P.BJERKSUND, H.RASMUSSEN,ANDG.STENSLAND
Abstract. Thepurposeofthispaperistwo-fold: Firstly,weanalyzeoption
value approximation of traded options inthe presence of a volatility term
structure. Theoptionsareidentiedas: \European"(writtenontheforward
priceofafutureowdelivery);and(ii)Asian. Bothtypesareinfactwritten
on(arithmetic)priceaverages. Secondly,adoptinga3-factormodelformarket
risk whichiscompatiblewiththe valuationresults,wediscuss riskmanage-
mentintheelectricitymarketwithintheValueatRiskconcept.Theanalysis
is illustrated by numericalcases from the Norwegian electricity derivatives
market.
1. Introduction
Historical time series, implicit volatilities of quoted option prices, as well as
theexperienceofprofessionaltradersand brokers,clearlyindicatethepresenceof
a volatility term structure in the Norwegian electricity derivatives market. The
purpose of this paper is to analyse the implications of this volatility term struc-
ture for: (i)valuation of themostfrequently traded options;and (ii)market risk
management.
Our startingpointis to representthe electricityforwardmarket atdate t bya
forwardprice function f(t;T), which maybe interpreted as the forwardprice at
date t ofa hypotheticalcontractwith deliveryat date T (i.e., with aninnitese-
mal deliveryperiod). Intheelectricityforwardmarket,theunderlyingquantityis
deliveredasaowduring aspecic futuretime period. Thiscontractmaybein-
terpretedasaportfolioofhypotheticalsingle-deliverycontracts,hencetheforward
pricefollowsfromthefunction f(t;T)byno-arbitrage.
Assuming lognormality, we representtheuncertaintyin theforwardmarket at
date t bya volatility function ( t;T t), which correspondsto the Black'76
implicit volatilityofaEuropeanoptionwithtimeto exercise t written onthe
future forwardpricef(t;T)withtimetodeliveryT t.
However, the traded \European" electricity option is written on the forward
price of a contract with deliveryas aconstantow during a specic future time
period. Following Kemna and Vorst (1990), we adopt the Black'76 concept for
approximatingtheoptionvalue,andobtainthetheoreticalforwardpriceaswellas
anapproximatedplug-involatility.
The traded Asian optionis written on theaverage spot priceobserved during
aspecic period. Theexercisedateof theoptiontypicallycoincides withthelast
observationdate. WeobtainthetheoreticalforwardpriceandtheBlack'76plug-in
volatility.
Date: February2.2000.
BjerksundandStenslandarebothprofessorsattheNorwegianSchoolofEconomicsandBusi-
nessAdministration(NHH) inBergen. RasmussenispartnerofVizRiskManagementServices
inBergen,andholdsaDr.Oecon. degreefromNHH.
We thankP.E.Manneand participants ofthe XVIIFIBEannual conference inBergenfor
usefulcomments.
Next, weturnto riskmanagementwithin theValueat Riskconcept. Theidea
of Value at Risk is to quantify the downside risk of the future market valueof a
givenportfolioatachosenhorizondate. Werepresentthemarketriskbya3-factor
model which is compatible with ourforwardpricedynamics assumption. We use
Monte Carlo simulation in order to generate the probability distribution of the
future portfoliomarketprice.
Theadvantageofintegratingvaluation andriskmanagementis: (i)themarket
riskexposureofafuturepositionisconsistentwiththecurrentforwardandoption
prices;and(ii)wemayuseouroptionvaluationapproximationresultstocalculate
conditional futureoptionvalues.
2. The model
2.1. The forward market. Research onvaluationofcommodityderivativesand
managementofcommoditymarketriskhasbeenanexpandingareawithinnance
during the last decade. At the sametime, the use of various bilateral OTC ar-
rangements in the industry has increased, and new commodity derivatives have
beenintroducedin thenancialmarketplace.
