Pricing Electricity Derivatives on an Hourly Basis

Pricing Electricity Derivatives on an Hourly Basis

Nicole Branger∗ Oleg Reichmann§ Magnus Wobben‡

First version: May 29, 2009

This version: April 9, 2010

∗ Finance Center Munster, Westfalische Wilhelms-Universitat Munster, Germany. E-mail:[email protected]§ Seminar for Applied Mathematics, Eidgenossische Technische Hochschule Zurich, Switzerland.

E-mail: [email protected]‡ Funded by DFG (Deutsche Forschungsgesellschaft), Department of Economic The-

ory, Westfalische Wilhelms-Universitat Munster, Germany. E-mail: [email protected]




Abstract

The purpose of this paper is to develop a framework for pricing electricityderivatives on an hourly basis. We do not - in contrast to most current ap-proaches - focus exclusively on spot models which primarily reflect empiricalspot price dynamics, but also ensure a straightforward applicability to thevaluation of electricity derivatives. We show that a model with a jump and aspike component can be calibrated to both the time-series of hourly spot pricesand the cross-section of futures prices, once we allow for time-dependent jumpand spike parameters. Furthermore, we illustrate the importance of derivativepricing in electricity markets and present some examples of options on futuresand hourly spot-options, such as operating reserves and physical transmissionrights.

Keywords: electricity derivatives, hourly risk premiums, option pricing, op-erating reserve, physical transmission rights

JEL: G13, Q41


Nicole Branger∗ Oleg Reichmann§ Magnus Wobben‡



Abstract

The purpose of this paper is to develop a framework for pricing electricityderivatives on an hourly basis. We do not - in contrast to most current ap-proaches - focus exclusively on spot models which primarily reflect empiricalspot price dynamics, but also ensure a straightforward applicability to thevaluation of electricity derivatives. We show that a model with a jump and aspike component can be calibrated to both the time-series of hourly spot pricesand the cross-section of futures prices, once we allow for time-dependent jumpand spike parameters. Furthermore, we illustrate the importance of derivativepricing in electricity markets and present some examples of options on futuresand hourly spot-options, such as operating reserves and physical transmissionrights.

Keywords: electricity derivatives, hourly risk premiums, option pricing, op-erating reserve, physical transmission rights

JEL: G13, Q41

∗ Finance Center Munster, Westfalische Wilhelms-Universitat Munster, Germany. E-mail:[email protected]§ Seminar for Applied Mathematics, Eidgenossische Technische Hochschule Zurich, Switzerland.

E-mail: [email protected]‡ Funded by DFG (Deutsche Forschungsgesellschaft), Department of Economic The-

ory, Westfalische Wilhelms-Universitat Munster, Germany. E-mail: [email protected]

1 Introduction

The liberalization of electricity markets has lead to more volatile prices and to an

increased trading in electricity derivatives. The pricing of electricity derivatives is

demanding due to its unique characteristics as compared to stocks and commodities.

The most important aspect in pricing electricity derivatives is the fact that electricity

must be produced in the same quantity as is consumed in real time, in order to

avoid network collapses. This can be achieved by the feed-in of electricity generated

by conventional or renewable power plants as well as by the feed-in of electricity

provided by storage facilities. However, the cost of storing electricity largely depends

on the stored quantity, since low-cost storage opportunities, such as artificial and

natural water reservoirs, are limited.1 This makes electricity trading more restrictive

than stock or commodity trading.

In disaggregated wholesale markets (unlike pool markets), electricity is traded either

bilaterally (“over the counter”) or centrally on an exchange. Market participants are

producers and consumers of energy who are interested in physical delivery, but also

intermediaries (banks, speculators and energy trader, for example) who trade in the

market to earn money by speculation or by providing insurance, but want to avoid

any physical delivery. There is a fundamental difference between the forward mar-

ket, where the time between trading and delivery can range from two days to several

years, and the spot market, where electricity is traded from one day before deliv-

ery up to an intra-day gate closure, which is the last opportunity for a settlement

of the trader’s electricity portfolio. Figure 1 visualizes the time line of electricity

1In other words, the marginal cost of holding and not consuming (potential) power during a

time interval [t, t + dt] depends on the quantity of stored electricity at the point in time t, so

that the assumption of the linearity of prices does not hold any more. Note that the slope of the

marginal cost function of storing electricity is higher in countries with a low potential for artificial

and natural water reservoirs, such as Germany, than in markets with more advantageous natural

storage conditions, such as the Scandinavian electricity market.

1

trading. Portfolios with open positions after gate closure have to be balanced by the

transmission system operator (TSO) using balancing power. In order to provide ade-

quate balancing power, the TSO has to obtain a predetermined quantity of operating

reserve in earlier regulated auctions.

day ahead gateReserve capacityForward market activities day‐aheadauction

gateclosure delivery

Reserve capacityauctions

Forward market activities

T – 1day T ‐ 45min T (1 hour)T – x daysT – x years/months

Figure 1: The time line of electricity trading in disaggregated wholesale markets.

Operating reserves can be interpreted as option contracts on hourly electricity

prices.2 Whereas the literature provides a detailed survey of option pricing on stock,

foreign exchange, interest and commodity markets, literature on electricity options,

especially on electricity spot options is quite rare at present. Benth et al. (2008) pro-

vide an overview of derivative pricing in electricity and related markets and survey

the relevant literature on this topic. They discuss several approaches of electricity

price modeling and option pricing, but focus, in contrast to our analysis, on con-

tracts that are traded on power exchanges, i.e. European and spark spread options

on futures as well as Asian options on spot prices.

Standard approaches dealing with lognormal spot price dynamics (see e.g., Lucia

and Schwartz (2002), Geman and Roncoroni (2006) or Cartea and Figueroa (2005)),

are not applicable for the pricing of options on futures. Furthermore, they are not

able to model negative prices. In contrast to these classical models, Benth et al.

(2007) consider a sum of Ornstein-Uhlenbeck processes, each having a different mean

reversion behavior. Therefore, they are able to separate jumps and spikes. To avoid

negative prices, they neglect negative jumps. In our model, we account for both

2Besides operating reserves, we discuss and evaluate physical transmission rights in Section 5,

since cash-settled option contracts written on the hourly spot price are not traded so far.

2

negative jumps and a separation of jumps and spikes.3

The paper is organized as follows: Section 2 introduces standard electricity deriva-

tives, while Section 3 presents the setup of the spot market model. In Section 4, we

discuss the data as well as the empirical results. Section 5 contains some numerical

examples and presents some regulatory issues. Section 6 concludes.

2 Standard Electricity Derivatives

In a competitive electricity market, the supply function is given by the so-called

merit order model, in which all available power plants offer electricity in an increasing

series of their variable costs. The market clearing price thus corresponds to the

variable costs of the marginal power plant, i.e. the last power plant in the merit

order needed to satisfy the demand.4

In nearly all European electricity markets, there is a reference (short-term) price for

electricity which is determined in exchange-organized, day-ahead auctions. In these

auctions, remaining capacities and remaining electricity demand clear the market.5

These residual bids are often inelastic, which causes a high volatility in the so-called

spot markets. Besides the resulting spot prices, we consider forward prices, which

can also serve as underlyings for electricity derivatives.

3On December 22, 2008, electricity consumers received between 9,98 and 101,52 e/MWh from

the producers between midnight and 6 am on the European Power Exchange (see http://www.eex.

com/en). On October 04, 2009, even the daily average spot price (Phelix Day Base) was negative.

The positive probability for the occurrence of negative prices implies risk for non-flexible base-load

power plants.4Although electricity suppliers are power plants as well as load shedding suppliers in reality,

the marginal supplier is simply called the marginal power plant.5In general, spot markets are markets for residual capacities and residual electricity demand,

since a bulk of the expected load is already contracted via forward contracts. We discuss forward

contracts in Subsection 2.2 below.

3

2.1 Spot markets

The reference spot market price serves as an underlying for most electricity deriva-

tives. Two important characteristics of electricity spot prices are the occurrence of

spikes and a strong connection to seasonal levels (mean reversion). Similar to stock

prices or interest rates, electricity prices can be subject to jumps. However, there

are also so-called spikes, i.e. abrupt or unanticipated price peaks that cross a certain

threshold for a usually very short period of time. Economically, the spiky nature of

electricity prices, in contrast to other commodity prices, can be attributed to the

limited storage potential. With a large part of the demand being inelastic, randomly

occurring outages of generation and network capacities have an extreme effect on

electricity spot prices which is not observed for other commodity markets.

