc
implied correlation
(C)ρimp surface (Scenario I)
7.5 910.5 12 0
1
2
0.5
0.6
0.7
0.8
T
hc
implied correlation
(D)ρimp surface (Scenario II)
FIGURE 9. Implied correlation analysis for various parameters and scenarios
6.3. Implied Correlation Analysis. We next analyse ‘implied correlation’ ρim p
p,i, meaning the value
of ρp,ifor which Margrabe’s formula reproduces the stack model price. As Figures 6 and 7 suggest,
for high (positive) values of ρin the stack, it may be impossible for Margrabe to reproduce the price,
for any ρp,i∈[−1,1]. In such cases, implied correlation does not exist. However, ρim p
p,itypically
exists for most values of ρ, and can be understood as a convenient way of measuring (or quoting)
the gap between Margrabe and the stack model price.
In Figures 9a-9b, we investigate implied correlation ‘smiles’ (against hc) for a dark spread option
in Scenario I and with ρ=0. In Figure 9a we first vary the relative capacities of coal and gas (with
¯
ξ=1 throughout). In all cases Margrabe overprices the spread since ρimp
p,c>0, but the difference
is much larger when coal is the dominant technology. As we approach the case of a dark spread
in a fully gas driven market ( ¯
ξgnear 1), Margrabe approaches the stack price (ρimp near 0). In

24 REN ´
E CARMONA, MICHAEL COULON, AND DANIEL SCHWARZ
Figure 9b we assume ¯
ξc=¯
ξg=0.5, but instead vary µd(with σdnow 0.12). We see that the
implied correlation has a slight downward (upward) skew if demand is high (low), and a fairly
symmetric ‘frown’ for µd=0.5. Figures 9c-9d plot implied correlation as a function of both hand
Tfor Scenarios I and II. When given fuel forward curves as inputs (Scenario II), we can observe a
distinctive tilt in the implied correlation surface for long maturities.
6.4. Power Plant Valuation. We conclude this analysis with an investigation into the bid stack
model’s predictions for power plant valuation under our chosen scenarios. A generating unit of fuel
type i∈I, with heat rate hi, can be approximated as a sum of spread options on spot power (cf.
[19])). Letting {Tj:j∈J}represent all future hours of production over the plant’s life, the plant
value (V P
t)for t∈[0,T]is then
V P
t=∑
j∈J
exp(−r(Tj−t))EQP
Tj−hiSi
Tj+
Ft
While this approximation technique ignores complicated operational constraints, it is consistent with
our approximation of the electricity price setting mechanism itself, since a plant bidding at cost every
day receives exactly this string of payoffs in our model.
0.2 0.5 0.8
10
x 108
µd (avg demand level)
Coal or gas plant value
stack model (ρ=−0.5)
stack model (ρ=0)
stack model (ρ=0.5)
Margrabe (ρ=−0.5)
Margrabe (ρ=0)
Margrabe (ρ=0.5)
(A) Scenario I
0.2 0.5 0.8
1
2
x 109
µd (avg demand level)
Coal plant value
stack model (ρ=−0.5)
stack model (ρ=0)
stack model (ρ=0.5)
Margrabe (ρ=−0.5)
Margrabe (ρ=0)
Margrabe (ρ=0.5)
(B) Scenario II (Coal Plant)
0.2 0.5 0.8
1
2
x 109
µd (avg demand level)
Coal plant value
stack model (ρ=−0.5)
stack model (ρ=0)
stack model (ρ=0.5)
Margrabe (ρ=−0.5)
Margrabe (ρ=0)
Margrabe (ρ=0.5)
(C) Scenario III (Coal Plant)
0.2 0.5 0.8
10
x 108
µd (avg demand level)
Gas plant value
stack model (ρ=−0.5)
stack model (ρ=0)
stack model (ρ=0.5)
Margrabe (ρ=−0.5)
Margrabe (ρ=0)
Margrabe (ρ=0.5)
(D) Scenario III (Gas Plant)
FIGURE 10. Power Plant Value (3yrs, 1000MW) as a function of µdfor various scenarios
In Figure 10 we value a 1000 MW power plant with a life of three years in the Margrabe case
and stack model, exploiting our closed-form formulas for reasonable computation time. Instead
of making an arbitrary assumption about the important periodicities of power demand (which vary
from market to market) we use a fixed mean µdfor all hours in each calculation, but investigate
the resulting plant value as a function of µd. We also fix hi=exp(ki+mi¯
ξi/2)throughout. In
Scenario I (Figure 10a), Margrabe prices consistently higher than the stack model as expected.
Figure 10b considers Scenario II where fuel forward prices pull future gas and coal bids in opposite

ELECTRICITY PRICE MODELING AND ASSET VALUATION: A MULTI-FUEL STRUCTURAL APPROACH 25
directions, and Margrabe matches the distribution of P
Tonly from history. Remarkably, for long
enough maturities and high enough µd, Margrabe sometimes underprices a coal power plant relative
to the stack model. Here the stack captures that gas is likely to be the marginal fuel in the future,
with coal plants operating near full capacity.
