exchange is probably one of the reasons theNordic market became so successful. Although Iwill concentrate on the Nordic power market,most of this will also be relevant for other devel-oping power markets, particularly because sever-al exchanges try to build on the Nordic model.

After the establishment of Nord Pool it wasnot long before large US energy companies likeEnron and TXU became members of theexchange. In Nord Pool’s last peak period Enronwas one of the largest, if not the largest player atNord Pool with an active market making andposition taking operation in Oslo. Even today alot of international players are members of theexchange, from the member list (http://www.

nordpool.no/) we can see that well known bankslike Morgan Stanley, Merrill Lynch, Barclays, J.Aron (the commodity arm of Goldman Sachs)and also well known hedge funds like Tudor andD.E. Shaw are exchange members. At the time ofwriting more than 300 firms are exchangemembers of Nord Pool. Most of the large USenergy companies that used to be members ofthe exchange left with the collapse of Enron.That most of the trading in the Nordic market atthat time was on a well functioning exchangeand not OTC meant Enron (and everybody else)had to pay margin deposits on their contracts.This resulted in no credit losses in the Nordicmarket for anyone that had traded indirectly

48 Wilmott magazine

IntroductionThe Nordic electricity market with its exchangeNord Pool is today one of the most activeexchanges in the word for physical electricity andelectricity derivatives. The Norwegian electricitymarket started to deregulate in 19911 beforewhich it was an inter-utility market. In 1994 anOTC market for trading in electricity forwardsand options developed, at about the same timethat the exchange Nord Pool was established.New electricity legislation in Finland and Swedenopened up the deregulation of their markets in1996 and 1998, and in 1999 Denmark followed.The early deregulation and establishment of an

THE COLLECTOR

This paper was written while taking a sabbatical from trading and spending some time at the Norwegian University of Science and Technology. I would like to thank Stein Erik Fleten for some very useful comments. Needless to say, I remain solely responsible for any errors that still remain.

Espen Gaarder Haug

In this paper I look at the practical valuation of power derivatives from a trader’s perspective. Most peo-ple that have written about valuation of power derivatives are academics or quants working in theresearch departments of large organizations far away from the trading desk. Most of them havenever traded a single power option. In general there is nothing wrong with that as some ofthe greatest practical research in quantitative finance has come out of academia andresearch departments, we just have to mention the Black-Scholes-Merton model to remind usof that. This also brings to mind some big swinging traders who have told me that they don’tcare about the theory, and the very next second were looking at the Black-Scholes-Mertondelta on their Bloomberg screen to hedge some options. Anyway, when it comes to electricityderivatives most academics have made simple things too complex and at the same time haveforgotten simple things that have great importance. The Black-Scholes-Merton model or its bino-mial equivalent is a great example of making things simple enough, but not simpler than that, atleast when it comes to equity, futures or currency options. Still, as we will see the Black-Scholes-Merton model, or rather the formula will not necessarily do without some modifications whenapplied to the electricity market. This article was written during a research sabbatical from trading,to be honest most of this article was written from a bar in the town of Trondheim, one of the great-est university towns on this planet. To write or read about formulas and abstract mathematics isin my experience best done in a relaxing atmosphere. A frozen margarita could certainly helpyou absorb this article once you have finished reading it, or better still before.

Key words: Electricity forwards/swaps, power swaptions, hybrid pay-off options, Nord Pool, NordicMarket.

Practical Valuation of Power DerivativesBare Blåbær

^

with Enron through Nord Pool. However, therewere some considerable losses for players thathad done OTC transactions with Enron. In theUS there were basically no well-functioningexchanges for power derivatives and many play-ers there had large credit exposures to Enron.The result was a chain reaction of losses. As aresult the power trading in the US market driedup almost completely. At Nord Pool there wasderivatives trading as usual although the vol-umes fell quite dramatically as even Nordic com-panies got scared from losses in the OTC market,and also because some of the biggest marketmakers and position takers like Enron simplywere gone.