Another important aspect of spot prices in disaggregated wholesale markets is that

they can theoretically range from −∞ to ∞ e/MWh.6 Surprisingly, negative elec-

tricity prices make technical and economical sense, and can include important incen-

tive signals for load-shifting. They allow producers to pay consumers when electricity

demand is very low while production is relatively high at the same point in time.

Paying market participants for consuming electricity is then cheaper than temporar-

ily shutting down the plants.7

Typical spot market contracts are day-ahead auctioned contracts for delivery during

certain blocks (normally hours).8 Strictly speaking, the spot price is thus a discrete-

time process. Futures contracts, on the other hand, can be sold at any point in time,

6This is true when prices are exclusively determined by the demand side and can be explained

by the inelasticity of the supply bid curve outside a certain interval of quantities. See also Figure

2 in Section 4.7These situations can occur, for example, due to an unexpected feed-in of wind power during

the night or during national holidays. The European Energy Exchange (EEX) in Leipzig, Germany,

introduced negative electricity prices for the first time in October 2008.8Due to the implementation of the gate closure, a ”real” spot market (i.e. there is no time gap

between trading and delivery) does not exist for electricity and spot prices are consequentially

4

so that futures prices can best be described by continuous-time processes. With few

exceptions, we therefore have to consider discrete-time spot markets and continuous-

time forward markets. Due to its mathematical tractability (see Section 3), we use

a continuous-time process for the spot price, too.

Rigorously, the dynamics of observed spot prices follow from the dynamics of the

true spot prices. Let (Ω,F ,Ft≥0,P) be a complete filtered probability space and let P

be a continuous-time stochastic process on [0, T ]×Ω which describes the unobserved

instantaneous spot price of electricity at any given point in time. If we assume that

the smallest tradable unit for electricity is one hour, we can define the price of an

hourly spot market contract, which is established in the day-ahead auction, as

P di = E

[∫ tdi+1

tdi

P (u) du|Ftd−1j

], (1)

where tdi is the start of hour i on day d. Ftd−1j

describes the information up to hour9

j on the day d − 1 before delivery. If we ignore the time span between td−1j and tdi

and furthermore approximate the integral by its left integrand, we get

P di ≈ E

[P (tdi )|Ftdi

]= P (tdi ) =⇒ P (t) ≈ P (t).

2.2 Forward market

Forward markets can be regarded as insurance markets for risk-averse hedgers who

physically produce or need electricity and aim at transferring spot price risks to

insurers. Speculators, on the other hand, act as insurers and intend to generate

speculative profits from taking these price risks. Electricity forward market contracts

unobservable.9Normally it holds true that j = 12. In addition we assume that expectation is taken under

the physical measure P, although there is a gap of at least twelve hours between the auction and

delivery.

5

can either prescribe the physical delivery of electric power during a certain delivery

period [T1, T2], or can be financially settled (with the same virtual delivery period).

The bulk of exchange-traded futures is cash-settled against the mean of the spot

prices during the delivery period. From an economic point of view (and ignoring all

contract-specific settlement procedures), the payoff at time T2 is given by∫ T2

T1

ω(T, T2)(PT − F [T1,T2]

t

)dT.

The function ω(T, T2) allows to take different payment dates into account. It is equal

to 1 if the futures contract is settled at maturity, whereas ω(T, T2) = exp(r(T2−T ))

if it is settled continuously during the delivery period. As pointed out by Benth and

Koekebakker (2008), the electricity futures contract is thus closely related to a swap

contract, i.e. a continuum of forward contracts.10

Since electricity cannot be stored (at acceptable costs), the cost-of-carry formula

does not hold, i.e. there is no deterministic relation between the spot price today

and the futures price. Instead, the futures price follows from the expected future

spot price, where the expectation is calculated under the risk-neutral measure Q.

This gives

F[T1,T2]t =

∫ T2

T1

EQ[P (T )|Ft] · ω(T, T1, T2) dT, (2)

where the weighting function ω(T, T1, T2) is

ω(T, T1, T2) =ω(T, T2)∫ T2

T1ω(u, T2)du

=

1

T2−T1for ω(T, T2) ≡ 1

rer(T2−T )

er(T2−T1)−1for ω(T, T2) = er(T2−T ).

From an economic point of view, the futures price depends on the expected future

spot prices and on the risk premiums paid in the market for the various risk factors.

10We assume that interest rates are deterministic, so that there is no difference between futures

prices and forward prices. We rather use ’forward contract’ to denote a contract written on the

spot price at a future point in time, while the term ’futures contract’ refers to a contract written

on the spot prices over some future time interval.

6

The expected future spot price, in turn, depends on expected future demand and

supply. While expected demand is essentially influenced by forecasts of weather,

business cycles, political conditions and consumer behavior, the expected supply

depends primarily on future fuel and carbon costs. As the latter are by far the most

volatile impact factors over the medium- and long-term, fuel and carbon costs can

be regarded as the main drivers of electricity futures prices (see also Janssen and

Wobben (2008, 2009)).

3 Model Setup

We model the spot price Pt as the sum of a (deterministic) seasonal component11 f , a

jump-diffusion component X, and a spike component Y . The Jump-Diffusion-Spike

(JDS) model is given by

Pt = f(t) +Xt + Yt

dXt = −κXt dt+ σ dW Pt + JXt dNX

t (3)

dYt = −γ Yt dt+ JYt dNYt

where W P is a Wiener process, and where NX and NY are two inhomogeneous

Poisson processes with deterministic intensities hPt,X and hP

t,Y , respectively. P denotes

the physical (or real-world) measure. JXt and JYt denote the time-dependent jump

sizes, and the respective densities are denoted by gPt,X and gP

t,Y .

Both X and Y are additive (non-Gaussian) OU-processes which exhibit time de-

pendencies and mean reversion towards zero. The use of two processes instead of

11A deterministic seasonal function f should be at least an element of C1T , where T is the

observed time period, respectively. Usually it is modeled as a finite sum of trigonometric functions.

From a modeling point of view, the estimation of a stochastic seasonal component could be more

convenient, since it captures more flexible seasonal movements for future periods. To keep our

model tractable, however, we assume a deterministic component.

7

one jump-diffusion process allows for a different speed of mean-reversion for the two

components. We assume that diffusive shocks and jumps in X vanish over a longer

time period, while jumps in Y vanish much faster. The latter thus capture spikes,

i.e. extreme, but short deviations from the long-run level, which are positive. On the

other hand, X allows to model normal jumps in both direction.

3.1 Forward contracts and option pricing

The dynamics of X and Y under a risk-neutral measure12 Q are given by

dXt = κ

(−λtσ

κ−Xt

)dt+ σ dWQ

t + JXt dNXt (4)

dYt = −γ Yt dt+ JYt dNYt (5)

where WQ is a Wiener process, NXt and NY

t are two Poisson processes with Q-

intensities hQt,X and hQ

t,Y , and the distributions of the jump sizes JXt and JYt under

Q are given by the density functions gQt,X and gQ

t,Y , respectively.

The market price for diffusion risk is denoted by λt, i.e. the compensation for taking

one unit of diffusion risk W Pt is equal to λtdt. We assume that this compensation

is time-dependent.13 Furthermore, we allow for a premium for the jump intensity

and for the distribution of the jump size. We assume that the jump intensities and

the mean jump sizes under Q only depend on time, so that there is no change in

the model structure when switching from the physical measure to the risk-neutral

measure.

12Due to its non-storability, electricity is not a traded asset. Therefore, the discounted spot price

P is in general not a martingale under the risk-neutral measure, while the futures price of course

is a Q-martingale. Furthermore, note that the risk-neutral measure need not be unique.13If we allowed for a market price of risk that is proportional to X, then the change of measure

would lead to a different mean-reversion speed instead of a different mean-reversion level (see also

Benth et al. (2009)).