Another interesting case to consider is Scenario III, in which there is very little overlap between
coal and gas bids to begin with, and coal is likely to remain below gas in the merit order. Figures 10c-
10d reveal the result of this change. Unsurprisingly, for a gas power plant (Figure 10d) the deviation
between the stack model and Margrabe is large, since the gas plant has little chance of being called
upon to produce power. On the other hand, the difference between Margrabe and the stack model
is much less for the coal plant, and the stack model price appears to converge to Margrabe for high
demand (similarly to Figure 9a for high ¯
ξg). The reason for this is that when coal is always below
gas in the stack and demand always high, then the power price can be approximated by the gas stack
alone. Hence, power price should be close to lognormal and the correlation between power and coal
close to that of gas and coal. Furthermore, as there is no mismatch of mean or variance (as there
was in 10b), under such a scenario, Margrabe’s formula should give a very similar price to the stack
model.
Finally, it is important to remember the heavy-tailed nature of most electricity spot prices, which
makes a lognormal distribution for P
Thighly questionable. As discussed in Section 3.3, the bid
stack model allows for a straightforward extension to capture spikes (or negative prices) consistently
without limiting the availability of derivative pricing formulae. To illustrate the impact of spikes,
Figure 11 plots the same scenarios as in Figures 10a and 10d, except using our extended stack
model6with parameter ms=50 (and σdreduced from 0.2 to 0.1). The impact of spikes is to shift
the stack model valuation closer to Margrabe, but only in the case of high µd. This is intuitive of
course because spikes are only present for high enough demand. Since the stack model produces a
heavier-tailed distribution for P
Tthan the lognormal, very high µdcan lead to stack model valuation
even above Margrabe.
0.2 0.5 0.8
15
x 108
µd (avg demand level)
Coal or gas plant value
stack model (ρ=−0.5)
stack model (ρ=0)
stack model (ρ=0.5)
Margrabe (ρ=−0.5)
Margrabe (ρ=0)
Margrabe (ρ=0.5)
(A) Scenario I
0.2 0.5 0.8
15
x 108
µd (avg demand level)
Gas plant value
stack model (ρ=−0.5)
stack model (ρ=0)
stack model (ρ=0.5)
Margrabe (ρ=−0.5)
Margrabe (ρ=0)
Margrabe (ρ=0.5)
(B) Scenario III (Gas Plant)
FIGURE 11. Power plant value with stack model extended to include spike regime
7. CONCLUSION
The valuation and hedging of both physical and financial assets in electricity markets is a com-
plex and highly state-dependent challenge, particularly for medium to long term problems in markets
6This implies a spike of about $150 in the event Xt=1.1¯
ξ, and so is a fairly conservative choice. Note also that we still
match the mean of the Margrabe formula (i.e., calibrate to the stack model’s electricity forward curve), but only match the
variance to the case of no spikes, since otherwise the fit to the new variance (with spikes) becomes infeasible for large µd.

26 REN ´
E CARMONA, MICHAEL COULON, AND DANIEL SCHWARZ
driven by multiple underlying fuels. As we have seen in the examples of §6, it is important to be
aware of the current merit order and resulting bid stack shape, as well as likely or possible changes
to this order in the future. Observed fuel forward curves can provide valuable information for this
purpose, but cannot be incorporated easily into traditional reduced-form models for power prices.
On the other hand, a structural approach maintains a close link with the physical characteristics of
the electricity market, allowing for the inclusion of a variety of forward looking information, such
as demand forecasts, or changes in the generation mix of the market, a pertinent issue in many coun-
tries nowadays. The piecewise exponential bid stack model proposed here achieves this link, while
crucially retaining closed-form expressions for forwards and spread options, as presented in Sec-
tions 4 and 5. In this way, it enjoys the benefits of a simple reduced-form model, while mimicking
the complex dependence structure produced by a full production cost optimization model, for which
derivative pricing is typically a computationally infeasible task. Furthermore, the availability of ex-
plicit expressions for forwards and options is highly beneficial for many other practical purposes,
such as calibration to observed market quotes, the calculation of Greeks and for risk management
applications (e.g. simulating price changes for a portfolio of physical assets). Although we focused
primarily here on a simple coal and gas based market, we believe that our general framework has
the flexibility to be adapted to various market conditions, as illustrated for example by our simple
extension to capture extreme spikes at times of high demand, an important feature of most power
markets. As we have shown through many examples, the delicate interplay between demand, capac-
ity, and multiple fuel prices is embedded into our approach, thus providing an intuitive framework
for understanding complicated interdependencies, while also helping to bridge the prevalent gap
between mathematical tractability and the economics of supply and demand.