Back to the “beginning”; in 1996 I resignedfrom my job as a market maker in fixed incomeoptions in Chemical Bank (today J.P. MorganChase Bank2) to set up my own company doingresearch, software and consulting for the fixedincome and particularly the electricity market. Iwanted to be part of the New New Thing: the fastgrowing electricity market. Many traders in thenewly developed electricity market had quitegood quant skills, many with backgrounds inengineering, others had moved over from fixedincome trading, FX trading or the stock market,but very few had the combination of solid optiontrading background with good quant skills.There were a lot of different opinions on how tovalue electricity forwards and options at thattime, one player explained to me how they usedsome type of regression analysis to value options.I was wondering if they had ever heard aboutarbitrage-free pricing models? Or maybe arbi-trage principles had no value here, because onecould not store electricity? A few very quantita-tively oriented players were looking into MonteCarlo models trying to take into account season-ality, mean reversion, temperature, precipitationand a lot of other factors in one massive supersimulation. The few people with option tradingexperience from other markets that recently hadswitched to the electricity market seemed to sim-ply apply the Black-76 model. I was skeptical to allapproaches, but as a trader I always had greatrespect in particular for other experiencedtraders. I got hold of some spot price data andlooked at the historical distributions. The spot

Wilmott magazine 49

price had extremely fat tails, further it had sever-al statistical measures indicating mean reversionand seasonality. Also how could one even usemodern finance theory on electricity that wasnot easy to store. A lot of the academics musthave thought the same; soon over the next fewyears papers on mean reversion and seasonalitygot published in relation to electricity deriva-tives. When I think back to that time, even thecontract descriptions from the OTC andexchange market were not specific on all thedetails that could be important from a valuationperspective. I spent days and months contactingbrokers and other people actively involved in themarket to find out what really was the standard.That several people active in the market gavecompletely different answers did not makethings simpler, exactly when was the delivery,and did you get delivery of the forward or was itall completely cash settled, or cash settled plusdelivery of the forward, and was the forwarddelivered at the strike price or the settlementprice? In the early days of a completely new mar-ket there are also limits in how much data youcan back-test your ideas on. It had all the charac-teristics of a new fast growing market whichincludes some confusion about some “details”,details that can give opportunities for arbi-trageurs. The big differences in opinion abouthow to value the derivatives contracts in the mar-ket told me I should get involved as a traderrather than as a consultant, and soon enough Iwas trading the New New Thing.

The physical market was basically limited tothe physical players; the power producers, largeretailers and energy intensive industry. TheNordic market is dominated by hydro-electricpower, later when Sweden became a member italso meant some nuclear power and to somedegree oil, gas and coal generators. To value theforward price is partly arbitrage pricing, becausehydro-electric producers can at least in theorystore their electricity by letting their water reser-voirs fill up, but as storage is limited by severalfactors, the pricing of forward contracts alsoinvolve expectations (knowledge and gambling)about temperature, precipitation, ice melting, youname it. As I mainly worked with the financialplayers, first with my own consultant software

firm and later as a proprietary trader, I decidedearly on that I would not compete with the physi-cal players in their knowledge about supply anddemand in the physical market. I could not evenstore electricity without buying a hydro-electricpower plant, some financial players got involvedin this, but most financial players only gotinvolved in the derivative market where every-thing was financially settled against the spotprice. In such a situation my view was that theedge one possibly could get over the physical play-ers was through understanding exactly how onecould value derivatives against other derivativescontracts, and this is what this article is all about.That is forwards against other forwards andoptions against forwards (“dynamic” hedging).

The most actively traded contracts soonbecome the seasonal and annual forward con-tracts, today Nord Pool have switched from threeseasonal forwards to four quarterly forwardscontracts per year, but this make little or no dif-ference for the formulas we soon will look at. Inthe delivery period these exchange traded for-wards have daily financial settlement againstthe spot price. The spot is a physical auction mar-ket with a daily fixing price. For example, afourth quarter 2006 forward will have its deliv-ery period starting October 1 ending December31. For every day in the delivery period the con-tract is financially settled against the daily fix-ing price of the physical spot market. Each “day”is 24 hours, so the fourth quarter 2006 quarterlycontract will have 2208 MWH (megawatthours),or actually 2209, because of the adjustment fromsummer time in October gaining us an addition-al hour. The prices at Nord Pool are today quotedin EUR (it used to be in NOK). What the marketcalled (and still call) forwards are in realityswaps. In the beginning of the market manyplayers were not really aware of this - or did notcare? One player I consulted for took advantageof this and arbitraged seasonal swaps againstannual swaps, locking in risk-free arbitrage. Themistake was that many players assumed the for-ward price was the same as the forward value, bynot recognize it as a swap they simply did not dothe correct discounting when comparing for-wards against forwards. Now let’s move on tosome formulas.

