Electricity Real Options Valuation

arXiv:physics/0608167 v1 16 Aug 2006

Ewa Broszkiewi z-Suwaj

HugoSteinhausCenter,InstituteofMathemati s andComputerS ien e Wro ªawUniversityofTe hnology

Wyspia«skiego27,50-370Wro ªaw,Poland Ewa.Broszkiewi z-Suwajpwr.wro .pl

Inthis paperareal optionapproa hfor thevaluationof real assetsis presented. Two ontinuoustime models used for valuation aredes ribed: geometri Brownianmotionmodelandinterestratemodel. Thevaluation forele tri ityspreadoptionunderVasi ekinterestmodelispla edandthe formulasfor parameterestimatorsare al ulated. The theoreti alpart is onfrontedwithrealdatafromele tri itymarket.

PACSnumbers: 02.30.Cj,02.50.Ey,02.70.- 1. Introdu tion

Theliberalizationofele tri itymarket ausedthatmodelingonthis mar-ketbe ameveryimportant skill. Ithelps usto minimizelossand hedgeour position. It is a very interesting fa tthat spread option ould be used for valuationsomereal assetsaspowerplants ortransmissionlines. Butbefore thatwe need to knowthe spread optionpri e formula. Verypopularmodel used for option valuation is the geometri Brownian motion model but it is not very e ient. In the 2005 the idea of modeling domesti ele tri -ity market using interest rate model was introdu ed by Hinz, Grafenstein, Vers hue and Wilhelm [1℄. They valuated European all option written on powerforward ontra tunderHeathJarrowMortonmodeland,inthisway, reated very interesting lassof models.

Theaimofmyworkisthevaluationand alibrationofele tri ityspread option underinterestratemodel appliedfor ele tri itymarket. Istart with assumption of Vasi ek model and using martingale methodology [2℄ valu-ate spread option. Using the maximum likelihood fun tion methodology I estimate model parameters. I ompare onstru ted model with geometri Brownian motion modelbyapplying both models to real option valuation. I make simulations toshowthedieren e between to dis ussed models.

Mypaperisorganizedinthefollowingway. Atthebeginning(Se tion2) Ides ribe whatdoesitmeanthatwe bought aspreadoption,next(Se tion 3)Iintrodu e thereaderintorealoptions world. InSe tion4Ides ribe val-uationmethodologyforspreadoptionunderinterestratemodelandpresent also option pri e formula for geometri Brownian motion model. The al-ibration methods for both models are des ribed in Se tion 5. At the end, in Se tion 6, all theoreti al deliberation are onfronted withreal data and some simulationresults arepresented.

2. Ele tri ity Spread Options

Inthis se tiontwo interesting ross ommodityderivativesonele tri ity marketaredes ribed. Therstoneisthesparkspreadoption,whi hisbased on fa tthat somepowerplants onvertgasinto ele tri ity. Theunderlying instrumentisthedieren ebetweenthegasandele tri itypri es(thespark spread). The basi parameter onne tedwiththis kindof instrument isthe heat rate, the ratio whi hdes ribesthe amount of gasrequired to generate 1MWhofele tri ity. Thedenitionofsu haninstrumenthasaform[3,4℄: AnEuropean sparkspread alloptionwritten onfuel

F

,atxedswap ratio

K

,givesits holderthe right,but notthe obligation topayKtimestheunit pri e of fuel

F

at theoptions maturity

T

and re eive thepri e of one unit of ele tri ity.

