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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∧infξ∈(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∈IbiSi

t∨sup(p∈R:∑

i∈I

ˆ

b−1

ip,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∈Ibisi∨sup(p∈R:∑

i∈I

ˆ

b−1

ip,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

Eexp1

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∈MSi

tαi!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=1n

i"n−i

∑

j=0n−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,¯

ξgand 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)≤bgSg

t{c} {∅}

Sg

texp(kg+mgDt)bgDt,Sg

t≤bc(Sc

t){g} {∅}

Sc

texpkc+mcDt−¯

ξg bcDt−¯

ξg,Sc

t>¯

bgSg

t{c} {g}

Sg

texpkg+mgDt−¯

ξc bgDt−¯

ξc,Sg

t>¯

bc(Sc

t){g} {c}

(Sc

t)αcSg

tαgexp(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):=bcξ−¯

ξg,scI{bc(ξ−¯

ξg,sc)>¯

bg(sg)}+bgξ−¯

ξc,sgI{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+expmsx−¯

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=expl1+q2

1

2Φ2

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

biξ,xiΦ1(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+Φ1Ri+(ξ,0)/σ

+bcg(ξ,x)exp −αcαgσ2/2Φ1Ri+ξ−¯

ξi−,¯

ξi−/σ+αi−σ−Φ1Ri+(ξ,0)/σ+αi−σ,

fhigh (ξ,x) = ∑

i∈I

biξ−¯

ξj,xiΦ1−Riξ−¯

ξj,¯

ξj/σ

+bcg(ξ,x)exp −αcαgσ2/2"−1+∑

i∈I

Φ1Riξ−¯

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+logFj

t−logFi

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

tiFti.

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)

logSg

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=EQSi

TFt=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=max0,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

expm2

iσ2

d

2(biµd,Fi

tΦ2×1

2 " ¯

ξi−µd

σd−miσd

−µd

σd−miσd#,Ri(µd,0)−m2

iσ2

d

σi,d

;miσd

σi,d!

+biµd−¯

ξj,Fi

tΦ2×1

2 " ¯

ξ−µd

σd−miσd

¯

ξj−µd

σd−miσd#,−Riµd−¯

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#,Riµd−¯

ξj,¯

ξj+αjσ2−γmiσ2

d

δiσi,d

;miσd

δiσi,d!)

+Φ1−µd

σd∑

i∈I

bi0,Fi

tΦ1Ri(0,0)

σ+Φ1 µd−¯

ξ

σd!∑

i∈I

bi¯

ξi,Fi

tΦ1 −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

expms(µd−¯

ξ) + 1

2m2

sσ2

dΦ1 µd−¯

ξ

σd

+msσd!−Φ1 µd−¯

ξ

σd!

−exp−mnµd+1

2m2

nσ2

dΦ1mnσd−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≤expkc+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

tFti.

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=max0,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)expm2

cσ2

d

2Φ2×2

2 ¯

ξca3

a4a2,Rc(µd,0)−m2

cσ2

d

σc,d

;mcσd

σc,d

+bcµd−¯

ξg,Fc

texpm2

cσ2

d

2Φ2×2

2 a8a6

a7a5,−Rcµd−¯

ξg,¯

ξg+m2

cσ2

d

σc,d

;−mcσd

σc,d!

+bgµd−¯

ξc,Fg

texp m2

gσ2

d

2!Φ2×1

2 a8

¯

ξc,−Rgµd−¯

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,−Rcµd−¯

ξg,¯

ξg−αgσ2+γmcσ2

d

σc,d

;−mcσd

σc,d!

+Φ2×1

2 a8

¯

ξc,Rgµd−¯

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

tΦ1 −Ri¯

ξi,¯

ξj

σ!−hcFc

t(1−Φ1(a7) + Φ1(a6)−Φ1(a5))),

where

˜

Ri(z):=z+logFj

t−logFi

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

expms(µd−¯

ξ) + 1

2m2

sσ2

dΦ1 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) = κiλi−(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 logFp

t/hiFi

t+σ2

p,i/2

σp,i!−hiFi

tΦ1 logFp

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 dWp,Wit=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))EQP

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=max0,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

i0,Fi

texp1

2(n2−n)σ2

iΦ1 R(n)

i(0,0)

σ!

+Φ1 µd−¯

ξ

σd!∑

i∈I

bn

i¯

ξi,Fi

texp1

2(n2−n)σ2

iΦ1 −R(n)

i¯

ξi,¯

ξi

σ!

+∑

i∈I

bn

iµd,Fi

texpζ(n)

iΦ2×1

2 " ¯

ξi−µd

σd−nmiσd

−µd

σd−nmiσd#,R(n)

i(µd,0)−nm2

iσ2

d

σi,d

;miσd

σi,d!

+∑

i∈I

bn

iµd−¯

ξj,Fi

texpζ(n)

iΦ2×1

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)expη(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

2γ2σ2

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)expσ2

αcαg1−Φ1(Rc(ξ,0)/σ−αgσ)−Φ1˜

Rg(h(ξ))/σ−αcσ,

vhigh,2(ξ,x) = bg(ξ−¯

ξc,xg)Φ1−Rgξ−¯

ξc,¯

ξc/σ−hcxcΦ1˜

Rc(h(ξ))/σ

+bcg (ξ,x)exp−αcαgσ2

2Φ1Rgξ−¯

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