electricity derivatives

Electricity Derivatives

Giovanni Barone-Adesi1 Andrea Gigli2

This Draft: 02, May 2002

Abstract: In this paper we propose an algorithm for pricing derivatives written onelectricity. A discrete time model for price dynamics which embodies the mainfeatures of electricity price revealed by simple time series analysis is considered. Weuse jointly Binomial and Monte Carlo methods for pricing under a risk-neutralmeasure of which we prove the existence. Moreover, to reduce the time spent insimulations we derive a sufficient condition for early option exercise in order to gaincomputational efficiency.

1 E-mail address: [email protected] E-mail address: [email protected]

1 IntroductionElectricity markets are becoming a popular …eld of research amongst academicsbecause of the lack of appropriate models for describing electricity price behaviorand pricing derivatives instruments. Models for price dynamics must take intoaccount seasonalities and spiky behavior of jumps which seem hard to model bystandard jump processes. Without good models for electricity price dynamicsit is di¢cult to think about good models for futures, forward, swap or optionpricing. As a result, very often pratictioners are forced to employ models basedon the cost-of-carry for their needs.

There are several reasons which discourage the use of cost-of-carry mod-els (see H. Geman and O. Vasiceck, 2001 and H. Geman and A. Roncoroni,2001). Basically the most important arguments against them are related to thephysical and temporal constraints of electricity as a non-storable commodity.Non-storability implies that arbitrage arguments cannot be used in de…ning apricing model when electricity is the underlying of a derivative contract. Sec-ondly, electricity transportation is based on the availability of line connectionswhich can be damaged by rare and extreme events. Probably this kind of riskshould be included in a good pricing model.

In this paper we do not try to solve all of the above problems but we simplypropose an algorithm for pricing derivative instruments which is based on someintuition from market data. Our main idea is to include in a discrete timemodel for electricity price dynamics those features of electricity price behaviorrevealed by simple time series analysis. We use jointly Binomial and MonteCarlo methods for pricing under a risk-neutral measure. The existence of sucha measure is discussed in the appendix. Moreover, to reduce the time spent insimulations we show how the inclusion of a su¢cient condition for early optionexercise allows us to gain computational e¢ciency.

The paper is organized as follows. In the next section we show the intuitionswhich are on the basis of our model, in section III we describe in detail how toimplement the pricing and in section IV we present some simulation results forAmerican call options. The last section remarks our main …ndings and somestill open issues.

2 A discrete time modelIn the following, the electricity price at time point t > 0 is denoted by Et andspikes are assumed to occur at random times and with random durations aspercentage changes on the previous price level. A spike is de…ned as a positiverandom percentual change in the previous price level of electricity and it isdenoted as St when it happens at time t. The function D(S¿ ) : R+ ! N+

returns the duration of the most recent spike occurred at time ¿ < t: We alsode…ne Q(S¿ ) : R+ ! N+ as the time at which the spike will break down.The di¤usion is Xt, its parameters are ¹ and ¾ and depends on "t » N(0; 1).Electricity prices are assumed to follow

¢Et = ¢f(Xt; t) + WtStEt¡1 ¡ ZtS¿Et¡1; E0 = X0 (1)

where¢Xt = ¹¢t + ¾

p¢t"t

2

0

40

80

120

160

200

240

280

320

360

99:01 99:07 00:01 00:07 01:01 01:07 02:01

Netherlands market

0

10

20

30

40

50

60

70

80

99:01 99:07 00:01 00:07 01:01 01:07 02:01

NORDPOOL market

Figure 1: Electricity spot prices from Netherland and NordPool markets.

Wt =½

1 with probability qt0 with probability 1 ¡ qt

Zt =½

1 if t = Q(S¿ )0 otherwise

The function f(Xt; t) may allow for seasonalities in the di¤usion, otherwise wemay choose f(Xt; t) = Xt. The probability qt is the probability that a spike isrealized at time t, provided t > Q(S¿ ). The duration of spike occurred at timet(i), D(St(i)), is assumed to be independent of spike magnitude and price level.