8

The forward price is given by

F Tt = EQ

t [PT |Ft]

= f(T ) + e−κ(T−t)Xt − σ∫ T

t

e−κ(T−s)λs ds+

∫ T

t

e−κ(T−s)EQ[JXs ]hQX,s ds

+ e−γ(T−t)Yt +

∫ T

t

e−γ(T−s)EQ[JYs ]hQY,s ds. (6)

In line with intuition, a very high mean-reversion speed γ for the spike component

Y implies that current spikes, i.e. very high values of Y , have a small impact on the

forward price. The dynamics of the forward price are given by

dF Tt = e−κ(T−t) (σdW P

t + λtσdt)

+ e−κ(T−t) (JXt dNXt − E

Qt [JX ]hQ

t,Xdt)

+ e−γ(T−t) (JYt dNYt − E

Qt [JY ]hQ

t,Y dt). (7)

For the pricing of European-style options, we use Monte-Carlo simulations of the

spot (4), (5) and the forward dynamics (7). In order to be complete and for an effi-

cient computation of the option’s Greeks, we also give the Partial Integro Differential

Equation (PIDE) for the value of a European option in Appendix A. Therefore, we

give a sketch of the existence proof for a strong solution of the PIDE in Appendix

A.1, while Appendix A.2 presents a finite difference method (FDM) for the numerical

solution of the PIDE.14

3.2 Options on Futures

For the pricing of options on electricity futures, it is often more convenient to use

a model without spikes. Although the resulting process for the spot price P does

not fit the true behavior of the spot price, the spike-dependency of futures prices

given in Equation (6) decreases very fast with increasing T − t, due to the very

large mean-reversion speed γ. The impact of Y on futures prices and thus also on

14For a detailed discussion of FDM for parabolic PIDE and the pricing of American options

using FDM we refer to Cont and Voltchkova (2006) as well as Burger et al. (2010).

9

option prices is very small if we consider times to maturity for the underlying futures

contract beyond one or two months.

If we also omit jumps in X, we get the model of Lucia and Schwartz (2002)

Pt = f(t) +Xt, (8)

dXt = −κXt dt+ σ dW Pt . (9)

The market price for diffusion risk is λt, and the dynamics of X under the risk-

neutral measure are the same as in the last section. The price of a forward that

delivers at time T is then given by

F Tt = f(T ) + e−κ(T−t)Xt − σ

∫ T

t

e−κ(T−s)λs ds.

The price of a futures with a certain delivery period [T1, T2] is given by

F[T1,T2]t =

∫ T2

T1

ω(T, T1, T2)F Tt dT (10)

=

∫ T2

T1

ω(T, T1, T2)

(f(T ) + e−κ(T−t)Xt − σ

∫ T

t

e−κ(T−s)λs ds

)dT.

where the deterministic weighting function is given by

ω(T, T1, T2) =ω(T, T2)∫ T2

T1ω(T, T2) dT

.

If the future is settled at the end of delivery ω(T, T2) is equal to one, while it is

equal to er(T2−T ) if the contract is settled continuously over the delivery period. The

futures price (10) is normally distributed with mean and variance given by

a = EQ[F

[T1,T2]t2 |Ft1

]= F

[T1,T2]t1

b = VQ[F

[T1,T2]t2 |Ft1

]=

σ2

2κ3(T2−T1)2(e2κt2 − e2κt1)

(e−κT2 − e−κT1

)2for ω(T, T2) ≡ 1

σ2e−2rT2

2κ(r−κ)2d(e2κt2 − e2κt1)

(e(r−κ)T2 − e(r−κ)T1

)2for ω(T, T2) = er(T2−T ).

10

The price of a futures call option at the point in time t1 with maturity in t2 and

strike price K is then given by

Ct1 = e−r(t2−t1)

√b

1√2π

e− 1

2

(K−a√

b

)2

+ (a−K)

[1−N

(K − a√

b

)]. (11)

Thus, the value of the futures option depends on the (observable) futures price as

well as on the volatility and the mean-reversion speed of the diffusive component X.

If we allow for jumps in X and assume a normal distribution for JX and a constant

jump intensity under Q, then we can calculate the price along the lines of Merton

(1976).

3.3 Risk Premia

The risk premium for the forward contract follows from the market prices of risk.

In general, both the diffusion component and the jump components are priced. The

market price of risk for the Wiener process is given by λt, and we assume that it is

a deterministic function of time at most. The pricing of jump risk depends on the

jump intensity and the jump size distribution under the physical and the risk-neutral

measure.

The expected gain from a short position in a forward contract with forward price

F Tt is given by15

F Tt − EP

t [PT ] = EQt [PT ]− EP

t [PT ]

= −σ∫ T

t

e−κ(T−s)λs ds

+

∫ T

t

e−κ(T−s) (EQt

[JXs]hQs,X − E

Pt

[JXs]hPs,X

)ds (12)

+

∫ T

t

e−γ(T−s) (EQt

[JYs]hQs,Y − E

Pt

[JYs]hPs,Y

)ds.

15The left hand side can be estimated with historical spot and futures prices, while the right

hand side follows from the difference between the (calibrated) physical and risk-neutral measures.

11

The first term is the premium for diffusion risk, while the second and third term

are the premiums for jump risk and spike risk, respectively. The difference between

the forward price today and the expected spot price can thus be explained by the

market price for diffusion risk, the market prices for jump intensity risk and spike

intensity risk, and the risk premiums for jump size risk and spike size risk. Since a

forward contract has a linear payoff function, premia for higher moments of jumps

and spikes do not have any impact on the premium.16

In the case of commodities, the risk premium is driven by the preferences and the

hedging needs of investors in the market. Producers, who sell electricity for future

delivery, want to hedge against low or even negative prices by selling forward con-

tracts. Hence, they are willing to accept a negative risk premium in Equation (12).

Consumers, on the other hand, hedge against high prices and in particular against

price spikes by taking a long position in forward contracts. They are willing to accept

an expected loss on this long position and thus a positive risk premium in Equation

(12). Depending on whether the hedging needs of producers or consumers dominate,

the resulting risk premium will be negative or positive.

4 Empirical and Computational Results

In the following, we concentrate on the model calibration as well as the description

of electricity data and empirical risk premiums for the German electricity market.

In Section 5, we are going to evaluate some typical spot and futures options.

16For a discussion of the impacts of different jump risk premiums on option prices, see Branger

et al. (2009).

12

4.1 Model-Estimation under the Physical Measure

Unlike most electricity markets, the German wholesale market is a disaggregated

market, where everyone is permitted to trade electricity either via the German power

exchange EEX (European Energy Exchange) or bilaterally for short- and long-term

delivery. There is no capacity mechanism to explicitly pay for fixed cost recovery of

generation capacity (i.e. an energy-only market).17

1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

x 104

-3000

-2000

-1000

0

1000

2000

3000

Quantity

Pric

e

X: 1.559e+004Y: -2.16

Demand

Supply

1.5 1.51 1.52 1.53 1.54 1.55 1.56 1.57 1.58 1.59 1.6

x 104

-30

-25

-20

-15

-10

-5

0

5

10

15

X: 1.559e+004Y: -2.16

Quantity

Pric

e

Demand

Supply

Figure 2: Exemplary EEX-bid curves from November 09, 2008, hour 7.

For the empirical analysis and the estimation procedure, we use data from the EEX

day-ahead market and the EEX forward market for the period from July 01, 2002

to December 24, 2008. The calibration is based on all information up to the end of

the day before valuation.

In the day-ahead market, physical contracts for the 24 hours of the following day are

traded in a double-sided sealed-bid auction at 12.00 pm on each exchange trading

day. In order to procure or sell electricity in the auction, market participants submit

hourly bids until 11.55 am. These bids allow buyers and sellers to place different

quantities at different prices, ranging from -3000 to 3000 e/MWh. The individual

17For more information on the German electricity market, see for example Janssen and Wobben

(2008, 2009).

13

1. Nov 10. Nov 20. Nov 30. Nov

0

50

100

150

200

250

300

350

400

450

500

Day

Pric

e

Figure 3: History of hourly power prices

in November 2008, emphasized: hour 7 on

November 09, 2008.