Formanycommodities,theforwardpricesindicate anon-constantconvenience
yield(e.g.,seasonalpattern). Moreover,thecommodityoptionmarketpricesclearly
indicate that the constant volatility assumption of Black'76 is violated for most
commodities. Typicallytheimplicitvolatilityadecreasingandconvexfunctionof
time tomaturity.
GibsonandSchwartz(1990)developatwo-factormodelforoilderivatives,where
the commodity spot price is geometric Brownian, and the instantaneous conve-
nienceyieldratefollowsamean-revertingOrnstein-Uhlenbeckprocess. Withinthis
model, closed form solutionsexist forthe forwardprice aswell asEuropean calls
(see Bjerksund (1991)and Jamshidianand Fein (1990)). Hilliardand Reis(1998)
investigateseveralalternativemodels,includingthecasewherethespotpriceisa
mixedjump-diusionprocess. Forasurveyonalternativemodelsforvaluationand
hedning,seeSchwartz(1997).
Models where assumptions on spot price and convenience yield dynamics are
starting points will typically predict forward prices which are dierent from the
ones observedinthemarket. UsingthegeneralHeath,Jarrow,andMorton(1992)
approach, Milterson and Schwartz (1998) develop a general framework for com-
modityderivativesvaluationandriskmanagementwithstochasticinterestratesas
wellas stochastic convenience yield. Thismodelcan be calibratedto the current
forwardmarket.IntheirGaussianspecialcasethecalloptionvalueessentiallyboils
downtoageneralisedversionofBlack'76.
Our model assumptions may be considered as a special case of the gaussian
Miltersen-Schwartz model. Complicating the picture in the case of electricity
derivatives,however,is thefact that thephysical "underlyingasset"isaconstant
ow received during aspecic time period, rather than one "bulk"delivery at a
specic date.
Turningtoourmodel,werepresenttheforwardmarketatdatetbyacontinuous
forward price function, where f(t;T) denotes the forward price at date t on a
contractwith deliveryat date T t. Considera forward contractwith delivery
date T, and assume the following forward price dynamics at date t T (with
respecttotherisk-adjustedmartingaleprobabilitymeasure)
df(t;T)
=
a
+c
dW
(t); (1)
wherea,b,andcarepositiveconstants,anddW (t)istheincrementofastandard
Brownian motion with expectation E
t [dW
(t)] =0 and Var
t [dW
(t)] = dt. By
construction, the expectation of Eq. (1) is zero with respect to the martingale
measure.
The abovecorrespondsto the forwardpriceof this contractat thefuture date
2[t;T]beinglognormal,andgivenbythefollowingstochasticintegral
f(;T) = f(t;T)
exp (
Z
t
a
T s+b +c
dW
(s) 1
2 Z
t
a
T s+b +c
2
ds )
:
Observethat E
t [1
f(;T)] =f(t;T), which conrmsthat the forwardprice is a
martingalewithrespect tothe-probabilitymeasure.
Now,consider ahypotheticalEuropeancalloptionwith timeto exercise t,
writtenonthefutureforwardpricef(;T)onacontractwithtimetodeliveryT t.
It follows from the literature(see, e.g., Harrison and Kreps(1979) and Harrison
and Pliska(1981))that themarket valueofthe optioncanberepresentedby the
expected(usingthemartingalemeasure)discounted(usingtherisklessrate)future
pay-o. With thefuture forwardprice beinglognormal, thecallvalueisgivenby
theBlack'76formula
V
t h
1
(f(;T) K) +
i
= E
t h
e r( t)
(f(;T) K) +
i
= e r( t)
f(t;T)N(d
1 ) e
r( t)
KN(d
2
); (2)
where N()isthestandardnormalcumulativeprobabilityfunction,
d
1
ln(f(t;T)=K)+ 1
2
2
( t)
p
t
; (3)
d
2
d
1
p
t; (4)
s
Var
t
ln
f(;T)
f(t;T)
=( t) : (5)
Observethat thekey input of Black'76 is: (i)the forwardprice at date t of the
underlyingassetf(t;T);and(ii)theuncertaintyoftheunderlyingasset,represented
bythevolatility.