It is easy to imagine su h kind of option whi h better ts the Polish ele tri itymarket. Weshouldassumethatthe underlying instrument isthe dieren e between the arbon andele tri ity pri es. Butin this timethere is no possibility of valuation of su h an option be ause we don't have the representative arbon pri e. Generally, if we assume that

P

E

and

P

F

are respe tively future pri e of 1MWhof ele tri ityand the future pri eof the unitoffuelandKistheswapratio thanwe oulddes ribethepayoofthe European ele tri ity-fuelspread all optionas

C

F

(P

E

, P

F

, T ) = max[P

E

(T ) − KP

F

(T ), 0]

and the payoofthe European ele tri ity-fuelspread put option hasform

P

F

(P

E

, P

F

, T ) = max[KP

F

(T ) − P

E

(T ), 0]

The se ond derivative is the lo ational spread option. It is based on fa tthattransmissionofpowerfromonelo ationto anotherisverypopular transa tion. Itis normal,for transmission system,thatthepowerismoved fromthe pla eoflowerpri eto thepla eofhigher pri eandthis iswhythe transa tion is protable. The whole transa tion depends on the dieren e

oulduseoptions. Thiskindofinstrument ouldbedenedinfollowingway [3 ℄: An European all option on thelo ational spreadbetween the lo ation oneandlo ationtwo,withmaturity

T

,givesitsholdertherightbutnotthe obligationtopaythepri eofoneunitofele tri ityatlo ationoneattimeT and re eive thepri e of

K

units of ele tri ity at lo ation two. Assumethat

P

1

and

P

2

aretheele tri itypri esat therstlo ation andse ond lo ation respe tively. The payo of the European lo ational spread all option is given by

C

L

(P

1

, P

2

, T ) = max[P

1

(T ) − KP

2

(T ), 0]

Theputoptionisdened similarand thepayooftheEuropean ele tri ity-fuelspread put optionhasform

P

L

(P

1

, P

2

, T ) = max[KP

2

(T ) − P

1

(T ), 0]

3. Real Options

Supposethatfordes ribingthe ommodityweusethreequalities

(G, t, L)

: G - the nature of good, t - time when it is available, L - lo ation where it is available. We ould dene [5℄ a real option as te hnology to physi ally onvert oneor more input ommodities

(G, t, L)

into an output ommodity

(G

′

, t

′

, L

′

)

. For example most of powerplants are real option be ause they giveustherightto onvertfuelintoele tri ity. Thetransmissionlineisalso realoption. Itgivesustherightto hangetheele tri ityinonelo ationinto ele tri ityinse ondlo ation. TheworksofDeng,JohnsonandSogomonian [6 ,3℄ ontaintwoformulasdeninghowtovaluategenerationand transmis-sion assets. If we dene that

u

F

(t)

is a one unit of thetime-t right to use generation asset we ould say that it is the value of just maturing, time-t all option on thespread between ele tri ity and fuel pri es

C

F

(t)

and the one unitvalueof apa ityof powerplant using some fuel

F

isgiven by

V

F

=

Z

T

0

u

F

(t)dt =

Z

T

0

C

F

(t)dt

where

T

is thelength ofpower plant life. Similar ifwe dene that

u

AB

L

(t)

is aone unitthetime-trightto onvert one unitofele tri ityinlo ationA intoone unitofele tri ityinlo ation B we ould saythat it is the value of just maturing,time-t all option on the spread between ele tri itypri esinlo ationA andB

C

AB

L

(t)

. The oneunit valueof su h transmissionassetisgiven by

V

L

=

Z

T

0

u

AB

_L

(t)dt +

Z

T

0

u

BA

_L

(t)dt =

Z

T

0

C

_L

AB

(t)dt +

Z

T

0

C

_L

BA

(t)dt

4.Valuation methods

In this se tion I present the widely known geometri Brownian motion model andI valuate the all spread optionfor the new, interestrate model using martingalemethodology. All al ulationsaredes ribedbelow.