ESPEN GAARDER HAUG

Energy Swaps/ForwardsGiven the presence of traded contracts with quotedmarket prices, we can come up with a way to valuethe swap relative to other swaps. For example astrip of quarterly power swaps covering the wholeyear should have the same value as an annual swap.Otherwise there will be an arbitrage opportunity.All the forward contracts have financial settlementagainst the daily spot fixing that is set by the physi-cal market. That there are no physical deliverymakes it easier for financial players, nobody can forexample corner the forward market.

As already mentioned the electricity swapstraded in the Nordic power market are known asforwards, but are (from a valuation perspective)power swaps, a strip of one day electricity for-wards. To compare the value of different swapswe simply need to discount the cash flows. Theswap/forward market price is not the value ofthe swap contract, but only the contract price. Tocompare different power swaps with each otherwe need to find the value:

FValueToday = e−rb Tb

n

n∑i=1

F

(1 + rj,i/j)i(1)

where FValueToday is the swap value today and F isthe forward/swap price in the market. In thiscase ‘price’ should not be confused with ‘value’!

j is the number of compoundings per year(number of settlements in a one year forwardcontract). We assume here they are evenly spreadout. In practice there are no payments duringweekends, so every 5th payment do not have thesame time interval as the rest of the payments.However the effect of taking this into account isnot of economic significance, at least for month-ly or longer contracts.

n is the number of settlements in the deliveryperiod for the particular forward contract. NordPool uses daily settlement, so this will typicallybe the number of calendar or trading days in theforward period.

rj,i is a risk-free interest swap rate starting atthe beginning of the delivery period and endingat the i the period. Further it has j compound-ings per year.

Tb is the time to the beginning of the forwarddelivery period.

rb is a risk-free continuously compoundedzero coupon rate with Tb years to maturity.

Equation (1) is what a couple of market par-ticipants figured out early in the game and wasin some cases able to arbitrage a strip of season-al contracts against annual contracts. We cansimplify equation (1) to

FValueToday = Fe−rb Tb

(1 − 1

(1 + rj/j)n

)rj

j

n, (2)

where rj now is the forward start-interest swaprate, starting at the beginning of the deliveryperiod and ending at the end of the deliveryperiod, with j compoundings per year set equalto the number of settlements per year.

ExampleConsider a quarterly electricity forward (swap)that trades at a price of 35 EUR/MWH (EUR permega watt hour), the delivery period is 2160hours or 90 days. It is six months to the start ofthe delivery period. Assume the forward startinterest swap rate, starting six months from nowand ending six months plus 90 days from now is5% converted to the basis of daily compounding.The six months continuous zero coupon rate is 4%.What is the present value of the power contractwhen using 365 days per year? F = 35, rj = 0.05,j = 365, n = 90, rb = 0.04, Tb = 0.5

FValueToday = 35e−0.04×0.5

×

(1 − 1

(1 + 0.05/365)90

)0.05

× 365

90

= 34.0940.

The present value of the power forward/swapis thus 34.0940 EUR per MWH. The total value ofone contract is found by multiplying the num-ber of hours in the contract period by the valueper MWH 2160 × 34.0940 = 73, 643.08 EUR.

ApproximationFormula (2) can be approximated by:

FValueToday ≈ Fe−rb Tb e−rd (Tm −Tb ),

where rd is the forward starting continuouslycompounded zero coupon rate for the delivery

period, multiplied by the time from the start ofthe delivery period Tb to the middle of the deliv-ery period Tm . This can be simplified further by

....FValueToday ≈ Fe−re Tm , (3)

where re is a continuously compounded zerocoupon rate from now to the end of the deliveryperiod. This approximation is reasonably accu-rate as long as we use consistent rates.