Theassumed dynamicstranslatesinto thevolatility beingafunction oftime
to exercise(oftheoption), t,andtimetodelivery(oftheunderlying forward),
T t,andgivenby
= ( t;T t)
= s
Var
t
ln
f(;T)
=( t) ; (6)
where 1
Var
t
ln
f(;T)
f(t;T)
= Var
t Z
s=
s=t
df(s;T)
f(s;T)
(7)
=
a 2
T s+b
2acln(T s+b)+c 2
s
s=
s=t :
In the following, werepresent theforwardmarket at date t bythe forward price
function f(t;T)andthevolatilityfunction ( t;T t).
3. European option
3.1. Forward on a ow delivery. Intheelectricityforwardmarket,theunder-
lying physical commodity is deliveredduring a specic time period [T
1
;T
2 ] as a
constantow(atarateof(T
2 T
1 )
1
unitsperyear). Weobservedeliveryperiods
on contractsranging from onedayto one year, depending on theremainingtime
to deliveryofthecontract.
Werepresenttheforwardmarketatdatetbytheforwardpricefunction f(t;s),
tsT. Byvalueadditivity,themarketvalueatdate tofreceivingoneunitof
thecommodityfromdatesT
1 toT
2
(atarateof1=(T
2 T
1
))issimply
V
t
"
Z
T2
T
1 1
s f(s;s)
T
2 T
1 ds
#
= Z
T2
T
1 e
r(s t) f(t;s)
T
2 T
1
ds; (8)
where t T
1
<T
2
. In arational market, theforwardpriceF(t;T
1
;T
2
)is deter-
mined such that themarket valueat datet of thepaymentsequalstherighthand
sideoftheequationjustabove. Indeed,inthehypotheticalcaseofup-frontpayment
at datet,theforwardpricewouldcoincide withtherighthandsidejustabove.
Now,supposethattheforwardpriceispaidasaconstantcashowstreamduring
thedeliveryperiod(atarateofF(t;T
1
;T
2 )=(T
2 T
1
)pertimeunit). Atdatet,the
net market valueofentering thecontractiszero, leadingto thefollowingforward
price
F(t;T
1
;T
2 )=
Z
T
2
T1
w(s;r)f(t;s)ds; (9)
where
w(s;r)= e
rs
R
T
2
T1 e
rs
ds
: (10)
Consequently, theforwardpriceF(t;T
1
;T
2
) may beinterpreted astheaverageof
theforwardpricesf(t;s)overthedeliveryperiod[T
1
;T
2
],withrespecttotheweight
function 2
which reectsthetimevalueofmoney.
3.2. Call option valuation. The European calls which are traded in the elec-
tricityderivatives market are typicallywritten on a forwardprice. Inparticular,
consider aEuropeancalloption written onthe pay-o F(;T
1
;T
2
)with strike K
andexercisedate T
1
. Observethattheexercisedateoftheoptionprecedesthe
deliveryperiodoftheunderlyingforwardcontract.
1
Toestablishtherstequality,applyIto'slemma
Var
t
ln
f(;T)
f(t;T)
=Var
t
"
Z
s=
s=t
df(s;T)
f(s;T)
Z
s=
s=t 1
2
df(s;T)
f(s;T)
2
#
;
inserttheassumedforwardpricedynamics,andobservethatthesecondintegralisdeterministic
asofdatet. ThesecondequalityfollowsfromthefactthatBrownianmotionshaveindependent
incrementsacrosstime.
2
Observethatw(s;r)>08s2[T
1
;T
2 ]and
R
T
2
T
w(s;r)ds=1.