1. Nov 10. Nov 20. Nov 30. Nov20

40

60

80

100

120

140

160

Day

Pric

e

Phelix Day Base

Phelix Day Peak

Phelix Month Base

Phelix Month Peak

Figure 4: History of the four main indices

for the valuation of futures delivering in

November 2008.

bid curves are then aggregated into market bid functions that determine the hourly

market clearing price and quantity for the next day. Figure 2 shows the EEX-bid

curves in the day-ahead auction for delivery in the seventh hour on November 09,

2008. The time series of the resulting prices for the seventh hour during November

2008 is shown in Figure 3.

The average of all clearing prices from the hourly auctions for one day is referred

to as the Physical Electricity Index (Phelix) Day Base. The average of the hourly

prices during peak load times (08.00 am to 08.00 pm) is referred to as the Phelix Day

Peak. These indices constitute the reference power prices in Germany (see Figure 4

for the Phelix Day Base and Peak during November 2008).

For a reliable estimation of the spot market model discussed in Section 3, we use

hourly prices. Based on these estimates, we will then rely on futures prices to esti-

mate the market prices of risk.18

The frequency of jumps and spikes in hourly electricity prices varies significantly

during the day. To get an idea of this time-dependence, we identify spikes at precisely

18We describe the EEX futures contracts in the next subsection.

14

2 4 6 8 10 12 14 16 18 20 22 240

2

4

6

8

10

12

14

16

18

20

Delivery hours

Cum

ulat

ive

num

bers

of s

pike

s

Time-dependent spike frequency

Figure 5: Positive large price increases (spikes) exceeding three times the standard deviation

(July 2002 – December 2008).

those points in time where price increases exceed three times the standard deviation

of the (deseasonalized) time series. Figure 5 gives the resulting cumulative numbers

of spikes for the different delivery hours. It can be seen that there are no spikes

during night hours and that spike frequencies are greatest during the twelfth and

the eighteenth hour.

To see whether the intensity of normal jumps is also time-dependent, we look at the

normal probability plots for the (deseasonalized) price differences for hourly con-

tracts. Figures 6 and 7 give these plots for the fourth and twelfth hour, respectively.

Both series are not normally distributed. The significant kurtosis rather leads to

the assumption that there are a few, mainly negative, jumps during the night (see

Figure 6) and that there are more, mainly positive, jumps during the peak-hours.

We assume that the jump and spike intensities are time-dependent both under the

physical measure P and under the risk-neutral measure Q. The same holds true for

the market prices of intensity risk which link the intensities under P and under Q.

For the distributions of the jump and spike size, we also assume that all parameters

15

−15 −10 −5 0 5 10

0.001

0.003

0.01 0.02

0.05

0.10

0.25

0.50

0.75

0.90

0.95

0.98 0.99

0.997

0.999

Returns (Hour 4 − Hour 3)

Pro

babi

lity

Normal Probability Plot

Figure 6: Empirical (deseasonalized) price

differences (Hour 4 - Hour 3).

−40 −20 0 20 40 60 80 100

0.001

0.003

0.01 0.02

0.05

0.10

0.25

0.50

0.75

0.90

0.95

0.98 0.99

0.997

0.999

Returns (Hour 12 − Hour 11 )

Pro

babi

lity

Normal Probability Plot

Figure 7: Empirical (deseasonalized) price

differences (Hour 12 - Hour 11).

can be time-dependent under Q. For tractability reasons, however, we restrict the

analysis to constant parameters under P. Furthermore, we assume that all time-

dependencies are driven by a smooth functional of the seasonal component f(t).

We now describe the estimation procedure in more detail. Irrespective of the model

under consideration, we always follow the same steps to estimate the model under

the physical measure P:

• Estimation of the deterministic seasonal component f from historical data

using an iterative least square fit.

• Estimation of the resulting spike part Y via the autocorrelation-function and

a maximum-likelihood estimation (MLE).

• Estimation of the jump-diffusion part X via Markov Chain Monte Carlo.

Estimation of f and Y

First of all, we calibrate f to historical spot price data using an iterative least square

fit. This is necessary due to the spiky nature of electricity prices and the fact that

spikes should not have an influence on the deterministic level. After each least square

fit, we apply a recursive filter to the deseasonalized hourly prices which identifies

16

spikes at precisely those points in time where price increases exceed a multiple of the

standard deviation of the remaining time series (see Cartea and Figueroa (2005)).19

We assume the following functional relation for the historical mean reversion level

f(t), where the unit for t is one hour:

f(t) = fyear(t) + fday(t).

The yearly and daily seasonal components are defined as

fyear(t) =12∑i=1

1Month(i)(t)m(i) + j1 · t+2∑i=1

1CO2(i)(t) · j2(i)

fdayi(t) = k1,i sin(k0,i · t) + k2,i cos(k0,i · t)

+ k3,i sin(2k0,i · t) + k4,i cos(2k0,i · t)

+ k5,i sin(3k0,i · t) + k6,i cos(3k0,i · t)

+ k7,i sin(4k0,i · t) + k8,i cos(4k0,i · t) + k10,i,

where we set m1 and k10,1 to zero.

The yearly seasonality fyear contains a linear trend, twelve dummies for each month

and two dummies for the two phases of the EU-Emissions Trading Scheme (2005 –

2007 and 2008 – 2012). The intra-daily seasonality fday is modeled as the sum of

trigonometric functions with a seasonality of 24, 12, 8 and 6 hours. If the model is

calibrated to daily data instead of hourly data, fdayi(t) is reduced to the constant

k10,i. The sum over the daily effects can then be replaced by the function fweek(t)

which is a sum of these seven dummy variables.

After estimating the seasonal level and separating the spikes of the spot price model,

we have to estimate the intensity of the spikes and the distribution of the spike size.

The latter is estimated via MLE. We assume an exponential distribution for the

19In addition to this tolerance threshold for the spike identification, we have to specify an abortion

criterion. Here, we use an upper bound on the difference between the seasonal functions of two

sequential iterations.

17

spike size, and as stated above, the mean jump size under P is modeled as constant.

The spike intensity, on the other hand, is modeled as a time-dependent function. In

particular, we set it equal to

ht,Y = jY · ht,Y , where ht,Y =f(t)−min

s∈Tf(s)

1#T ·

∑s∈T f(s)−min

s∈Tf(s)

. (13)

Here, jY is the total number of spikes in the observation period T divided by the

number of observations #T . This average spike intensity is then multiplied by an

affine function of the seasonality component f . As a result, the spike intensity is

large whenever the seasonal component is large, and vice versa. This special choice

for the spike intensity results in a higher spike activity during peak hours than

during off-peak hours.

To estimate the mean-reversion speed γ of the spike process Y , we consider the

empirical autocorrelation function for stationary OU processes

ACFY (s) =Cov(yt, yt+s)√

Var(yt)√

Var(yt+s)= e−γs.

Here, (y)≥0 is the filtered spike data. For the estimation of γ, we use a least

square fit of the vector [e−γ·1, e−γ·2, . . . , e−γ·24] to the empirical autocorrelation vector

[ACF (1), ACF (2), . . . , ACF (24)], i.e. we choose s = 1, 2, . . . , 24.

The jump diffusion process X can now be estimated from the remaining spike-less

and deseasonalized prices via a Markov Chain Monte Carlo algorithm. In order to

explicitly separate the jump and diffusion behavior of electricity spot prices, we

choose an NIG-distribution for the jump size.20 For modeling the time-dependent

jump frequency ht,X , we again rely on the affine function ht,Y (see Equation (13))

of the seasonal component f

ht,X = jX · ht,Y · Γ(t),

Γ(t) = 1 + ρ · cos

(t

12 · 365π

).

20For more information on the NIG-distribution, see Appendix B.1.

18

Analogous to above, jX is the average jump intensity. The function Γ accounts

for higher jump frequencies during the winter, where the scarce reserve capacities

usually have fast increasing marginal costs (see also Janssen and Wobben (2008,

2009)).

Estimation of X

Given the estimated seasonal components and the spikes, we get the time series

of X. We now use Markov chain Monte Carlo (MCMC) to estimate the jump-

diffusion process X in equation (3) under the physical measure P. The basic idea of

MCMC is to simulate a Markov chain of the (unknown) parameters with a stationary

distribution that coincides with the probability distribution of these parameters

given the data. We refer to Gilks et al. (1996) for more details on MCMC methods.