ExampleConsider the same input as in last example. Tomake the examples equivalent we have to find arate re that is consistent with rb = 0.04 andrj = 0.05. To find re from the example above we first need to convert rj to a continuously compounded rate rcj = 365 ln(1 + 0.05/365) =0.04999658 . Based on no arbitrage opportunitieswe must have

e−re Tm = e−rb Tb e−rcj (Tm −Tb ),

re = rbTb + rcj(Tm − Tb)

Tb + (Tm − Tb),

re = 0.04 × 0.5 + 0.049997 × 90/365/2

0.5 + 90/365/2

= 0.04198,

and we can now approximate the value of theforward price

FValueToday ≈ 35e−0.04198 ×(0.5+90/365/2) = 34.0961,

The approximate value is thus not very dif-ferent from 34.0940 calculated by the moreaccurate formula.

Power OptionsAs I have already indicated in the introduction,the spot price in the Nordic electricity marketdefinitely has some seasonality, further statisti-cal tests indicate it follows some type of meanreversion. The returns also have very fat tails. Inthe Nordic market almost all options are notdirectly on the spot price, but rather on quarter-ly and annual power swaps. If the spot price isseasonal and mean reverting this should alreadybe reflected in the forward price in a efficientmarket. For example in the winter months

50 Wilmott magazine

where there is peak demand for electricity in theNordic countries (heating) if the low demand inthe summer months (there is not much air condi-tion in the Nordic markets) not already is reflect-ed in the summer months forward then marketparticipants can do very good relative value tradesby selling forward contracts for the summermonths and for example buying the next winterforwards against it. Then with a high probabilitythe spot price would come down in the summerand give profit on the short summer contracts.

I don’t doubt one needs a seasonal model,possibly in combination with a mean revertingmodel and many other factors to value the fairforward price from the spot price, but in an effi-cient market we must assume this already ismore or less reflected in the forward prices. Ifyou don’t think so you should forget the optionmarket and concentrate on doing relative valuetrades in spot against forwards and particularforward against forwards. When valuing optionsbased on arbitrage theory and dynamic hedgingthe only thing that is important is the optionvalue relative to the underlying, that is the sto-chastic process and the distribution of thereturns of the underlying asset (in a risk neutralworld). Many academics have concluded that ageometric Brownian motion and Black-Scholes-Merton type model are useless for electricityderivatives, because they observe seasonality and

mean reversion in the spot price, see Lucia andSchwartz (2002).

If we look at the distribution and some statis-tical tests of the spot price returns they are right.This we can get a good idea about simply fromlooking at Figure 1 that plots the histogram forthe Nord Pool electricity spot using data fromNovember 5 2001 to November 3 2004. From thefigure we can see the extremely fat tails and thelarge peak, which also is confirmed with aPearson kurtosis of 11.12 (leptokurtic distribu-tion), remember the normal distribution only

has kurtosis of 3. As already mentioned theoptions trading at Nord Pool are Europeanoptions on the quarterly and annual forwardcontract and not directly on the spot. Figure 2shows the return distribution on a annual for-ward. This is hardly comparable to the spot data,it is much closer to a normal distribution. It stillhas some fat tails as also is evident from aPearson kurtosis of 4.9, but this is not very dif-ferent from what we observe in a lot of stockswhere the Black-Scholes-Merton model or it’sbinomial equivalent is actively used by many

Wilmott magazine 51

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Figure 1: Electricity Spot Daily Returns. Daily data from Nov 5-2001 to Nov 3-2004Volatility 114.99%, Skewness 0.96, Pearson kurtosis 11.12

(most?) practitioners. Not convinced? Just lookat Figure 3 which is the return distribution ofAmazon.com, the distribution has much fattertails (Pearson kurtosis of 13.29) than the annualelectricity forward, and I know for sure manyoption traders used Black-Scholes-Merton typemodels to value options on such stocks. As weknow, option traders are naturally not using theBlack-Scholes-Merton model in it’s naive waywhen valuing options on stocks, they fudge themodel to work in such a market by pumping upthe volatility for out-of-the money options ver-sus at-the money options, we will get back tothis later. My point is simply that based on sta-tistical analysis one can basically conclude thatthe distribution on quarterly and annual elec-tricity forwards returns is not very different

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Electricity Spot Price. Daily data from Nov 5-2001 to Nov 2-2004

ESPEN GAARDER HAUG

Using Ito’s lemma we get the following PDE

[∂ f

∂ t+ 1

2

∂2 f

∂F2σ 2F2

]dt = rf .