APPENDIX A. MOMENTS AND COVARIANCES
If demand at maturity satisfies
DT=max0,min ¯
ξ,XT,
with XT∼N(µd,σ2
d)independent of ST, then for t∈[0,T], the n-th moment of P
Tis given by
E[Pn
T|Ft] = Φ1−µd
σd∑
i∈I
bn
i0,Fi
texp1
2(n2−n)σ2
iΦ1 R(n)
i(0,0)
σ!
+Φ1 µd−¯
ξ
σd!∑
i∈I
bn
i¯
ξi,Fi
texp1
2(n2−n)σ2
iΦ1 −R(n)
i¯
ξi,¯
ξi
σ!
+∑
i∈I
bn
iµd,Fi
texpζ(n)
iΦ2×1
2 " ¯
ξi−µd
σd−nmiσd
−µd
σd−nmiσd#,R(n)
i(µd,0)−nm2
iσ2
d
σi,d
;miσd
σi,d!
+∑
i∈I
bn
iµd−¯
ξj,Fi
texpζ(n)
iΦ2×1
2 " ¯
ξ−µd
σd−nmiσd
¯
ξj−µd
σd−nmiσd#,−R(n)
i(µd−¯
ξj,¯
ξj) + nm2
iσ2
d
σi,d
;−miσd
σi,d!
+∑
i∈I
δibn
cg (µd,Ft)expη(n)(−Φ2×1
2 " ¯
ξi−µd
σd−nγσd
−µd
σd−nγσd#,Ri(µd,0) + nαjσ2−nγmiσ2
d
δiσi,d
;miσd
δiσi,d!
+Φ2×1
2 " ¯
ξ−µd
σd−nγσd
¯
ξj−µd
σd−nγσd#,Ri(µd−¯
ξj,¯
ξj) + nαjσ2−nγmiσ2
d
δiσi,d
;miσd
δiσi,d!),

ELECTRICITY PRICE MODELING AND ASSET VALUATION: A MULTI-FUEL STRUCTURAL APPROACH 27
where j=I\ {i},δi= (−1)I{i=i+}as before and
R(n)
i(ξi,ξj):=kj+mjξj−ki−miξi+log(Fj
t)−log(Fi
t)−(n−1
2)σ2
i−1
2σ2
j+nρσiσj,
η(n):=n2
2γ2σ2
d−αcαgσ2,and ζ(n)
i:=1
2(n2−n)σ2
i+1
2n2m2
iσ2
d.
A more general formula for E[Pn
T(Sc
T)nc(Sg
T)ng|Ft]can be obtained similarly, allowing us to calculate
for example covariances between electricity and fuels.
APPENDIX B. TE RM S IN SPR EAD FO RM ULA
The integrands in Proposition 4 correspond to the dark spread price in various demand intervals,
in the case of a given or known demand value. These terms resemble those for forwards in Proposi-
tion 2, but can be categorized by whether the option is always, never or sometimes in the money for
each case. For the last of these three cases, we have
vlow,1(ξ,x) = bc(ξ,xc)Φ1(Rc(ξ,0)/σ)−hcxcΦ1˜
Rc(h(ξ))/σ
+bcg (ξ,x)expσ2
αcαg1−Φ1(Rc(ξ,0)/σ−αgσ)−Φ1˜
Rg(h(ξ))/σ−αcσ,
vhigh,2(ξ,x) = bg(ξ−¯
ξc,xg)Φ1−Rgξ−¯
ξc,¯
ξc/σ−hcxcΦ1˜
Rc(h(ξ))/σ
+bcg (ξ,x)exp−αcαgσ2
2Φ1Rgξ−¯
ξc,¯
ξc/σ−αcσ−Φ1˜
Rg(h(ξ))/σ−αcσ,
and vmid,2,c=vlow,1,vmid,2,g=vhigh,2, where h(ξ) = (loghc−β−γ ξ )/αg. Then
vmid,1(ξ,x) = fmid (ξ,x)−hcxc,vhigh,1(ξ,x) = fhigh (ξ,x)−hcxc
correspond to cases that are always in the money, while the following are never in the money:
vlow,2(ξ,x) = vmid,3(ξ,x) = 0
ACK NOWL EDG EM ENT S
This work was partially supported by the National Science Foundation grant DMS-0739195.
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BENDHEIM CEN TE R FO R FINA NC E, D EP T. ORFE, UNIVERSITY OF PR IN CE TO N, PRI NC ETO N NJ 08544, USA
E-mail address:rcarmona@princeton.edu
ORFE, UNIVERSITY OF PRINCETON, PRI NC ET ON NJ 08544, USA
E-mail address:mcoulon@princeton.edu
OXFORD-MAN IN ST IT UT E, UNIVERSITY OF OXFORD, OXFORD, UK
E-mail address:schwarz@maths.ox.ac.uk