Methods for the estimation of NIG random variables using an MCMC algorithm are

also available in the literature. Karlis and Lillestol (2004) use a data augmentation

approach that involves inversion of matrices of size (N×N), where N is the number

of observations. This approach is not feasible for the estimation of a model based on

hourly prices as it involves inversion of huge matrices which is computationally very

expensive. We thus rather rely on using Gibbs-sampling and the Metropolis-Hastings

algorithm (see for example Gamerman and Lopes (2006) p.191 ff).

In order to obtain adequate prior distributions for the MCMC algorithm we fit the

expected values of the priors to the empirical moments of X. We simulated 100,000

iterations with a burnin period of 10,000. The calibrated parameters are given in

column P in Table 1. Note that the jump- and the spike-frequency jX and jY are

average values under P and that the mean reversion speed for spikes is more than

seven times higher than the mean reversion speed for jumps. In addition, Appendix

B.2 gives some empirical results and a comparison of simulated and empirical hourly

spot prices.

19

Table 1: Result of the estimation of the spot price model.

Q- and P-parameters

descriptions Q P

X: mean reversion rate κ 0.0969 0.0969

X: volatility σ 9.0359 9.0359

X: diffusion risk premium −λσ/κ 1.7102 0.0000

X: average jump frequency jX 0.4169 0.4238

X: seasonal jump parameter ρ 0.9151 0.9151

X: NIG-tail heaviness α 0.0185 0.0185

X: NIG-symmetry parameter β 0.0011 0.0011

X: (average) NIG-location µ -0.3028 0.0590

X: NIG-scale parameter δ 5.111 5.111

X: NIG-skewness parameter χ 0.0568 0.0568

X: NIG-normality parameter ξ 0.9559 0.9559

Y : mean reversion rate γ 0.7072 0.7072

Y : average spike frequency jY 0.0046 0.0007

Y : (average) spike-size η 372.1397 308.94

Estimation errors

Since we calibrate our model to hourly data, which are much more complex than

daily data, we can not fully capture the daily mean-reversion behavior of the em-

pirical data (see Figure 8 and Figure 9)21 and deseasonalized data are still non-

stationary.22 The latter naturally affects the robustness of the calibration results.

To avoid these problems, one can either shorten the period of observations, choose

an even more complex deterministic function, or try to put more weight on the spike

21In order to model the daily mean reversion behavior correctly, one would need 24 dimensions

for the X-process, i.e. one mean-reversion speed for each hour. However this would complicate the

model structure and the numerical treatment of the pricing PIDE.22Both, the Augmented Dickey-Fuller unit root test and the Phillips-Perron unit root test for

trend-stationary auto-regressive models reject the null hypothesis in our case.

20

process Y by reducing the tolerance threshold for the spike identification.

0 10 20 30 40 50 60 70 80-0.2

0

0.2

0.4

0.6

0.8

Lag in hours

Sam

ple

Aut

ocor

rela

tion

Sample Autocorrelation Function (ACF), empirical data

Figure 8: Empirical ACF from hourly

EEX-prices from July 2002 – December

2008.

0 10 20 30 40 50 60 70 80-0.2

0

0.2

0.4

0.6

0.8

Lag in hours

Sam

ple

Aut

ocor

rela

tion

Sample Autocorrelation Function (ACF), simulated data

Figure 9: Simulated ACF from 10,000 sim-

ulations of the JDS-model under P (same

length as empirical data).

4.2 Analysis of the Risk Premiums and Model Calibration

under a Risk-Neutral Measure

In the next step, we determine the parameters under the risk-neutral measure Q.

The parameters under the physical measure P, which have already been estimated,

put some restrictions on the equivalent measure Q.23 In particular, the volatility

σ of the diffusion component X and the time-dependent seasonality component f

have to be the same under both measures.

The model is calibrated to futures prices.24 Note that electricity is not traded, so

that the cost-of-carry formula for forward prices does not hold. Therefore, futures

prices indeed contain some information on the risk-neutral measure.

23For a discussion of these restrictions in the context of index option pricing, see e.g. Pan (2002a),

Broadie et al. (2007).24Due to the very small liquidity of option trading at electricity exchanges, we do not use the

option prices for the calibration.

21

As explained above, we allow for time-dependent jump and the spike intensities hQX,s

and hQY,s. We also allow for time time-dependent parameters of the distributions of

the spike and jump size. Since we rely on futures prices, i.e. on linear contracts,

we can only identify the expected jump size EQ [JXs ] and the expected spike size

EQ [JYs ]. To estimate the higher moments of the jump and spike size distribution

under Q, too, we would need to consider non-linear contracts like options.

To formally capture the time-dependence of the expected jump- and spikes-sizes, we

use linear functionals of the seasonal component f(t) for the NIG location µ and the

spike size η. For the jump intensities, we rely on the (deterministic) function ht,Y

again. As a result, the ratio of the jump intensities under the risk-neutral and the

physical measure is constant. With constant expectations under P, the market prices

of risk are thus time-varying, too, and we can capture time-dependent risk premia.

Economically, the time-dependence of the risk premia in pricing electricity risk can

be explained by physical-driven shifts in demand and supply preferences, due to

individual hedging needs of consumers and retailers on the on hand and producers

on the other hand.25

The futures prices stem from the EEX, which operates a forward market with fi-

nancial futures for the delivery of electricity in the current month, the next nine

months, the next eleven quarters and the next six years. We focus on prices of Phelix

Month Futures during 2002 − 2008. The underlying of these futures is the Phelix

Month index, which is equal to the average Phelix Day prices over the respective

month (see Figure 4). The Phelix Month Futures is settled daily before and during

the delivery period. At the last delivery date, the futures price equals the Phelix

Month index. Hence, the futures contract can be regarded as a swap and, thus, op-

tions on futures are closely related to fixed strike Asian spot options. Economically

(and without taking compounding into account), the long position in this future will

25For an analysis of time-dependent risk premia in interest rate and stock markets, see for

instance Engle et al. (1987) and Pan (2002b).

22

1 1.5 2 2.5 3 3.5 4 4.5 5-2

-1

0

1

2

3

4

5

6Term structure of the risk premiums

time to maturity in months

risk

prem

ium

PeakBaseOff-peak

Figure 10: Term structure of the risk pre-

miums over the following 5 month.

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec-4

-2

0

2

4

6

8

10

risk

prem

ium

Risk premia evolution during a year

typical inspection period

load peak in January

negativ riskpremiums

Figure 11: Local risk premiums of the base

futures during a year.

buy electricity over the respective month not at the Phelix Month index (which is

the average spot price), but at the futures price agreed upon at the initial point in

time.

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec-4

-2

0

2

4

6

8

10

model risk premiumsempirical risk premiums

1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1

1.5

2

2.5

3

model risk premiumsempirical risk premiums

Figure 12: Results of the model fit to risk premiums for monthly base futures.

In order to identify the term structure of the risk premia, we assume that the length

of the delivery period is one month. Figure 10 gives the term structure of risk premia

of the base, the peak and the off-peak futures over all delivery months for times to

maturity between one and five months. For short-term contracts, the risk premium

is positive, i.e. the futures price is larger than the expected future spot price. This

23

implies that consumers who take a long position in futures to hedge their exposure

are willing to pay a premium, which is earned by the producers, since the hedging

needs of the consumers are larger than those of the producers. The risk premia

decrease with increasing times to maturity and can become negative. For longer

times to maturity, the hedging needs of the producers increase, and they start to

pay a premium to the consumers in order to hedge against sudden price drops. This

hedging need is stronger for the off-peak futures, for which there is a higher risk of

seeing price drops or even negative prices.

Figure 11 shows the local risk premia for delivery in the different months, i.e. the

average risk premia over all times to maturity for delivery in January, February, and

so on. The risk premium is highest for contracts with delivery in winter, when there

is additional demand for heating and consumers have a higher incentive to hedge. It

decreases during the summer and can even be negative. The large positive premium

in August can be explained by the inspection period of nuclear plants, during which

the supply is reduced and the risk for price spikes is thus larger.