FollowingKemnaandVorst(op.cit.),weapproximatetheoptionvaluewithinthe
Black'76framework. Wehavealreadyobtainedthetheoreticalforwardpriceofthe
underlying uncertainpay-o, F(t;T
1
;T
2
). In addition, weneed an approximated
volatilityparameter. ApproximatetheforwardpricedynamicsfortT
1 by
3
dF(t;T
1
;T
2 )
F(t;T
1
;T
2 )
Z
s=T
2
s=T1 1
T
2 T
1 df(t;s)
f(t;s) ds
=
a
T
2 T
1 ln
T
2 t+b
T
1 t+b
+c
dW
(t): (11)
Next,obtaintheapproximatedvariance
Var
t
ln
F(;T
1
;T
2 )
F(t;T
1
;T
2
= Var
t Z
t
dF(s;T
1
;T
2 )
F(s;T
1
;T
2 )
ds
=
a
T
2 T
1
2 Z
t
ln T
2 s+b
T
1 s+b
2
ds (12)
+
2ac
T
2 T
1 Z
t ln
T
2 s+b
T
1 s+b
ds+c 2
Z
t ds;
where therstandthesecondintegralsare
Z
t
ln T
2 s+b
T
1 s+b
2
ds = h
(x+)( ln(x+)) 2
2(x+)ln (x+)ln (x )
+4aln(2)ln
x
2
4dilog
x+
2
(13)
+(x )(ln (x )) 2
4 i
X()
X(t)
;
Z
t ln
T
2 s+b
T
1 s+b
ds = [(x+)ln (x+)
(x )ln(x ) 2]
X()
X(t)
; (14)
where wedene
1
2 (T
2 T
1
); (15)
X(s) b+ 1
2 (T
2 +T
1
) s; (16)
andwhere thedilogarithmfunction isdenedby 4
dilog(x)= Z
x
1 ln (s)
1 s
dswherex0 (17)
see, e.g.,AbramowitzandStegun(1972).
Now,consideraEuropeancalloptionwithexercisedate writtenontheforward
price F(;T
1
;T
2
), where t < T
1
< T
2
The option value at date t can now
3
Theapproximationproceedsinthefollowingtwosteps
dF(t;T1;T2)
F(t;T
1
;T
2 )
Z
s=T
2
s=T
1 w(s;r)
df(t;s)
f(t;s) ds
Z
s=T
2
s=T
1 w(s;0)
df(t;s)
f(t;s) ds:
4
Thefunctionisapproximatednumericallyby
dilog(x)= 8
<
: P
n
k =1 (x 1)
k
k 2
for0x1
1
2 ( ln(x))
2 P
n
k =1
((1=x) 1) k
k 2
forx>1
be approximatedby Black'76, using the forward priceF(t;T
1
;T
2
) above and the
volatilityparameterv
E
v
E
v
E
( t;T
1 t;T
2 t)
= s
Var
t
ln
F(;T
1
;T
2 )
F(t;T
1
;T
2 )
=( t) (18)
The volatility parameterv
E
associatedwith theEuropeanoptionisafunction of
the time to maturity of the option( t), thetime to start of delivery (T
1 t),
andthetimeto stopofdelivery(T
2 t).
4. Asian option
Asianoptionsarewrittenontheaveragespotpriceobservedduringaspecicpe-
riod[T
1
;T
2
],withexercisedate T
2
. Withcontinuoussampling,the(arithmetic)
averageofthespot pricesf(s;s)observedfromT
1 toT
2
isdened by
A(T
1
;T
2 )
Z
T
2
T1 1
T
2 T
1
f(s;s)ds: (19)
WeareinterestedinevaluatingacalloptionwithstrikeKandexercisedateT
2 ,
writtenonthearithmeticaverageA(T
1
;T
2
). Forsimplicity,wedealwiththecaseof
tT
1
rst. Withthefuturespotpricesbeinglognormal,thereisnoknownproba-
bilitydistributionforthearithmeticaverage. WithintheBlack'76framework,the
optionvalueapproximationproblemboilsdown tondingthetheoretical forward
priceandareasonablevolatilityparameter.