Equation (1) de…nes changes in electricity prices in an additive way. The…rst component represents a di¤usion, the second allows for positive percentualchanges when a new spike happens and the third one models the collapsing ofelectricity prices when a spike breaks down. The last two components a¤ectprices only when the indicator functions Wt or Zt are equal to 1. The formerdepends on the probability that a spike happens, which we assume to be zero ifthe previous spike is still a¤ecting prices, the latter is equal to one only at thetime the previous spike breaks down.

For modeling purposes it is useful to de…ne pt = ¸(t)¢t as the unconditionalprobability of spike arrivals in ¢t and to carry out the simulations using thismeasure. Indeed, we may simply simulate spikes in t > Q(S¿ ) using pt. Anotheradvantage of using pt to simulate spike arrivals is that it is easier to includeseasonal dependence and dependence from previous spike realizations in thespike frequency parameter.

For example, m multiple peridiodicities may be included de…ning the spike

frequency parameter like ¸(t) =mP

i=1!i sin

¡2¼°iti

¢, where time t is expressed

as a fraction of the year and !i and °i are parameters speci…ed to match peri-odicities. Alternatively, dependence on the previous n realizations of the pure

spike process can be introduced de…ning ¸(t) = ¸+nP

k=1µt¸(t¡k)I(t¡k), where

0 < µ < 1 and I(j ¡ k) is an indicator function which takes value 1 if a spikehas happened at time t ¡ k, and zero otherwise.

3

Another possible extension of our formulation regards the way the electricityprice reverts to the di¤usion process. If we call t(i) the time of the i-th spike, itwould be not di¢cult to model the decline of the electricity price over the periodD(St(i)) on a smoothed fashion rather than considering an istantaneous adjust-ment toward the di¤usion process after Q(St(i)). Anyway, in the remaining ofthis paper we consider the simplest case, with pt = ¸¢t.

3 Derivative PricingIn this section we show how it is possible to implement an algorithm that isconsistent with the model described in the previous section. The pricing ofcontracts such as Forwards or Swaps is not very complicate given that we canexpress the price in T of the former under a risk-adjusted measure as

FT = E£E(N¡1)¢t + ¢f(XN¢t; N¢t) + WN¢tSN¢tE(N¡1)¢t ¡ ZN¢tS¿E(N¡1)¢t

¤

where T = N¢t. The price of the latter in 0 is

SW0 =MX

i=1

(FTi ¡ L)e¡riTi

where ri is the discount rate for maturity Ti and L is the …xed price observed atthe inception of the swap. The current …xed price for maturity M is determinedsolving SW0 = 0.

The pricing of options is not immediate and in the remaining of this sectionwe will show how to implement our model for them. We de…ne options as rightsto either buy or sell a given quantity of energy at the current spot price.

On the basis of the previous considerations and for the sake of clearness welist below the speci…c assumptions used during the implementation.

² The distribution of the spike sizes follows a lognormal distribution. Weassume that the spiky behavior in electricity prices is governed by a ran-dom positive percentage variation in the level of the di¤usion process withmode SM and variance 0:04.

² Spikes happen ad random times but they cannot cumulate. We generatespikes as realizations of a random processes driven by pt’s but we putsome constraints to get a simulation under qt’s. For example, consideringa binomial tree with period ¢t, number of levels N = T=¢t, we may usethe condition that draws from the uniform distribution over the interval(0; 1) be smaller than ¸¢t to simulate N realizations of the variable It,taking value 1 if a spike happens and zero otherwise. Here ¸ serves asspike frequency, that is the expected number of spikes in a year underpt’s. In practice, simulation under qt’s is possible imposing that a non-zero realization of It be considered as a realization of a spike only if ithappens after the previous spike is broken down.