Where f is the value of the derivative security.Solving this for a European option with the bound-ary condition max[F − X, 0] where X is the strikeprice we get the well known Black-76 formula.

c = e−rT [FN(d1) − XN(d2)], (4)

p = e−rT [XN(−d2) − FN(−d1)], (5)

where

d1 = ln(F/X) + (σ 2/2)T

σ√

T,

d2 = ln(F/X) − (σ 2/2)T

σ√

T= d1 − σ

√T,

and r is the risk free rate, T is the years to matu-rity and N(·) is the cumulative normal distribu-tion function.

The experienced option traders that movedinto the electricity market early on using theBlack-76 formula were probably not far off at all,as we will see they were “just on the target”,except for their delta hedge. The Black-76 formu-la early on became more or less the standard inthe Nord Pool option market. But what about theboundary condition, didn’t I just tell you thatthe power forwards in reality were not forwardsbut swaps, so that the boundary condition andformula must be modified accordingly? By whatI think was a coincidence, the way Nord Pool hadformulated their option contracts actually “mag-ically” turned the valuation problem into anoption on a future/forward (expiring at sametime as the option) and not an option on a swap.The option contracts at Nord Pool were specifiedin such a way that at expiration they had cashsettlement equal to the difference between theforward price and the strike price, and in addi-tion you got delivery of a forward at the settle-ment price. The forward you got at maturityhad no impact on the valuation or hedgingproblem as you always had the opportunity toclose this out immediately at no cost, except thebid offer spread. However if you had deltahedged the option during its lifetime you werenow suddenly left with cash flows from yourhedging that did not start to materialize beforethe swap went into the delivery period. Thisoption may seem difficult to value at first.Things that seem confusing at first typicallyhave a very simple solution. They key was to fig-ure out the cash flows from just holding theoption was exactly the same as a standardEuropean option on a futures or forward4 con-tract, so the option value had to be equal to the

52 Wilmott magazine

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Figure 2: Year 2004 Forward/Swap Daily Returns. Daily data from Jan 3-2002 to Nov 3-2004, Volatility 15.4%, Skewness 0.15, Pearson kurtosis 4.9

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Figure 3: Amazon Daily Returns. Daily data from Nov 6-2001 to Nov 3-2004, Volatility61.33%, Skewness 1.17, Pearson kurtosis 13.29

from that on many stocks, except they aremaybe closer to the normal distribution thansome dot com stocks. In other words my guttells me that I can approximate by assuming theforward price (swap) follows a geometricBrownian motion

dF = σ Fdz

where σ is the volatility of the returns of theforward/swap price F and dz is a Wiener process.

^

54 Wilmott magazine

Black-76 formula. As mentioned earlier this soonalso became the market standard. Still what youactually was hedging with was not a future orforward but a power swap. To take this intoaccount one simply needed to adjust the deltaaccordingly. Many market participants got thiswrong for years. If the option value is given byBlack-76 then we could hedge the option usingthe Black-76 delta and a forward expiring atoption maturity. Now if the forward pricemove the Black-76 delta multiplied by the for-ward price will exactly offset the change in theoption value. However we are not hedging witha forward contract but with a swap, so we needan adjusted delta �H that multiplied by theswap value is exactly equal to the Black-76delta multiplied by the forward price. Letting�B76 = e−rT N(d1) be the Black-76 delta, the deltafor hedging with electricity forwards/swapsmust be given by:

�B76 F = �HFVaT

�B76 F = �HFe−rp (Tb−T)

(1 − 1

(1 + rj/j)n

)rj

j

n

�H = �B76 erp (Tb−T)nrj

j

(1 − 1

(1 + rj/j)n

)

�H = N(d1)e−rT erp (Tb−T)

nrj

j

(1 − 1

(1 + rj/j)n

) ,

�H ≈ N(d1)e−rT er(Tb−T)

nrj

j

(1 − 1

(1 + rj/j)n

) ,

where F is the swap/forward market price, and FVaT

is the swap value at option expiration T, rp is therisk-free rate from the option’s expiration to thebeginning of the delivery period. Tb is the timefrom now to the beginning of the delivery period,r is the risk free rate until option expiration.

d1 = ln(F/X) + Tσ 2/2

σ√

T,

and for a put

�put = N(−d1)e−rT erp (Tb−T)

nrj

j

(1 − 1

(1 + rj/j)n

) . (6)