Now,itfollowsfromthemartingalepropertyofforwardpricesthattheforward
price onacontractwritten on (the cash equivalentof) A(T
1
;T
2
) withdeliveryat
date T
2 is
F
t [A(T
1
;T
2
)] = E
t
"
Z
T
2
T1 1
T
2 T
1
f(s;s)ds
#
= Z
T2
T
1 1
T
2 T
1
f(t;s)ds: (20)
ObservethattheforwardpriceF
t [A(T
1
;T
2
)]simplyisthe(equallyweighted)arith-
meticaverageofthecurrentforwardpricesoverthesamplingperiod[T
1
;T
2 ]. This
forwardpricemaybeinterpretedasthecostreplicatingthiscontractinthemarket.
5
Turning tothe Black'76volatilityparameter,approximate thedynamicsof the
underlying forwardpriceatdate 2[t;T
2 ]by
dF
[A(T
1
;T
2 )]
F
[A(T
1
;T
2 )]
Z
s=T2
s=maxft;T1g 1
T
2 T
1
df(;s)
f(;s) ds
= 8
<
: n
a
T2 T1 ln
T
2 +b
T1 +b
+c o
dW
() when T
1
n
a
T2 T1 ln
T
2 +b
b
+ T
2
T2 T1 c
o
dW
() when >T
1
(21)
5
Assumeforthe momenta discretetimemodelwherethe deliveryperiod[T1;T2]isdivided
intontimeintervalsoftimelenghtt. Considerthefollowingstrategy: Attheevaluationdate
t, buye r(T
2 (T
1 +it))
(1=n) unitsforward foreachdelivery T1+it,i=1;:::;n. Astime
passesandthecontractsaresettled,invest(ornance)theproceedsatthe risklessinterestrate
r. AtthedeliverydateT
2
,thepay-ofromthestrategyis P
n
i=1 (1=n)f(T
1
+it;T
1 +i
t) P
n
1=1
(1=n)f(t;T1+it),wherethersttermrepresentsthedesiredspotprice,andthe
Obtaintheapproximatedvarianceby
Var
t
ln
A(T
1
;T
2 )
F
t [A(T
1
;T
2 )]
= Var
t
"
Z
=T2
=t dF
[A(T
1
;T
2 )]
F
[A(T
1
;T
2 )]
ds
#
=
a
T
2 T
1
2 Z
T1
t
ln T
2
+b
T
1
+b
2
d
+
2ac
T
2 T
1 Z
T1
t ln
T
2
+b
T
1
+b d+c
2 Z
T1
t
d (22)
+
a
T
2 T
1
2 Z
T
2
T1
ln T
2
+b
b
2
d
+
2ac
T
2 T
1 Z
T2
T
1 ln
T
2
+b
b T
2
T
2 T
1 d +c
2 Z
T2
T
1
T
2
T
2 T
1
2
d;
where therstand thesecond integralsareevaluated byinserting =T
2
in Eqs.
(13)-(16)above,andthefourthandthefthintegralsare
Z
T
2
T1
ln T
2
+b
b
2
d = b
h
y(ln(y)) 2
2yln (y)+2y i
y
1
(23)
Z
T
2
T1 ln
T
2
+b
b
T
2
T
2 T
1
d =
b 2
1
2 y
2
ln (y) yln(y)+y 1
4 y
2
y
1
T
2 T
1
(24)
where
y= T
2 T
1 +b
b
: (25)
TheBlack'76volatilityparameterv
A
isnowfoundby
v
A
v
A (T
1 t;T
2 t)
= s
Var
t
ln
A(T
1
;T
2 )
F
t [A(T
1
;T
2 )]
=(T
2
t): (26)
Observethatthevolatilityparameterv
A
isafunctionoftimetotherstsampling
date, T
1
t,andtimeto thelast samplingdate,T
2
t,wherethelattercoincides
withtime toexerciseoftheoption.