² Spike duration is independent of spike frequency. We use an exponentialrandom variable with parameter µ to draw spike durations, D(St). We settime in terms of years, i.e. µ = 0:01 implies an expected duration of thespike is about 3 days.

4

The pricing is based on a three-step procedure. Let ¢t = T=N , where N isthe number of intervals in which we discretize the time to maturity T :

1. In the …rst step we build a tree based on the di¤usion process with pa-rameter ¹ and ¾;

2. In the second we simulate 1,000 pathways. Each pathway includes N spikerealizations along the tree according to

It =½

1 if ut < ¸ £ ¢t0 otherwise

where ut » U(0; 1). For each time interval in which a spike arrives wedraw the percentage change in Xt, St, from a lognormal distribution withmode SM and variance 0:04. At the same time the duration of each spikeoccurrence is drawn from an exponential distribution with parameter µ.The process with spikes is obtained multiplying the nodes of the tree fromt(i) to T (St(i)) by 1+St(i) for the i-th spike, being careful to avoid multiplespikes in the same periods. An example is shown in …gure (2).

3. The last step requires the computation of the option price along each ofthe 1,000 trees.We then take their mean as a way to integrate over thedi¤erent spike process realizations.

European options prices are in‡uenced by the spike e¤ect only if a spikearrives or is not yet broken down in the last period. The prices on each tree canbe computed by the binomial method as the discounted option payo¤s weightedfor the risk neutral probabilities, given the strike price K. The average priceacross trees is the option price.

The American option case is computationally more intensive because it re-quire to go forward and backward along 1,000 trees. To speed up the optionvaluation procedure a su¢cient condition for option exercise can be used. Infact it can be shown that if

Ete¡¾p

¢t(N¢t¡t) ¡ K > 0 (2)

then it is rational to exercise the American call option in t, provided that noother spike happens later. Obviously when this condition applies we do not needto take into account nodes after t. The computational gain is greater especiallyif spikes are huge and rare, time to maturity is low and the di¤usion coe¢cientis small. In the section related to simulation results we show both the pricingusing this condition and the usual binomial tree pricing.

4 Simulation resultsFollowing the model just described we compute American call option pricesassuming that the price of Electricity is 100$ at time 0 and the risk free rateis 5%. In Table (1) we report American option prices under the usual Black-Scholes di¤usion model for comparison. In Tables (3) - (14) we have computedderivative prices considering di¤erent values for the time to maturity (1 yearand 6 months), strike (80, 100, 120), volatility (20% and 35%), spike magnitude

5

K = 80 K = 100 K = 120T = 1:0 ¾ = 0:20 N = 100 20:6834 7:6466 2:0647T = 1:0 ¾ = 0:20 N = 500 20:6797 7:6594 2:0546T = 1:0 ¾ = 0:35 N = 100 24:1465 13:3365 7:0071T = 1:0 ¾ = 0:35 N = 500 24:1413 13:3588 6:9810T = 0:5 ¾ = 0:20 N = 100 20:1149 5:5126 0:7068T = 0:5 ¾ = 0:20 N = 500 20:1168 5:5224 0:7046T = 0:5 ¾ = 0:35 N = 100 21:8756 9:6306 3:5556T = 0:5 ¾ = 0:35 N = 500 21:8630 9:6478 3:5445

Table 1: American call option prices obtained by standard binomial pricing forsome values of T , ¾, N and K. The starting value of the underlying price is 100and it is assumed to follow a simple di¤usion.

(lognormal mode equal to 10% and 50%), frequency (1 and 3 expected numberof spikes each year) and duration (0.01 and 0.03 years). The last three columnsof the tables shows the call price computed using the su¢cient condition, C0, theusual backward recursion on the binomial tree, C0 and a linear approximationC¤

0 .The number of periods in which we have divided the time to maturity is not

really important for pricing. In practice what really matter about the choiceof time steps is to avoid that several spikes with duration lower then ¢t beexcluded while valuation is carried out. In a more general setting, we may set¢t for each replication only after having determined the shortest duration ofspikes for a given replication.

