Introduction to WeatherDerivative PricingSTEPHEN JEWSON

STEPHEN JEWSON

is Director. Weather Risk,at Kisk Maiiai^L-iiieiK

Solutions ill London.

X @stephenjewson.i'om Over the last seven ye;irs weatherderivatives have emerged as anattractive new asset class, iincor-related with almost al! other kinds

ot iiivestniL-nt. and .1 number otlnsurancc com-panies, reinsurance companies, banks, hedgefunds, and energy companies have set upweather trading desks. The weather derivativeportfolios they hold might contain a few hun-dred contracts, and contract sizes (in terms ot̂the maximum payout) typically range from atew hundred thousand to tens of millions ofUS. dollars.

Trading strategies vary from company tocompany, and weather derivatives can be usedto create profitable investment portfolios in anuiiibcr ot wnys. For example:

• A diversified portfolio of wcatlicr deriv-atives can give gt><>d return for very lowrisk because ot the many different anduncorrclatcd weather nidices on wbichweather derivatives are based.

• A portfolio of weather derivative andcommodity trades can be high return butlow risk because of the correlationsbetween weather and commodiry prices.

• A portfolio of more standard investments(such as stocks and bonds) that containsa small number of weather derivatives cangive a lower risk than a portfolio of stan-dard investments alone because of the lackof correlation between the weather deriv-atives and tbc wider fuiaucial markets.

For those who trade and invest in weatberderivatives the methods used for pricing andrisk management are very important and thisarticle gives an overview of the pricing and risktnanagement methods that are used in industry.Also at tbe end of the article we briefly men-tion some pricing methods that bave been sug-gested in the academic literature.

We start witb the simplest case which istbe pricing ot weatber swaps. We then discusstbe pricing of weather options, and argue thatboth actuarial and arbitrage pricing can be rel-evant. We describe briefly the most importantaspects of portfolio modeling and risk man-agement. Finally we mention some otherimportant topics sucb as d.iily modeling oftemperature and seasonal forecasts.

PRICING OF WEATHER SWAPS

We start our discussion of tbe pricing ofW'-eatber deri\atives with tbe simplest case,which is the pricing ot weather swaps. Sucbcontracts are, technically speaking, either for-ward or future contracts. They are tradedwithout a premium and have a payoff that islinearly dependent on some weather index.Prices are quoted in terms of the strike levelof the index. Swaps as forwards (which aremostly capped, and so are not strictly linear)are traded OT( ' for a very wide range of loca-tions and indices. Swaps as futures (which don'thave caps) are traded 011 the Chicago Mer-cantile Exchange for monthly and seasonal

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contracts on 21 locations: 15 in the U.S., 5 in Europe,and 1 in Japan. All the exchange traded swaps are basedon daily tern pe rat tires, btit the daily temperatures are con-verted into the monthly or seasonal settlement index indifferent ways in different regions. In the U.S.. daily wintertemperatures are first converted into daily heating degreedays (a measure of cold based on the size of temperatureexcursions below a baseline) and daily summer temper-atures are converted into daily cooling degree days (a mea-sure of heat based on the size of temperature excursionsabove a baseline). Both heating and cooling degree daysare then summed across the period of the weather con-tract to create the settlement index. In Europe, daily wintertemperatures are converted into heating degree days inthe same way while daily summer temperatures aresummed across the period of the contract directly. In Japanthe monthly or seasonal index is formed as the averagetemperature across the period of the contract.

Actuarial Pricing

How should swap contracts be priced, i.e., howshould the strike level be determined? We first consideractuarial pricing. For many OTC weather contracts thisis the only possible method tor pricing since observablemarkets don't exist.

Actuarial pricing consists ot considering appropriatehistorical meteorological data and meteorological forecastsand using this data to derive a prediction for the distribu-tion of outcomes for the settlement index. The tirst stageof such an analysis is that the historical data has to be cleanedand corrected because of the presence of gaps {due to fail-ures in measuring equipment and data transmission sys-tems) and discontinuities (due to station changes): we don'tdiscuss these cleaning and correction processes in detail,but refer the reader to Boissonnade et al. [2()()2|.

Even after such cleaning and correction meteoro-logical data is generally not stationary but contains trendsdue to climate change and the urbanization ot weather sta-tions over time. These etTects mean that the weather overthe last 50 years is not a good mdication of the futureweather and the distribution of likely settlement levels ota weather contract. This problem can be tackled in twoways: either one can use only a short period of recentdata, such as 10 years, and assume that over this shortperiod the non-stationarit\' is small enough to ignore, orone can use a longer period ot data, such as 30 or 50years, and attempt to model the trend. Both these methodsare commoiily used in practice: the former has the advan-

tage that it is simple, but the predictions it gives are biased(because of the trend) and estimates ot probabilities in thetail of the distribution are very poor. The latter methodhas the advantage that the bias is lower and the tails otthe distribution are better estimated but the disadvantagethat the variance of errors is typically greater because ofparameter uncertainty on the parameters in the trendmodel. These issues, along with some empirical com-parisons of different detrending methods applied to U.S.temperature data, are described in detail in Jewson andBrix |20O4| and Jewson [20(Mh|.

The main goal of actuarial pricing tor swaps is toestablish the expectation ot the settle me nt index, whichis often called the tair strike. It a contract is traded manytimes at the fair strike then neither part\' will wm or loseon average. One can estimate this tair strike simply as theaverage of the (possibly detrended) historical values forthe settlement index. Because of the trends and the lim-ited number of years of data there is significant uncer-tainty about these estimates: simple methods for estimatingthe size of pricing uncertainty for both swaps and optionsarc given in Jewson [2003t].