For the calibration of the risk premia, we assume that the (constant) premium for

diffusion risk is equal to the average risk premium.26 The resulting parameters are

given in the column Q in Table 1. Note that under Q not only the jump- and spike-

frequencies, but also the jump- and spike sizes µ and η are average values and that

both, the spike size and the spike frequency increase under the change of measure.

Figure 12 gives the resulting risk premia in our model and in the data for different

delivery months (left panel) and different times to maturity (right panel) for base

futures.27 It shows that our model is able to match the risk premia observed in the

26Since this is kind of an ad-hoc assumption, we have done several robustness checks by us-

ing higher or lower values for this market price of risk. The resulting parameters do not change

dramatically.27The figures for peak and off-peak futures look quite similar and the sum of the peak and the

off-peak risk premium equals the base risk premium.

24

0 2000 4000 6000 8000-10

-5

0

5

10

15

20

25

hours

€/M

Wh

jump and diffusion risk

Jan Apr Jul Oct Dec-10

0

10

20

30

€/M

Wh

hourly risk premiums

October-4

-2

0

2

4

6

8jump and diffusion risk

October-4

-2

0

2

4

6

8

10hourly risk premiums

Figure 13: Results of the model fit to risk premiums on a hourly basis (including the constant

diffusion risk premium).

market rather well.

Figure 13 shows the hourly risk premia, i.e. the model-implied risk premia for a

forward contract evaluated in the middle of December with delivery in one selected

hour of the following year. The upper row gives the premia due to diffusion and jump

risk, i.e. due to the risk factors in X. The lower row gives the overall risk premium,

which also includes the premium for spike risk. As can be seen from Table 1, the

premium for diffusion risk is positive and equal to 1.71 e/MWh. The sign of the jump

risk premium depends on the time of the year and on the hour. It can be negative

during the summer and during off-peak periods, when producers have to hedge their

generation portfolio against negative jumps, which could cause prices that are lower

than the marginal costs of their inflexible base load power plants. Thus, the jump

25

October10

20

30

40

50

60

70

80

90

100

modeled HFCcommercial HFC

October10

20

30

40

50

60

70

80

90

100

modeled HFCcommercial HFC

Figure 14: Model results vs. commercial hourly forward curve in October, 2007. The spot

realization is included on the right side.

risk premium is closely related to the costly expense of shutting down these power

plants several times during the respective (off-peak) delivery period. Furthermore, in

off-peak hours, the (negative) jump risk premium decreases even more in times where

the feed-in of wind power increases, i.e. during the fall, when low or negative prices

become more probable. During peak hours, in contrast, the (positive) jump risk

premium increases, especially during the winter, when electricity load is expected

to be very high.

The average spike size is constant under P, but turns out to depend on time under

the risk-neutral measure Q. In contrast to the jump risk premium, the spike risk

premium is positive at any given point in time. It reaches its maximum in January,

when the buyers of electricity (mainly retail customers) fear price spikes more than

during the summer, while it is close to zero during off-peak hours, where spikes

hardly occurred in the past. Nevertheless, the jump risk premium dominates the

spike risk premium, since spikes are rare events.

Finally, we compare model-implied hourly forward prices, i.e. expected spot prices

under the risk-neutral measure Q, given by Equation (6) with commercial hourly

26

January0

50

100

150

200

250

300

350

modeled HFCcommercial HFCspot prices

Figure 15: Model results vs. commercial hourly forward curve in January, 2007.

forward prices (so-called hourly forward curves28), which are calculated with a fun-

damental model. The results of the comparison are shown in Figures 14 and 15.

Although we have got less information (for example concerning the Saturdays and

the evening peak hours in January), the model results approximatively reflect the

commercial hourly forward curve. The average values of the commercial and the

modeled forward curve are exactly the same, i.e. the value of the respective EEX

month futures. Therefore, we expect our model to produce realistic hourly forward

curves if it is combined with a good fundamental model for the derivation of the

seasonal component.

28We use a sequence of hourly forward curves, estimated on several valuation days in 2007. This

forward curve is provided by the energy company Alpiq.

27

5 Examples

In the following we discuss some numerical examples of futures and spot options,

which play an important role in liberalized electricity markets. First we price options

on futures and compare them to market prices. Then we consider more involved real

options such as operating reserves and physical transmission rights (PTR). The

pricing is done by Monte-Carlo simulation.

5.1 Options on Futures

50 55 60 65 70 751

2

3

4

5

6

7

8

9

10

11

strike prices

optio

n pr

ices

prices of call options on the Phelix Month (Apr 09) Futures, constant volatility

empirical option pricesBlack 76Lucia SchwartzJDS

50 55 60 65 70 75 800

2

4

6

8

10

12

strike prices

optio

n pr

ices

prices of call options on the Phelix Year 09 Futures, constant volatility


Figure 16: Model and empirical prices of call options on the Phelix April 09 Base Futures

(left) and on the Phelix Year 09 Base Futures (right) on November 17, 2008.

The literature on electricity options mainly deals with options on futures, which are

also traded at exchanges, while other options are traded OTC only. Note that these

options are options on swaps, since the future has a delivery period. Although Benth

et al. (2008) give closed-form solutions for options in arithmetic models with and

without jumps, we use a Monte-Carlo simulation of the forward dynamics, given in

Equation (7). If there are no jumps, the price of the European option on the future

is given in Equation (11).

28

First, we compare options on futures for different maturities, different strike prices

and different delivery periods in the model of Black (1976), the model (9) of Lucia

and Schwartz (2002) and our Jump-Diffusion Spike (JDS) model (3).

In particular, we look at the prices on November 17, 2008 for the European call

options on the Phelix Month (April 09) Base Futures (left graph) and the Phelix

Year 09 Base Futures (right graph). The futures price is 59.00 e/MWh for the Phelix

Month Future, while it is 62.54 e/MWh for the Phelix Year Futures. The times to

maturity are 127 and 24 days, respectively.

50 55 60 65 70 75-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

strike prices

log 10 of the price quotients of call options on the Phelix Month (Apr 09) Futures, constant volatility

empirical option pricesBalck 76Lucia SchwartzJDS

50 55 60 65 70 75 80-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

strike prices

log 10 of the price quotients of call options on the Phelix Year 09 Futures, constant volatility


Figure 17: Log10 of the quotient between model prices and empirical call option prices on

the Phelix April 09 Base Futures (left) and on the Phelix Year 09 Base Futures (right) on

November 17, 2008.

We calibrate all parameters in the models of Black (1976) and Lucia and Schwartz

(2002) directly to option prices. For the JDS model, we rely on the estimates in

Section 4 and only calibrate the jump and spike sizes to empirical option prices.

The results are given in Figures 16 – 18. The Lucia-Schwartz model seems to be

able to reflect prices for options on futures with a long time to maturity which are

in or at the money, but it fails to capture the prices for out-of-the-money options

and options on futures with a shorter time to maturity. Further numerical results

(not shown here) reveal that the pricing errors are the larger the shorter the time

29

to maturity. The Black 76 model gives large pricing errors for all options and is not

able to explain the time- and strike dependent volatility smile. If we allow for a time-

dependent volatility in the Black-model, however, its pricing performance improves

significantly, and it is better than the model of Lucia and Schwartz (2002) (see Figure

18). The JDS model captures the cross section of futures options best. Compared

to more complex models based on daily prices, the JDS model is competitive but

not dominant (see for instance Benth et al. (2007)). It is powerful for the pricing of

hourly derivatives and it should dominate less complex models for the valuation of

futures options. We discuss the valuation of selected hourly spot options in the next

two subsections.

50 55 60 65 70 75 80-0.15

-0.1

-0.05

0

0.05

0.1

strike prices

log 10 of the price quotients of call options on the Phelix Year 09 Futures, time-dependent volatility

BenchmarkBlack 76Lucia SchwartzJDS

Figure 18: Log10 of the quotient between model prices and empirical call option prices on

the Phelix Year 09 Base Futures, time-dependent volatility.

5.2 Physical Transmission Rights

A key factor for any international power trade is the ability to transfer power across

borders. Between the European power markets, the bulk of cross border net capaci-

ties is limited. It is distributed to market participants in the form of options. These

so-called physical transmission rights (PTR) can be interpreted as a bundle of Eu-

30

ropean call options on electricity price spreads between two zones. Details on the

contracts and on their pricing can be found in Wobben (2009) and the references

given there.

Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct2000

2500

3000

3500

4000

4500

5000

5500

month

pric

es in

€/M

W

call option priceshistorical PTR (Germany --> NL) prices

Figure 19: Model and historical prices of

PTR for the delivery of power from Ger-

many into the Netherlands.

Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct0

50

100

150

200

250

300

monthpr

ices

in €

/MW

put option priceshistorical PTR (NL --> Germany) prices

Figure 20: Model and historical prices of

PTR for the delivery of power from the

Netherlands into Germany.

In the following we concentrate on the monthly auctions between Germany and

the Netherlands. These monthly auctions are held on the 10th work day of each

month preceding delivery. In these auctions, user rights for the so-called available

transmission capacity (ATC) for the entire calendar month are auctioned. Bids must

be entered by noon and each bidder can enter multiple offers within the same auction.

The single (hourly) PTRs can then be traded up to four days before the actual

physical delivery by either selling them directly to a third party or by selling them

in the next daily auction. Each hourly PTR is an option which matures at 8:00 am

on the day before delivery. The ultimate payoff upon exercise is the spread between

the hourly spot prices for delivery on the next day. Since the spot auctions of the

German EEX and the Dutch APX are held at 12:00 pm, the option holder does not

know this spread upon exercise, but rather gets an hourly forward contract on this

31

spread.29 The monthly PTR can thus be interpreted as a portfolio of hourly forward

options on the spread.

We calibrate the valuation model to the spreads in monthly futures prices between

the Netherlands and Germany.30 The results of the computation are shown in Figure

19 and Figure 20.

The model prices for the PTRs are significantly larger than the prices observed in the

market. While this can be attributed to model misspecification, of course, it also

suggests that the market for cross-border transfer capacities is not fully efficient

(see also Wobben (2009)). First, the market is not complete, and the ability to

exploit pricing errors by following dynamic trading strategies is limited. Secondly,

the contracts are not firm. Cross-border capacities are not fully available if the TSO

holds a short position at the day of delivery, caused by unexpected shocks in the

available capacity, i.e. the installed capacity less the capacity needed for physical

transmission to guarantee the European net frequency of 50 Hz. Finally, the prices in

the monthly PTR auction differ substantially for one and the same product (RWE

Transportnetz, Germany - TenneT, the Netherlands and E.ON Netz, Germany -

TenneT, the Netherlands, see Wobben (2009)).

5.3 Operating Reserve

The German electricity market for operating reserve comprises the primary, sec-

ondary and tertiary reserve.31 Primary and secondary reserves are offered by plants

that are online during the whole day or at least during the peak or off-peak period,

while tertiary reserve can be provided within 4-hour blocks. Plants with shorter ex-

29All data are available at http://www.apxgroup.com/ and http://www.eex.com/en/.30The modeling and calibration of regional price spreads is described and analyzed in Wobben

(2009) for a slightly simplified version of our model.31Common international terms for these kinds of operating reserve are in turn frequency-response

reserve, spinning as well as non-spinning or supplemental reserve.

32

pected online times cannot offer these kinds of reserve as the response time is too

short for a cold start. In the following we will concentrate on the market for tertiary

reserve, which has a relatively competitive structure in Germany (there are over 20

market participants).

1 2 3 4 5 61

2

3

4

5

6

7

8

9

Bloc

rela

tive

diffe

renc

es

relative differences Kobs/Kthe

Figure 21: Relative differences of implicit

and observed strike prices (AP) of operat-

ing reserves for all 4h-blocs on November 17,

2008.

1 2 3 4 5 60

50

100

150

200

250

300

350

Bloc

Pric

e

K=40K=60K=80K=100

Figure 22: Some prototypes of power

plants offering operating reserve (LP),K de-

notes the marginal costs on the spot market

of these prototypes.

The market for tertiary reserve is a day ahead market where the market participants

are electricity producers as well as consumers on the supply side and the Transmis-

sion System Operator (TSO) on the demand side. In order to participate in the

auction for operating reserve, the (technically prequalified) suppliers have to submit

two bid prices instead of one bid price. The so-called Leistungspreis (LP) is paid in

e/MW if the supplier is preselected, i.e. if the supplier refrains from the opportunity

to sell electricity on the day-ahead market and holds off operating reserve instead.

The Arbeitspreis (AP) is paid in e/MWh if the supplier has to deliver balancing

power, i.e. if the TSO exercises its option.

The AP is the strike price of the option, while the LP is the price of the option. We

ignore all strategic considerations and assume that the AP is equal to the marginal

33

costs (MC) of the producer.32 Assuming that the interest rate is zero, we get the

following expression for the LP:

LP (T, t) = EQ [(PT −MC)+|Ft]∼ EQ [(PT − AP )+|Ft

], (14)

where t is the point in time where the auction is held and T is the delivery date.

As an example, we consider reserves for all four-hour-blocks on November 17, 2008.

Figure 21 gives the relation between observed (APs) and implicit strike prices, where

the implicit strike price is defined as the strike price for which the theoretical price

in Equation (14) is equal to the empirical LP. Due to the fact that an operating

reserve schedule requires significantly more flexibility than a spot market schedule,

it is not surprising that the observed strike prices are above the calculated prices.

The size of the difference is not that intuitive, as one would expect some technical

or calculatory costs explaining the difference. But an increase of 200% – 800% can

not be explained with technical arguments. Figure 22 shows real option prices for

different prototypes of power generating facilities. These price curves can be hardly

observed on the real market, since the empirical LP-bids are usually much lower

than the theoretical option premiums.33

Taken together the option premiums can not be explained from an option pricing

point of view. This discrepancy might be caused by strategic considerations of the

market participants caused by imperfections of the market design. The reason for

the strategic bids in the market for operating reserve is the selection procedure for

these bids. As the only criterion for being selected is the LP, it is beneficial to bid a

lower LP and compensate this with a higher AP. This leads to inefficiencies as due

to the different levels of information some bidders may be better informed about the

32Note that these costs will differ from the marginal costs observed in the spot market, since the

operating reserve schedule is fundamentally different from the spot market schedule.33See https://www.regelleistung.net/regelleistungWeb/?language=en for an overview of

the LP- and AP-bids for the German operating reserve markets.

34

LP which will be selected. Furthermore, there might be market barriers due to price

intransparancies. For proposals of an improvement of the market design we refer for

example to Perez (2008).

6 Conclusion

In this paper, we develop a framework for pricing electricity derivatives. We intro-

duce a model which accounts for the mean-reversion in electricity prices, jumps in

these prices, and spikes. In this model, we can price both spot and futures options.

The model can be calibrated to the time series of prices and to the cross section

of futures prices and (in an extension not considered here) also to option prices.

Since the cost-of-carry formula does not hold, futures prices already contain some

information on risk premia and thus on the relation between the physical and the

risk-neutral measure. The resulting risk premia vary over time, and a big part of

this variation can be explained by time-varying hedging needs of consumers and

producers.

We have shown that the model is able to price futures options. It can also be used

to derive theoretical prices of hourly spot options, like physical transmission rights

or operating reserves. A comparison between model prices and market prices hints

at a mis-specification of the model, or, as we argue, also at inefficiencies in these

markets.

Further research could first look at other markets. The US, the Nordic and the UK

markets have been liberalized significantly earlier than the German market. It would

thus be interesting to compare model prices and market prices for real options in

these markets to see whether the supposed higher liquidity and fewer market frictions

indeed lead to smaller price deviations.

Another interesting topic for further research is the analysis of spread options on

35

electricity and other (cor-)related commodities such as gas or coal. With a spot

market model, the value of a power plant can be determined by analyzing the implicit

(American) real options.

36

A Appendix 1

The two-dimensional PIDE for the price V = V (x, y; t) of a European-style optionis given by

∂V

∂t+(LD + LJ − r

)V = 0 (15)

where the differential operator LDV is given by

LDV =σ2

2

∂2V

∂x2+ κ(−λtσ

κ− x)

∂V

∂x

while the Integro Differential-Operator LJV for the jump part of P is given by

LJV = hQt,X

∫ ∞−∞

(V (x+ zX , y; t)− V (x, y; t)

)gQt,X(zX)dzX

+hQt,Y

∫ ∞−∞

(V (x, y + zY ; t)− V (x, y; t)

)gQt,Y (zY ) dzY − γy∂V

∂y,

where gQt,X and gQ

t,Y are the probability density functions of the jump- and spike-sizesunder the risk-neutral measure. The final condition for V follows from the terminalpayoff of the option contract.