At …rst look, prices obtained under our model are much higher then thosereported in Table (1). The di¤erence is due to the inclusion of the spike processand the di¤erences are particularly sensible to the degree of moneyness, time tomaturity, ¸ and SM . The di¤usion coe¢cient ¾ has little importance becausethe risk it represents is overwhelmed by those implied by the spiky behavior ofthe electricity prices.

Considering the prices obtained including the condition in Equation (2) itappears that these are close to C0 when we refer to in-the-money options andthe bias increases as long as moneyness decreases, reaching the highest levelwhen we are pricing out-of-the-money options, spikes are big and relativelyfrequent (¸ = 3). In these cases C0 is however signi…cantly greater than thecorresponding European value in Table (1) because of the possibility of spikesoccurring before the last period. That is di¤erent from the classical result in…nancial markets, that out-of-the money American options have values close tocorresponding European options.

In the tables reporting pricing results there is evidence that the strategyimplied in Equation (2) sometimes leads to a signi…cant loss of value on earlyexercise. Therefore Equation (2) is not always adequate for pricing options andwe need to correct the price in these cases. We have seen that our approximationperforms very well when T , ¸ and SM are small. In order to …nd a correctionfor this bias in all the other cases and maintaining the gain in computationale¢ciency using (2), we have expressed the bias standardized by the strike priceas a function of the spike intensity, the product of the frequency parameter and

6

t ¡ Statistic P ¡ values^̄

1 0:016370 0:004893 0:00111^̄

2 0:023631 0:000529 2:96E ¡ 45^̄

3 ¡0:018936 0:003423 5:65E ¡ 13

R2 95%F ¡ statistic 1022:514 0:000000

AIC ¡5:780151

Table 2: Estimation result from model (3).

time to maturity and the moneyness degree. The simple linear modelµ

C0 ¡ C0

K

¶

i= ¯1SM;i + ¯2Ti¸i + ¯3

µE0

K

¶

i+ ui (3)

has been estimated for pricing settings with SM ¸ 0:5 and N = 100. Results inTable (2) show that this simple speci…cation explains well the bias and it hasthe feature to be quite general to avoid over…tting, which is crucial to …nd agood correction. Now, using

C¤0 = C0 + K

µ^̄

1SM + ^̄2T¸ + ^̄

3E0

K

¶(4)

we have corrected C0 and obtained interesting results for several combina-tions of the parameters1 . The Mean Absolute Error is 1:19$ when SM = 0:5and 1:21$ when SM = 1 against 4:71$ and 6:26$ obtained without using the cor-rection2 . The highest errors (4$) occur when spikes are very frequent (¸ = 5).In our opinion this residual bias is due to the linear form of the correction in(4). The complete backward recursion should be used in these cases. Detailedresults are shown in Tables (3) - (14).

5 Concluding RemarksWe present a procedure to price derivatives written on electricity. The appli-cation proposed in detail is the pricing of American call options, assuming aconstant spike frequency, lognormally distributed spike magnitude and expo-nentially distributed spike durations. The model can be generalized consideringtime dependent parameters and include seasonalities and even multiple period-icities as we have exempli…ed for ¸.

Simulations results have shown that prices obtained under our algorithm arevery di¤erent from those obtained by standard option pricing models. Moreoverusing the condition in Equation (2) the time spent in computations can bereduced. This condition works almost perfectly if no other spike occurs after

1 In testing the correction we have considered combinations of the following parameters:¸ = 0:5; 1; 3; 5, T = 0:25; 0:5; 1; 1:5, SM = 0:5; 1,and K = 100; 105; 115; 120

2 We have we have not considered in-the-money options because their inclusion would havebiased toward zero the MAE estimate.