4.1. Geometri BrownianMotion Model

Supposethat the futurepri es of ommodity aredes ribed by following sto hasti dierential equations

dP

1

(t, T ) = µ

1

P

1

(t, T )dt + σ

1

P

1

(t, T )dW

t,1

,

dP

2

(t, T ) = µ

2

P

2

(t, T )dt + σ

2

P

2

(t, T )dW

t,2

,

where

W

t,1

= ρW

t,2

+

p1 − ρ

2

_W

′

t,2

and

W

t,2

,

W

′

t,2

are i.i.d. Brownian mo-tions. Itisknownfa t[5 ℄,thatthepri eofthespread all optionwithswap ratio

K

andtime to maturity

T

,written on futures ontra t withmaturity

U < T

is given by

C

1

(t) = e

−

r(T −t)

[P

1

(t, U )Φ(d

+

(t)) − KP

2

(t, U )Φ(d

−

(t))],

where

d

±

(t) =

ln

P

1

(t,U )

KP

2

(t,U )

±

σ

2

(T −t)

2

σ

√

_{T − t}

.

and

σ

2

= σ

2

₁

_{− 2σ}

1

ρσ

2

+ σ

2

2

.

4.2. Interest Rate Model

For domesti urren y, for example MWh, we denote two pro esses:

p

1

(t, T )

,

p

2

(t, T )

whi harethefuturepri es ofone unitof ommodity. The interest ratefun tionsfor su h pro essesarerespe tively

dr

t,1

= (a

1

− b

1

r

t,1

)dt + σ

1

dW

t,1

and

dr

t,2

= (a

2

− b

2

r

t,2

)dt + σ

2

dW

t,2

,

where

W

t,1

= ρW

t,2

+

p1 − ρ

2

_W

′

t,2

and

W

t,2

,

W

′

t,2

are i.i.d. Brownian mo-tions. We assume that there exist the savings se urity

N

t

(for example in USD), with onstant interestrate

r

,for whi h

P (t, T ) =

p(t, T )

e

−

rt

_N

is the USD pri e of future delivery of 1 unit of ommodity. We know [1℄ that thereexist a martingalemeasure

P

for whi h thedis ountedpro esses

p

1

(t,T )

N

t

,

p

2

(t,T )

N

t

aremartingales. We have

C

2

(0) = N

0

E

P

((p

1

(T, U ) − Kp

2

(T, U ))

+

N

T

−

1

|F

0

)/e

−

rt

_N

t

Wedene the new dis ountingpro essesfor i=1,2 as

dB

t,i

= B

t,i

r

t,i

dt,

where

B

0,i

= 1.

Ifwe hange the measurefrom

P

to

P

1

dP

dP

1

=

N

T

B

0,1

N

0

B

T,1

weknowthatpro esses

p

˜

1

(t, T ) =

p

1

(t,T )

B

t,1

,

p

˜

2

(t, T ) =

p

2

(t,T )

B

t,1

are

P

1

-martingales. From interestrate theorywe obtain

d ˜

p

1

(t, T ) = ˜

p

1

(t, T )n

1

(t, T )dW

t,1

,

where

n

1

(t, T ) = −

σ

1

b

1

(1 − e

−

_b

1

(T −t)

).

(1)

If we hange themeasure againfrom

P

1

to

P

2

dP

1

dP

2

=

B

T,1

B

0,2

B

0,1

B

T,2

we know that

p

ˆ

2

(t, T ) =

p

2

(t,T )

B

t,2

and

p

ˆ

1

(t, T ) =

p

1

(t,T )

B

t,2

=

˜

p

1

(t,T )B

t,1

B

t,2

are

P

2

-martingales and similarto theearlier situation,we have

d ˆ

p

2

(t, T ) = ˆ

p

2

(t, T )n

2

(t, T )dW

t,2

,

(2) where

n

2

(t, T ) = −

σ

2

b

2

(1 − e

−

_b

2

(T −t)

).

(3)

After simple al ulations we also have

d ˆ

p

1

(t, T ) = ˆ

p

1

(t, T )n

1

(t, T )d ˜

W

t,1

,

(4) where

W

˜

t,1

= ρW

t,2

+

p1 − ρ

2

_W

˜

′

t,2

, and

W

˜

′

t,2

= W

′

t,2

+

r

t,1

−

r

t,2

n

1

(t,T )

√

1−ρ

2

t

. We also assumethat

where

V

t

= ρ

1

W

t,2

+

p1 − ρ

2

1

W

′′

t,2

,

and

W

t,2

,

W

′

t,2

,

W

′′

t,2

are independent Wiener pro esses. For dis ounted pro essesfollowing equationistrue

P

i

(t, T ) =

p

i

(t, T )

e

−

_rt

_N

t

=

p

ˆ

i

(t, T )

e

−

_rt

ˆ

N

t

.