Next,considerthecasewheretheoptionisevaluatedwithinthesamplingperiod,
i.e., T
1
< t T
2
. It follows immediately from the denition of the arithmetic
averagethat
A(T
1
;T
2 )=
t T
1
T
2 T
1 A(T
1
;t)+ T
2 t
T
2 T
1 A(t;T
2
): (27)
Consequently,withT
1
<tT
2
,thecalloptionproblemisequivalentto
V
t h
1
T
2 ( A(T
1
;T
2 ) K)
+ i
= T
2 t
T
2 T
1 V
t h
1
T
2 (A(t;T
2
) K
0
) +
i
; (28)
where
K 0
T
2 T
1
T
2 t
K
t T
1
T
2 t
A(T
1
;t); (29)
i.e., aportfolioof T2 t
T
2 T
1
calloptions,eachwrittenontheaverageovertheremain-
ing sampling period [t;T
2
] where the strike is adjusted for the already observed
prices. Inthenon-trivialcase ofK 0
>0, thevalueof theadjusted optioncanbe
evaluated byinsertingT =t andK =K 0
in the evaluation procedure above. In
thedegeneratecaseof K 0
0,itwillalwaysbeoptimaltoexercisethecall,which
reduces theadjustedoptiontoaforwardwithcurrentvalue
V
t h
1
T
2 (A(t;T
2
) K
0
) +
i
=e
r(T2 t)
(T
2 t)
1 Z
T
2
t
f(t;s)ds K 0
!
: (30)
5. Valuation: An example
5.1. Current termstructure. TheNordicelectricitymarketNORDPOOLcon-
sistsofseveralforwardandfuturescontracts. Thetradedcontractandtheirmarket
pricesatDecember15. 1999arefoundin Exhibit1.
InsertExhibit1here
Based onthebid/askprices,weconstructacontinuous forwardpricefunction.
The forward function is given by the smoothest function that prices all traded
contractsonNORDPOOL within thebid/askspread. Theforwardpricefunction
at December151999isrepresentedbythecontinuousyellowcurveinFigure 1.
InsertFigure1here.
The redhorisontal lines in Figure 1correspond to thequoted forwardpriceof
eachtradedcontract.
5.2. Volatility. Thevolatilityin forwardpricesfalls rapidlyin this market. The
volatilityon asingledaydeliverystartingin oneweekmight be 80%, whereasa
similar deliverystartingin 6monthswill typicallyhavelessthan20%immediate
volatility.
InsertFigure 2here
Figure 2showstheforwardpricefunctionandthevolatilitycurveatDecember
151999forthefollowingcalendaryear(i.e.,2000).
5.3. Contract valuation. In thefollowing, we consider three valuation casesas
of December15. 1999 . The rstcase corresponds to the contract "FWYR-2000
Asian/M", see the rst line in Exhibit 2. The strike of the option is 120 and
the contract expiresat December31. 2000. The contract is subject to "monthly
settlements",whichmeansthatthecontractrepresentsaportfolio12monthlyAsian
options,where each optioniswritten onthemonthlypriceaverageand settledat
theendofthemonth.
InsertExhibit2here
The second caseis aEuropeanput optionwith strike120 andexpiration date
December311999,writtenontheforwardpriceontheforwardcontractondelivery
fromJanuary1.2000toJune30.2000. Thevalueoftheoptionandtheunderlying
contractarefoundinlines3and2in Exhibit2.
ThethirdcaseisaEuropeanputoptionwithstrike120andexpirationdateJune
30.2000,writtenontheforwardpriceontheforwardcontractondeliveryfromJuly
1.2000toDecember31.2000. Thevalueoftheoptionandtheunderlyingcontract
are foundin lines4and5.