7

ITM SM = 0:1 C0 C0 C¤0

T = 0:5 µ = 0:01 ¸ = 1 ¾ = 0:20 N = 100 29:8750 29:9848 29:0576T = 0:5 µ = 0:01 ¸ = 1 ¾ = 0:20 N = 500 29:5986 29:7658 28:7812T = 0:5 µ = 0:01 ¸ = 1 ¾ = 0:35 N = 100 30:5422 30:7317 29:7248T = 0:5 µ = 0:01 ¸ = 1 ¾ = 0:35 N = 500 30:1514 30:3610 29:3340T = 0:5 µ = 0:01 ¸ = 3 ¾ = 0:20 N = 100 37:6916 38:3850 38:7647T = 0:5 µ = 0:01 ¸ = 3 ¾ = 0:20 N = 500 36:9241 38:0500 37:9972T = 0:5 µ = 0:01 ¸ = 3 ¾ = 0:35 N = 100 37:8888 38:9504 38:9619T = 0:5 µ = 0:01 ¸ = 3 ¾ = 0:35 N = 500 37:7231 38:9613 38:7962T = 0:5 µ = 0:03 ¸ = 1 ¾ = 0:20 N = 100 29:5485 29:6705 28:7311T = 0:5 µ = 0:03 ¸ = 1 ¾ = 0:20 N = 500 29:9932 30:1945 29:1758T = 0:5 µ = 0:03 ¸ = 1 ¾ = 0:35 N = 100 30:5383 30:6589 29:7209T = 0:5 µ = 0:03 ¸ = 1 ¾ = 0:35 N = 500 30:2876 30:5165 29:4702T = 0:5 µ = 0:03 ¸ = 3 ¾ = 0:20 N = 100 37:7446 38:3995 38:8177T = 0:5 µ = 0:03 ¸ = 3 ¾ = 0:20 N = 500 37:1316 38:1575 38:2047T = 0:5 µ = 0:03 ¸ = 3 ¾ = 0:35 N = 100 37:8791 38:8375 38:9522T = 0:5 µ = 0:03 ¸ = 3 ¾ = 0:35 N = 500 37:0842 38:2996 38:1573

Table 3: American call option prices obtained by the algorithm using 1,000realizations of a spike process for some values of T , µ, ¸, ¾, N . The mode ofthe distribution of spike intensities is SM and the strike is 80.

ITM SM = 0:1 C0 C0 C¤0



8

ITM SM = 0:5 C0 C0 C¤0



ITM SM = 0:5 C0 C0 C¤0



9

ATM SM = 0:1 C0 C0 C¤0



ATM SM = 0:1 C0 C0 C¤0



10

ATM SM = 0:5 C0 C0 C¤0



ATM SM = 0:5 C0 C0 C¤0



11

OTM SM = 0:1 C0 C0 C¤0



OTM SM = 0:1 C0 C0 C¤0



12

OTM SM = 0:5 C0 C0 C¤0



OTM SM = 0:5 C0 C0 C¤0



13

the time of exercise. In practice we cannot forecast if the spike which veri…esEquation (2) is really the last one before the maturity. Therefore we need tocorrect prices obtained under this condition using Equation (4) when spikes arefrequent, their intensity is high and option time to maturity is long. Howeverprices from model (1) still remain valid and further studies should be addressedtoward improving computational aspects.

It is clear from the results that the most relevant determinants of the optionprice level are the parameters of the spike process. It would be useful to enrichthe model in this direction, adding seasonalities and stylized facts to this compo-nent. Moreover, our pricing models have been computed under the risk-neutralmeasure. The existence of this measure is discussed in the appendix but therelationship between the physical and the risk-neutral measures is an importanttopic for future research. We believe that spike risk is largely idiosyncratic andthis justi…es the binomial pricing method as advocated by Merton (Merton R.,1992), but only long series of market data may cast light on this issue.