A.1 Sketch of the existence proof for a strong solution ofthe PIDE for a European-style option

We switch to time-to-maturity τ = T − t and consider the PIDE (15) for w, wherew(x, exp(−γτ)y, τ) = V (x, y, τ). This leads to the following problem:

∂w

∂τ= A(τ)(w(τ)) +K(τ)(w(τ)) with (16)

A(τ)(w(τ)) =σ2

2

∂2w

∂x2(τ) + κ(µ(τ)− x)

∂w

∂x(τ) and

K(τ)(w(τ)) = rw + hY (τ)

∫R(w(x, y + z, τ)− w(x, y, τ))fY (exp(βτ)z) exp(βτ) dz

+hX(τ)

∫R(w(x+ x, y, τ)− w(x, y, τ))fX(x) dx,

with corresponding boundary and initial conditions. We can get rid of the non-homogeneous boundary conditions using the transformation v(x, y, τ) = w(x, y, τ)−g(x, y, τ). This leads to the following PIDE

∂v

∂τ= A(τ)(v(τ)) +K(τ)(v(τ)) + f(x, y, τ) with

f(x, y, τ) = −K(τ)(g(x, y, τ))− A(τ)(g(x, y, τ))− ∂g

∂τ(x, y, τ), (17)

37

with a corresponding initial condition. We define the operator A0 as:

A0v =σ2

2

∂2v

∂x2. (18)

Now we are able to prove the existence of a strong solution of the above problem(17) using (Amann, 1995, Theorem 1.2.1). Therefore, we have to show:

1. A0 generates an analytic semigroup,

2. A(t) = A0 + (A(t)− A0) for all t ∈ [0, T ] generates an analytic semigroup,

3. (A+K)(t) for all t ∈ [0, T ] generates an analytic semigroup,

4. t −→ (A+K)(t) is Lipschitz continuous and

5. f(·, x) : [0, T ] 7→ L2(Ω) is Lipschitz continuous for all x ∈ L2(Ω).

1., 2. and 3. follow using (Renardy and Roger, 2004, Theorem 12.22) and (Pazy,1983, Theorem 1.4.3), while 4 is trivial and 5 can be shown using the smoothness ofg.

A.2 Numerical solution of the PIDE

There are several approaches to solving the parabolic pricing PIDE numerically. Wesolve the pricing equation using a finite difference scheme. The considered problemreads

∂V

∂τ=

σ2

2

∂2V

∂x2− α(x)

∂V

∂x− βy∂V

∂y− rV

+hY (τ)

∫ ∞−∞

((V (x, y + z, τ)− V (x, y, τ)))fY (z) dz

+hX(τ)

∫ ∞−∞

((V (x+ z, y, τ)− V (x, y, τ)))fX(z) dz

on Ω = [0, T ] × [xmin, xmax] × [0,∞), where we impose zero Neumann boundaryconditions

∂V

∂x(xmin, y, τ) =

∂V

∂x(xmax, y, τ) = 0 ∀τ, y ∈ [0, T ]× [0,∞). (19)

Imposing homogeneous Dirichlet boundary conditions is only feasible, switching toexcess to payoff. For the discretization of the diffusion part we use an equidistant

38

grid and the following difference quotients:

α(xi)∂v

∂x(xi, yj, tk) ≈ min(α(xi), 0)

vki+1,j − vki,j∆x

+ max(α(xi), 0)vki,j − vki−1,j

∆x, (20)

σ2

2

∂2v

∂x2(xi, yj, tk) ≈

σ2

2

vki+1,j − 2vki,j + vki−1,j

(∆x)2and

βyj∂v

∂y(xi, yj, tk) ≈ βyj

vki,j − vki,j−1

∆y. (21)

The integral part of the equation is discretized explicitly to obtain a sparse systemmatrix. Here we give the scheme for the discretization of the spike part of the PIDE,the jump part is discretised analogously.∫ ∞

0

(V (xi, yj + z, tk)− V (xi, yj, tk))fJ(z)dz ≈P∑n=0

vtk−1

i,j+nνn, (22)

η

∫ ∆y(n+ 12

)

∆y(n− 12

)

fJ(z) dz = νn ∀n > 0, (23)

where ν0 = η∫ 1

2

0fJ(z) dz. Alternatively, we could use a FEM discretization and a

compression scheme to obtain a sparse system matrix as proposed e.g. by Matacheet al. (2004).

B Appendix 2

B.1 The NIG distribution

The Normal Inverse Gaussian (NIG) distribution is a special case of the generalizeddistribution, first introduced by Barndorff-Nielsen (1977). Let K1 be the modifiedBessel function of second kind of order one and let α > |β| > 0, δ > 0 and µbe some real constant parameters. The density function of a one-dimensional NIGdistributed random variable X is then given by

fNIG(x) =α

πexp(δ√α2 − β2 + β(x− µ)

) K1

(αδ√

1 +(x−µδ

)2)

√1 +

(x−µδ

)2,

39

whereas the characteristic function34 is given by

E[eiξ·X ] = etψNIG(ξ), ξ ∈ R,with ψNIG(ξ) = iξµ+ δ

(√α2 − β2 −

√α2 − (β + iξ)2

).

The moments of an NIG-distributed random variable X are

E[X] = µ+ δβ

γ(24)

V[X] = δα2

γ3(25)

Skew[X] = 3β

α√δγ

(26)

Kurt[X] = 3

(1 +

α2 + 4β2

δα2γ

), (27)

where γ =√α2 − β2. To represent the NIG parameters in the so-called shape

triangle, we transform the parameters by the following relations

ξ =1√

1 + δγand χ =

ξβ

α.

The new coordinates (χ, ξ) are located within a triangle defined by 0 ≤ |χ| < ξ < 1.The parameter χ is closely related to the skewness of X, while ξ measures thedeparture from normality, i.e. (χ, ξ) ∼ (0, 0) means that the NIG distribution is“close” to being normal. In other words, it holds that the skewness of the NIGdistribution is zero for β = 0 and that the kurtosis converges to 3 for α → ∞.Therefore, the variance is δ/α, while the expectation reduces to µ. with the boundaryconditions B(T ) = ν and A(t) = 0.

B.2 Empirical vs. simulated hourly spot prices

The following two Figures 23 and 24 compare empirical hourly spot prices providedby the German power exchange EEX with simulated hourly spot prices generatedby the JDS-model under the physical measure P, which is calibrated in Section 4.

In addition, Table 2 gives the four first central moments of the empirical data andone simulated X-path of the JDS-model.

34In general, the characteristic function of a Rd-valued random variable X is the function φX :Rd → R defined by

∀z ∈ R, φX(z) = E[eiz·X ] =∫R

eiz·x dµX(x).

40

0 0.5 1 1.5 2 2.5 3

x 104

0

200

400

0 0.5 1 1.5 2 2.5 3

x 104

0

1000

2000

3000

0 0.5 1 1.5 2 2.5 3

x 104

0

1000

2000

3000

simulation of X

original prices

simulated prices

Figure 23: Empirical and simulated hourly electricity prices.

Table 2: The four first central moments of the data and the model.

moments of the first differences of Xt

descriptions EP VP SkewP KurtP

moments of X -0.0001 124.9666 0.7623 66.1785empirical moments -0.0001 130.8103 0.3295 51.7494

41

3.2 3.22 3.24 3.26 3.28 3.3 3.32 3.34 3.36 3.38 3.4 3.42

x 104

-50

0

50

100

150

3.2 3.22 3.24 3.26 3.28 3.3 3.32 3.34 3.36 3.38 3.4 3.42

x 104

0

100

200

300

3.2 3.22 3.24 3.26 3.28 3.3 3.32 3.34 3.36 3.38 3.4 3.42

x 104

0

100

200

300

simulation of X

original prices

simulated prices

Figure 24: Empirical and simulated hourly electricity prices for the last observations.

42

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Documents

Pricing Electricity Derivatives on an Hourly Basis