Energy SwaptionsNot long ago (April 2005) Nord Pool changed thecontract description of their options on for-wards/swaps to have delivery of the underlyingswap at the strike price and not at the fixingprice. For this reason Nord Pool options are todayEuropean options on energy swaps (a.k.a. energyswaptions). If a call swaption is in-the-money atmaturity the option has delivery of a swap(named forward by the market) delivered to thestrike price. The pay-out from the option is thusno longer received immediately at expiration,but rather during the delivery period of theunderlying swap (forward). The energy call swap-tion formula is

c =

(1 − 1

(1 + rj/j)n

)rj

j

ne−rp (Tb−T) × Black − 76

=

(1 − 1

(1 + rj/j)n

)rj

j

ne−rp (Tb−T)

× e−rT [FN(d1) − XN(d2)]

=

(1 − 1

(1 + rj/j)n

)rj

j

ne−rb Tb [FN(d1) − XN(d2)],

(7)

where rp is the risk-free rate from the option’sexpiration to the beginning of the delivery peri-od. Tb is the time from now to the beginning ofthe delivery period. Moreover,

d1 = ln(S/X) + σ 2T/2

σ√

T, d2 = d1 − σ

√T.

For a put we similarly have

p =

(1 − 1

(1 + rj/j)n

)rj

j

ne−rb Tb [XN(−d2)

− FN(−d1)].

(8)

Good approximations for calculating the calland put swaption values are

c ≈ e−re Tm [FN(d1) − XN(d2)], (9)

p ≈ e−re Tm [XN(−d2) − FN(−d1)]. (10)

where re is the risk-free rate from now to the endof the delivery period and Tm is the time in years

from now to the middle of the delivery period,and

d1 = ln(S/X) + σ 2T/2

σ√

T, d2 = d1 − σ

√T.

ExampleConsider a call on a quarterly electricity swap,with six months to maturity. The start of thedelivery period is 17 days after the optionexpires and the delivery period is 2208 hours or92 days. The swap/forward trades at 33 EUR/MWH, the strike is 35 EUR/MWH. The number offixings in the delivery period is 92. The risk-freerate from now until the beginning of deliveryperiod is 5%. The daily compounding swap ratestarting at the beginning of the delivery periodand ending at the end of the delivery period is5%. The volatility of the swap is 18%. What is the option value? T = 0.5, Tb = 0.5 + 17/365 =0.5466, rb = 0.05, rj = 0.05, j = 365, n = 92, andσ = 0.18 yields

d1 = ln(33/35) + 0.5 × 0.182/2

0.18√

0.5= −0.3987,

d2 = −0.3987 − 0.18√

0.5 = −0.5259,

N(d1) = N(−0.3987) = 0.3451,

N(d2) = N(−0.5259) = 0.2995,

c =

(1 − 1

(1 + 0.05/365)92

)0.05

365

92e−0.05×0.5466

× [33N(d1) − 35N(d2)] = 0.8761.

To find the value of an option on one swap/forward contract we need to multiply by thenumber of delivery hours. This yields a price of2208 × 0.8761 = 1, 934.37 EUR. Alternatively wecould have found the option value using theapproximation (9), using time from now to themiddle of delivery period Tm = 0.5 + 17/365+92/2/365 = 0.6726 and assuming the rate fromnow to the end of the delivery period isre ≈ 0.05:

c ≈ e−0.05×0.6260 [33N(d1) − 35N(d2)] = 0.8761.

At four decimals accuracy the approximationevidently gives the same result as the more accu-rate formula.

ESPEN GAARDER HAUG^

ESPEN GAARDER HAUG

W

56 Wilmott magazine

Put-call parityFor a standard put (a.k.a. receiver swaption) orcall option (a.k.a.payer swaption) the put-callparity is:

p = c + (X − F)

(1 − 1

(1 + rj/j)n

)rj

j

ne−rb Tb , (11)

and of course

c = p + (F − X)

(1 − 1

(1 + rj/j)n

)rj

j

ne−rb Tb . (12)

The put-call parity can be used to constructsynthetic put or calls as well as synthetic swaps/forwards from traded puts and calls. The syn-thetic/implied forward price from a put and acall is given by

F = c − p(1 − 1

(1 + rj/j)n

)rj

j

ne−rb Tb

+ X.