6 AppendixLet F be the forward price for delivery at T , where T is any relevant time.De…ne F ¤ as the forward price of a contract that delivers electricity if no spikeis in progress at T . If a spike is in progress F ¤ delivers against an additionalpayment equal to the spike intensity3 . Let M be the expected intensity of thespike at T . De…ne

¼ =(F ¡ F ¤)

Mthen a risk-neutral measure exists if and only if 0 < ¼ < 1.

PROOF: From the de…nition of ¼.

F ¤ = ¼M + (1 ¡ ¼)0 +Z

Xd eP

where X is the spot price at time T and eP is the risk neutral measure for thedi¤usion process without spikes.

3 Capped forwards, traded in the United States, have prices similar to F¤.

14

SM = 0:1 SM = 0:5K = 120 K = 100 K = 120 K = 100

T = 0:5 µ = 0:01 ¸ = 1 ¾ = 0:20 1:1375 7:0594 1:9645 7:3608T = 0:5 µ = 0:01 ¸ = 1 ¾ = 0:35 4:3546 11:2197 4:7711 11:4557T = 0:5 µ = 0:01 ¸ = 3 ¾ = 0:20 1:3416 7:3741 2:3670 8:1194T = 0:5 µ = 0:01 ¸ = 3 ¾ = 0:35 4:5267 11:6207 5:4795 12:8859T = 0:5 µ = 0:03 ¸ = 1 ¾ = 0:20 1:2208 7:4976 2:0696 8:8299T = 0:5 µ = 0:03 ¸ = 1 ¾ = 0:35 4:6356 11:5720 5:4623 12:6670T = 0:5 µ = 0:03 ¸ = 3 ¾ = 0:20 1:6977 8:1432 4:9431 11:9138T = 0:5 µ = 0:03 ¸ = 3 ¾ = 0:35 4:9571 12:0864 7:9667 15:0492T = 1:0 µ = 0:01 ¸ = 1 ¾ = 0:20 3:3493 10:5975 3:7466 11:2411T = 1:0 µ = 0:01 ¸ = 1 ¾ = 0:35 9:0318 16:2801 9:6239 17:0490T = 1:0 µ = 0:01 ¸ = 3 ¾ = 0:20 3:6694 11:1631 5:2833 13:1754T = 1:0 µ = 0:01 ¸ = 3 ¾ = 0:35 9:3584 16:7175 10:6889 17:9083T = 1:0 µ = 0:03 ¸ = 1 ¾ = 0:20 3:6148 11:0441 5:0901 12:4233T = 1:0 µ = 0:03 ¸ = 1 ¾ = 0:35 9:2296 16:5490 10:6972 17:9176T = 1:0 µ = 0:03 ¸ = 3 ¾ = 0:20 4:0952 11:6818 7:8177 14:4743T = 1:0 µ = 0:03 ¸ = 3 ¾ = 0:35 9:6601 17:4878 12:6447 20:9894

Table 15: European call option prices obtained by the algorithm using 1,000realizations of a spike process for some values of T , µ, ¸, ¾. The mode of thedistribution of spike intensities is SM and N = 100.

7 Bibliography

References[1] H. Geman and O. Vasiceck (2001) Forward and future contracts on non

storable commodities: the case of electricity, Preprint.

[2] H. Geman and A. Roncoroni (2001) A class of marked point processes formodeling electricity prices, ESSEC working paper.

[3] L. Julio and E. Schwartz (2002) Electricity prices and power derivatives:evidence from the Nordic Power Exchange, Review of Derivatives Research5, 5-50.

[4] R. Merton (1992), Continuous time …nance, Blackwell, Cambridge, USA.

15

Figure 2: The …gure show three examples of spike processes realizations overbinomial trees. The number of levels a¤ected by the spike is random, as well asthe intensity of the spike magnitude.

16

electricity derivatives

Economy & Finance