(6)

Having thene essarysto hasti dierentialequations we ouldpri ethe option. We hange themeasurefrom

P

2

to

Q

infollowing way

dP

2

dQ

=

B

T,2

p

2

(0, U )

B

0,2

p

2

(T, U )

.

Pro ess

X(t, T ) =

p

1

(t,T )

p

2

(t,T )

=

ˆ

p

1

(t,T )

ˆ

p

2

(t,T )

is

Q

-martingale. From Ito lemma we know that

dX(t, T ) = X(t, T )(n

1

(t, T )d ˆ

W

t,1

− n

2

(t, T )dW

t,2

),

where

ˆ

W

t,1

= ρW

t,2

+

p1 − ρ

2

W

ˆ

t,2

and

ˆ

W

t,2

= ˜

W

t,2

′

+

n

2

2

(t,T )−n

2

(t,T )ρn

1

(t,T )

n

1

(t,T )

√

1−ρ

2

t

. Now, for al ulation of the option pri e, we ould use the Bla k-S holes formula

C

2

(0) =

p

2

(0, U )e

r0

N

0

E

Q

((

p

1

(T, U )

p

2

(T, U )

−K)

+

|F

0

) = P

2

(0, U )E

Q

((

P

2

(T, U )

P

1

(T, U )

−K)

+

|F

0

) =

P

2

(0, U )E

Q

((X(T, U ) − K)

+

|F

0

) = P

2

(0, U )(X(0, U )Φ(d

+

) − KΦ(d

−

)) =

P

1

(0, U )Φ(d

+

) − KP

2

(0, U )Φ(d

−

),

where

d

±

=

ln

X(0,U )

_K

_±

σ

2

(0,U )

₂

σ(0, U )

,

σ

2

(t, T ) =

Z

T

t

(n

2

₁

_{(u, T ) − 2n}

2

(u, T )ρn

1

(u, T ) + n

2

2

(u, T ))du,

and

Φ

isthe normal umulative distribution fun tion. For every timepoint

0 ≤ t ≤ T

the option pri e with swap ratio

K

and time to maturity

T

, written on futures ontra t withmaturity

U < T

is given by

C

2

(t) = P

1

(t, U )Φ(d

+

(t)) − KP

2

(t, U )Φ(d

−

(t)),

where

d

±

(t) =

ln

P

1

(t,U )

KP

2

(t,U )

±

σ

2

(t,U )

2

σ(t, U )

.

This methodology ould be used dire tlyfor lo ational spread options and also for fuel-ele tri ity spread options ifwe assumethat theswap ratio

be-5. Histori al Calibration

Inthisse tionwedes ribehowtotourmodelsforreal,histori aldata. At the beginning we assume that we are given histori al pri es of future ontra ts

P

1

(t

k

, T

j

)

and

P

2

(t

k

, T

j

)

,

k = 0, . . . , n

,

j = 0, . . . , m

, in dis rete timepoints

t

0

< t

1

< . . . < t

n

and

T

0

< T

1

< . . . < T

m

,where

t

k+1

− t

k

= dt

and

T

j+1

− T

j

= ∆T

.

For geometri Brownian motion model the alibration methodology is not very ompli ated. Weanalyzethereturnsoffuturepri esof instrument and

µ

isits mean,

σ

2

isits varian eand orrelationparameter issimplythe orrelationbetween returnsof two instruments. Butforinterest ratemodel the alibration is quite ompli ated, espe ially for multidimensional HJM model [7℄. Calibrationfor dis ussed Vasi ek modelispresentedbelow. Letus onsider following pro ess

η

i

(t, T

j

) =

ˆ

p

i

(t, T

j

)

ˆ

p

i

(t, T

j+1

)

=

P

i

(t, T

j

)

P

i

(t, T

j+1

)

.