InsertExhibit3here
Exhibit 3 considers the rst case in moredetail. Each line corresponds to an
Asianoptionwithstrike120writtenonamonthlypriceaveragewithexpirationat
theendofthemonth. ObservethatasseenfromDecember15.1999,thevolatility
oftheunderlyingmonthlypriceaverageisadecreasingandconvexfunctionofthe
deliverymonth (e.g., January43.8%; June30.3 %;December 24.4%). Byvalue
additivity, the value of each monthly option adds up to the value of the quoted
6. Value atRisk
The idea of Value at Risk (VaR) is to focus onthe downside market risk of a
givenportfolio atafuture horizondate. Foradiscussionon VaR,see Hull(1998)
andJorion(1997).
Evidencesuggeststhateventhoughaone-factormodelmaybeadequateforval-
uationinamulti-factorenvironment,ittypicallyperformspoorlyasatoolforrisk
management(e.g., dynamic hedging). Inthe following, we discuss athree-factor
ValueatRisk(VaR)model,whichisconsistentwiththevaluationandapproxima-
tionresultsabovefollowingfrom Eq.(1)above.
In order to obtain a richer class of possible forward price functions, assume
the followingforwardprice dynamics (with respect to the martingale probability
measure)
df(t;T)
f(t;T)
= a
T t+b dW
1 (t)+
2ac
T t+b
1
2
dW
2
(t)+cdW
3
(t); (31)
where a, b, and c are the positive constants from Eq. (1) above, and dW
1 (t),
dW
2
(t), anddW
3
(t)areincrementsofthree uncorrelatedstandardBrownianmo-
tions. Observethat theinstantanous dynamics of Eq. (31) just above is normal
with zeroexpectation andvariance
Var
t
df(t;T)
f(t;T)
= (
a
T t+b
2
+ 2ac
T t+b +c
2 )
ds; (32)
whichisconsistentwiththedynamics ofEq.(1)above.
It follows that the forward price function f(;T) at the future date is the
stochastic integral
f(;T) = f(t;T)exp (
Z
t a
T s+b dW
1 (s)
1
2 Z
t
a
T s+b
2
ds )
exp 8
<
: Z
t
2ac
T s+b
1
2
dW
2 (s)
1
2 Z
t
2ac
T s+b ds
9
=
;
(33)
exp Z
t
c dW
3 (s)
1
2 Z
t c
2
ds
:
Inaddition,theforwardmarketatthefuturedate isrepresentedbytheassociated
Black'76implicitvolatilityfunction( ;T ),where2[;T]istheexercise
date oftheoption,andT isthedeliverydateoftheunderlying forward.
Consideraportfolio ofelectricity derivativesat thefuture date . Theideaof
VaR is to analyse the downside properties of the probability distribution of the
future portfoliovalue. Weapplythesimulationmethodologyinorder to generate
this probability distribution, from which Value at Risk can be calculated. The
procedureconsistsofthefollowingsteps(whicharerepeated): First,usearandom
generatorto drawapossiblerealisationforthefuture forwardpricefunction con-
sistent withEq. (33)above. Second, use theabove valuation and approximation
resultstocalculatetheassociatedmarketvalueofeachposition,conditionalonthe
realised forward price function (as well asthe future implicit Black'76 volatility
function). Thirdly, calculate the conditional market valueof the portfolio (which
followsimmediatelyfrom valueadditivity). Now,foralargenumberofiterations,
we approximate the probability distribution of the future portfolio value by the
7. Value at Risk: Anexample
7.1. Price path simulations. Eq. (33) describes how the future forward price
function issimulatedfrom currentmarketinformation. Thef(t;T)function isthe
forwardpriceattimetfordeliveryattimeT. Theparametersa,b,andcareinputs
to thevolatilityfunction.