Energy Swaption Greeks

Delta:

�call =

(1 − 1

(1 + rj/j)n

)rj

j

ne−rb Tb N(d1), (13)

�put = −

(1 − 1

(1 + rj/j)n

)rj

j

ne−rb Tb N(−d1). (14)

Vega:The vega is the swaptions sensitivity with respectto change in volatility.

Vegacall ,put =

(1 − 1

(1 + rj/j)n

)rj

j

ne−rb Tb Fn(d1)

√T.

(15)

It is necessary to divide by 100 to express vegaas the change in the option value for a 1% pointchange in volatility.

Gamma:Gamma for swaptions:

�call ,put =

(1 − 1

(1 + rj/j)n

)rj

j

ne−rb Tb n(d1)

Fσ√

T. (16)

Rho:

ρcall =

(1 + rj/j)−n−1

rj−

(1 − 1

(1 + rj/j)n

)r2j

j

n

−T

(1 − 1

(1 + rj/j)n

)rj

j

n

× e−rb Tb [FN(d1) − XN(d2)],

(17)

ρput =

(1 + rj/j)−n−1

rj−

(1 − 1

(1 + rj/j)n

)r2j

j

n

−T

(1 − 1

(1 + rj/j)n

)rj

j

n

× e−rb Tb [XN(−d2) − FN(−d1)].

(18)

Still, What About Fat TailsEven when using a modified Black-Scholes-Mertonmodel that takes into account that the underlyingis a power swap one still has the same problem asin any other market, the observation of fat tails inthe returns. Most traders prefer to get around thisby fudging the model, inputting some type ofvolatility smile. In the early development of theNordic market one could often see a flat volatilitysmile, giving opportunity for good relative valuetrades. An alternative is naturally extending theapproach to stochastic volatility models, an inter-esting model in this respect is the SABR model byHagan, Kumar, Lesniewski and Woodward (2002),as this model only requires two additional inputparameters; the volatility of volatility and the cor-relation between the underlying asset and the

volatility. As in other markets, in the electricitymarket we observe jumps from time to time, jump-diffusion or more likely jump-diffusion in combi-nation with stochastic volatility models is possiblya way to go. Personally I always prefer to have awhole set of models in front of me on a hugespreadsheet. From this I can at least quickly findout how sensitive a given strike and maturity are todifferent input parameters for volatility of volatili-ty or jumps using models with different assump-tions. This will also help me fudge my basic model.In other words I am using a Black-Scholes-Mertontype model without actually using it.

When it comes to options directly on the spotone typically has to return to modeling the spotdirectly, here seasonality, mean reversion, and veryfat tails can all be of great importance, howeverthis is only a small fraction of the derivatives mar-ket at least in the Nordic market and I have con-centrated on the main contracts actively trading.

1. See Westerberg (1999) for more information on the his-tory of the Nordic electricity market, well you better under-stand Norwegian to read it, or you can hire me as a con-sultant and I can read it for you.2. Chemical Bank took over Chase Manhattan Bank andchanged name to Chase, then Chase took over J.P.Morgan and changed name to J.P. Morgan Chase, thenJ.P. Morgan Chase took over Bank One but still kept thename J.P. Morgan Chase.3. Naturally more than what is presented here.4. With a forward expiring at the same time as the option.

■ Black, F. (1976): “The Pricing of Commodity Contracts,”Journal of Financial Economics, 3, 167–179.■ Black, F., and M. Scholes (1973): “The Pricing ofOptions and Corporate Liabilities,” Journal of PoliticalEconomy, 81, 637–654.■ Hagan, P. S., D. Kumar, A. S. Lesniewski, and D. E.Woodward (2002): “Managing Smile Risk,” WilmottMagazine, September, 1(1).■ Lucia, J., and E. Schwartz (2002): “Electricity Prices andPower Derivatives: Evidence from the Nordic PowerExchange,” Review of Derivatives Research.■ Merton, R. C. (1973): “Theory of Rational OptionPricing,” Bell Journal of Economics and ManagementScience, 4, 141–183.■ Westerberg, G. (1999): “Utviklingien i Det NordiskeKraftmarkedet,” Derivatet, 11, 3–7.

FOOTNOTES & REFERENCES