From It lemmawe know that

dη

i

(t, T

j

) = η

i

(t, T

j

)[(n

2

i

(t, T

j+1

) − n

i

(t, T

j+1

)n

i

(t, T

j

))dt+

+(n

i

(t, T

j

) − n

i

(t, T

j+1

))dW

t,i

]

We ouldwritethat

n

i

(t, T ) = n

i

(T − t)

be ausethisfun tiondependsonly fromthe dieren ebetween thematuritytime

T

andthetimepoint

t

. Ifwe then onsider thepro ess

s

i

(T − t) =

dη

i

(t, T )

η

i

(t, T )

,

we know that

s

i

(T − t)

isnormallydistributedwithmean

α

i

(T − t) = (n

2

i

(T + ∆T − t) − n

i

(T + ∆T − t)n

i

(T − t))dt

and varian e

β

i

2

(T − t) = (n

i

(T − t) − n

i

(T + ∆T − t))

2

dt.

Knowing theform offun tions

n

i

(T − t)

(1 ),(3) we seethat

β

_i

2

_{(T − t) =}

 σ

i

b

i

e

−

b

i

(T −t)

_{[1 − e}

−

b

i

∆T

_]



2

and

β

2

_i

_{(T − t)}

β

2

i

(T − t + dt)

= e

2b

i

dt

(8) After dis retisation and for assumption that

dt = ∆T = 1

,

T − t = p∆T

and

j = 1, . . . , m

we ouldsaythattheestimator of

β

hastheform

ˆ

β

i

2

(p∆T ) =

1

m

m

X

j=1

(s

2

i,j

(p∆T ) − s

i,j

(p∆T ) ¯

s

i

),

where

¯

s

i

=

1

m

m

X

j=1

s

i,j

(p∆T )

and we put

s

i,j

(p∆T ) =

P

i

(T

j

−

p∆T,T

j

)

P

i

(T

j

−

p∆T,T

j

+∆T )

−

P

i

(T

j

−

(p+1)∆T,T

j

)

P

i

(T

j

−

(p+1)∆T,T

j

+∆T )

P

i

(T

j

−

p∆T,T

j

)

P

i

(T

j

−

p∆T,T

j

+∆T )

.

So using(7 ),(8) we have

ˆb

i

=

1

2∆T

ln

ˆ

β

i

2

(p∆T )

ˆ

β

i

2

((p + 1)∆T )

,

ˆ

σ

i

=

ˆ

β

i

2

(p∆T )ˆb

i

e

−ˆ

_b

_i

_{(p∆T )}

[1 − e

−ˆ

_b

_i

_∆T

]

.

Itiseasytonoti ethatthe orrelationparameterbetweenpro esses

s

1

(T −t)

and

s

2

(T − t)

is

ρ

,sowe have

ˆ

ρ =

P

m

j=1

(s

1,j

(p∆T ) − ¯

s

1

)(s

2,j

(p∆T ) − ¯

s

2

)

q

P

m

j=1

(s

1,j

(p∆T ) − ¯

s

1

)

2

q

P

m

j=1

(s

2,j

(p∆T ) − ¯

s

2

)

2

.

At the end we should al ulate also parameters onne ted withpro ess

N

t

. From equation(6) weknowthatfor i=1,2

ξ

i

(t, T ) =

ˆ

p

i

(t, T )

ˆ

N

t

= e

−

rt

P

i

(t, T ).

dξ

2

(t, T ) = ξ

2

(t, T )[v

2

− vn

2

(t, T )ρ

1

]dt + ξ

2

(t, T )[(n

2

(t, T ) − vρ

1

)dW

t,2

−

−vp1 − ρ

2

1

dW

′′

t,2

].