InsertFigure 3here
Inordertosimulatepossiblepricepaths,weuseEq. (33)repeatedly. InFigure3
we present 100 simulated week prices based on this model. In each simulated
path thefollowingprocedureis followed. First,theforwardfunction nextweekis
simulated,integratingthiscurvefromzeroto7daysgivestherstweekprice. Next
weuse this newforwardcurve in combination with thevolatilitycurveto obtain
the forwardcurvein the next stepand so on. In this wayweobtain the correct
andlargeshort-termvolatilityinpricesinadditiontothemuchsmallervolatilityin
pricesasseenfromtoday. Weobservethatthesimulationmodelgivesasubstantial
meanreversioninprices.Thisisinaccordancewithempiricaldata. Theadvantage
ofthis methodisthatcurrentinformationaboutthevolatilitycurveandtheterm
structure ofpricesissuÆcienttoperformthissimulation.
7.2. Valueat Riskcalculation. Inthefollowing,wefocusonthedownsiderisk
of a given nancial portfolio of forwards and options. Assume that we want a
probability distributionwhich represents thepossiblefuture market valuesof the
portfolio in one week. First we simulate the termstructure starting in oneweek
using Eq. (33). For each simulation we nd the market value of all instruments
in the portfolio. By assigning equal probability to each simulation, this gives a
distributionoffuturemarketvalues.
InsertFigures4,5,and6here.
Wehavechosenaverysimpleexampleportfolio. Itconsistsofaforwardcontract
for therst6monthsin year2000and aput optionwithexercisedate atthelast
dayof1999,writtenonthesameforward. Thestrikeontheoptionis120. Figure4
gives the distribution in one week for the forward contract. Figure 5 gives the
similar information for theput option. In Figure 6 we give thestatistics for the
totalportfolio. Theexampleillustratestheriskreductioneectfromtheoptionon
thetotalportfolio.
8. Conclusions
Thepurposeofthispaperistoderiveadecisionsupportmodelforprofessionals
in theelectricitymarketforvaluationandriskmanagement. Thepaperappliesre-
sultsandmetodsfromnance,andincorporatesthefactthatelectricityderivatives
are writtenonacommodityowratherthanabulkdelivery.
The electricity derivatives market is represented by a forward price function
following from the quoted priceson traded contracts. Themarket uncertaintyis
modelledbyavolatilityfunctionbeingadecreasing(andconvex)functionoftime.
Thepaperpresentsvalueapproximationresultsfor"European"aswellasAsian
calloptions. The3-factormarketriskmanagementmodelpresentedinthepaperis
compatible withtheseresults,andcanbeusedforquantitifyingthefuture market
riskofgivenportfolios(includingVaR).
Appendix
This appendix evaluates Eq. (12) above. Dene the new integration variable
x = 1
(T
2 T
1
) s with upper and lower limitsX(t) b+ 1
(T
2 +T
1
) t and
X()b+
2 (T
2 +T
1
) ,andtheconstant
2 (T
2 +T
1
),andwriteEq. (12)as
Var
t
ln
F(;T
1
;T
2 )
F(t;T
1
;T
2
=
a
T
2 T
1
2 Z
X(t)
X()
ln
x+
x
2
ds
+ 2ac
T
2 T
1 Z
X(t)
X() ln
x+
x
ds+c 2
( t)
Observethat with b>0and t< T
1
<T
2
,wehavex+>0and x >0
forx2[x;x ]Now,usethefollowingtworesults:
6
Z
ln
x+
x
2
dx = (x+)( ln(x+)) 2
2(x+)ln (x+)ln (x )
+4aln(2)ln
x
2
4dilog
x+
2
+(x )( ln(x )) 2
4;
Z
ln
x+
x
dx = (x+)ln (x+) (x )ln (x ) 2;
where
dilog(x) Z
x
1 ln(s)
1 s ds:
Substitute theresultsintothevarianceexpression,toobtainthedesiredresult.
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6
Itisstraightforwardtoverifytheseresult,usingthefactthat
@
@x
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E-mail address: petter.bjerksund@nhh.no
E-mail address: contact@viz.no