Weknow thatthepro ess

y(T − t) =

dξ

_ξ

2

(t, T )

2

(t, T )

isnormallydistributedwithmean

(v

2

_−vn

2

(t, T )ρ

1

)dt

andvarian e

(n

2

2

(t, T )−

2vn

2

(t, T )ρ

1

+ v

2

)dt

sowe havethat for

dt = 1

ˆ

v

2

= ˆ

n

2

₂

(p∆T ) + 2¯

_{y −}

P

m

j=1

(y

j

(p∆T ) − ¯y)

2

m −

P

m

j=1

(y

j

(p∆T ) − ¯y)

and

ˆ

ρ

1

=

ˆ

v

2

_{− ¯y}

ˆ

vˆ

n

2

(p∆T )

where

y

j

(p∆T ) =

e

−

_r∆T

P

i

(T

j

− p∆T, T

j

) − P

i

(T

j

− (p + 1)∆T, T

j

)

P

i

(T

j

− (p + 1)∆T, T

j

)∆T

and

¯

y =

1

m

m

X

j=1

y

j

(p∆T ).

6. Simulation and Con lusion

ForsimulationIuseddatafromNewYorkMer antileEx hange(NYMEX). I onsidered histori al quotation of future natural gas (Henry Hub) and ele tri ity(PJM) ontra tssin e January, 2004 until Mar h,2006. The pa-rameters were al ulated using alibration methods des ribed before. All estimatedparametersarepresentedinTable6 . Iassumedthatthe onstant interest rate is

r = 0.05

. For valuation of gasred power plant I assumed that the life-time of the power plant is

T = 15

years and

P

E,0

= 55.750

USD,

P

F,0

= 6.3080

USD.

InFigure1. weseethevalueofpowerplantfortheheatraterangingfrom 5to15forbothpresentedmodels. We ouldnoti ethatthereisdieren ein hangesdynami foranalyzedmodels. Thevalueofpowerplant forinterest rate modelis mu h more smallerthan for GBM modeland it tendsto zero

TABLEI Estimated parameters for GBM model and for interest rate model using histori al datafromNew YorkMer antileEx hange.

Geometri Brownian Motion InterestRateModel

σ

e

1.0945

σ

e

0.0678

σ

g

1.2943

σ

g

0.0042

µ

e

4.4098

b

e

3.7515

µ

g

4.8145

b

g

1.8205

ρ

0.8688

ρ

0.1892

ρ

1

0.7266

v

0.0668

0

5

10

15

20

25

30

35

40

45

0

20

40

60

80

100

120

Time to maturity (weeks)

Future Price

Gas MMBtu

Electricity MWh

5

6

7

8

9

10

11

12

13

14

15

0

50

100

150

200

250

300

350

Heat Rate

Power plant price

GBM

Interest Rate

Fig.1. Toppanel: Futurepri esofnaturalgasandele tri ityfor ontra tmaturing inMar h,2006. Bottompanel: Simulatedunitvalueofgasredpowerplant,with life length15years,forbothmodels.

value of powerplant for heatrate greater than

P

E,0

P

F,0

≈ 9

underGBMmodelisusuallytoohigh,soalsointhisaspe ttheinterestrate model givesbetter results.

REFERENCES

[1℄ J. Hinz, L.Grafenstein, M. Vers huere,M. Wilhelm, Quantitative Finan e 5,49,(2005).

[2℄ M. Musiela, M. Rutkowski Martingale Methods in Finan ial Modelling, Springer(1997).

[3℄ S. Deng, B.Johnson,A. Sogomonian,Pro eedings of the Chi ago Risk Man-agement Conferen e (1998).

[4℄ A. Weron,R. Weron, Power Ex hange: Tools for Risk Management, CIRE, Wro ªaw,(2000).

[5℄ E. Ronn, Real Optionand Energy Management: Using OptionsMethodology toEnhan eCapitalBudgetingDe isions,RiskBooks(2004).

[6℄ S. Deng: POWERpapers (2000).