Market Pricing

We ncnv consider cases in which there is an observ-able market tor the weather swap. For certain locations,particularly London, Chicago, and New York, one mightconsider that the weather swap market trades frequentlyenough (currently several trades per day) that this marketis likely to be reasonably ctticicnt as a mechanism tor tairstrike discovery. For valuation of currently held positionsthese market strikes can then be used instead ot results otaT) actuarial analysis. One can also compare these marketprices with the results of actuarial analysis: in most casesone tinds that the market prices lie within the range otpossible values given by the actuarial analysis and onlyoccasionally do the prices tor commonly traded contractsmove out of this range, presumably because of supply anddemand imbalance.

As a weather swap contract progresses toward expiry,market price inevitably converges onto tundaniental priceand the two are equal at expiry.

The Role of Forecasts

Immediately bctorc and during the measurementperiod tor a swap contract weather torccasts play an impor-tant role in helping predict the likely outcome ot the con-

58 INTRODUCTION TO WEATHEK DEKIVATIVE I'IUCING 2UO4

tract, and the movements in the prices ot commonly tradedcontracts arc almost entirely driven by changes in weatherforecasts. Weather forecasts can be incorporated into esti-mates of the fair strike of most weather swap contracts ina very simple fashion. The starting point is to ensure tliatthe forecast being used represents tbe expcctatioii of futuretcmperatttres. Tins is not always the case for commerciallyavailable forecasts and methtxls to cnstire tbat a particularweather forecast really does represent an expectation aregiven in Jew^son [2002]. Then, for a contract based on thesum of temperatures over the contract period, or a con-tract on degree diys for whicli there is no cbance of crossingthe baseline, tbe results from tbe weatber forecast cansimply be added to an actuarial analysis for the remainderof the contract. Other cases, such as contracts based ondegree days in situations where there is some cbance ot thetemperature crossing the baseline, are more complex andshould be priced using probabilistic forecasts of future tem-peratures. Methods for making such forecasts from thesingle and ensemble meteorological forecasts available com-mercially are discussed in Jewson [2004c|. How proba-bilistic forecasts should be used in pricing is discussed injewson and Caballero |2(H)3b|.

PRICING OF WEATHER OPTIONS

We now move on to discuss the pricing of weather(3ptit]ns, which are contracts with a non-linear ratbcr thanlinear payofF, and whicb are exchanged tor a premium.Tbe hrst case we consider is that of a weather option ona location for which there is no weather swap market.Tbis would be the case for most OTC weather optionsstructured for end users witb weather risk. In tbis casethe only pricing method available is to perform an actu-arial analysis based on historical meteorological data. Theanalysis is slightly more complicated than that for swapcontracts since one is now interested in the distributionot possible values for the settlement index, rather thanjust the expected value. Tbe concept of primary interestis the expected payoff of the option (with expectations cal-culated under tbe objective measure), and this is oftenreferred to as tbe fair price of tbe option. An option con-tract traded many times witb the premium set at this fanprice will neither make nor lose money on average.

The simplest way to estimate this expected payoffuses so-called burn analysis in which we calculate how tbeoption would bave performed in previous years. A slightlymore complex approach involves fitting a distribution to

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the historica] mdex v;ilues ;ind c;ilculating thte expectedpayoff from this distribLitit)n. Botli of these methods ]iavebeen in use since the begnimngs of the weather marlcet,although the earliest reference we have been ab]e to fnidis Goldman Sachs [1999], f-ittmg a distribution may pos-sibly give slightly more accurate estimates of the fair pre-mium in some cases, and ma\' well give more accurateestimates of the greeks and the risk of extreme events {seeJevvson [2003e] for a quantitative comparison betweenthe burn and index modehng approaches), hi the case ofthe normal and kernel distributions, a num[ier ot closed-torm expressions have been puhhshod (see Mclntyre[1999] and Henderson [2(.)()2[ for inciividual examples,and Jewson |2()()3a, 2(H)3b, 2(K)3d| for a comprehensivelist of results tor all common contract types). Normal dis-tributions are generally appropriate tor seasonal contracts,but not necessarily lor standard monthly contracts, and cer-tainly not tor exotic contracts based on the number otextreme weather events such as tlie number ot treczingdays overwinter In jewson [2()l)4t] we give an ana]ysisot tlie etiectiveness ot t]ie norma] distribution tor seasona]and mont[i]\- indices based on U.S, temperatures. For dis-tributions otiier tlun tlie norma] or kernel closed-tormsolutions may also be possible, and tor all distributionsMonte Carlo simulations or numerical integration can beused to calculate tlie fair price and other diagnostics.

The amount of liistorical data available does not pindown the distribution ot the settlement index very pre-cisely, and tiiere are often several distributions that cannotbe rejected using statistical testing. For options with strikesnear the mean and when calculating the expected payotl̂one can choose any of these distributions tor pricing sincethe choice of distribution does not have a major etfect onthe results: the choice of number of years or trend is muchmore important. For options with strikes tar trom themean, the choice of distribution becomes more important,however (see jewson |2(K)4e| fora detailed comparison otthe relative importance ot trends and distributions).

Having calculated the expected payotl ot an option.the seller may wish to add a premium to compensate forthe unhedgeable risk being taken on. Such a premiumwould usually be calcu]ated to be proportiona] to somemeasure of risk; e,g,, one might add 30% of the standarddeviation ot the payotfs. Closed-torm expressions tor thestandard deviation ot the payotTs of options on normaland kernel distributions are given in jewson [2003cJ. Thestandard deviation of payoffs for other distributions canbe calculated using Monte Carlo methods.

Use of Weather Forecasts

The use ot weather torecasts in the pricing of weatheroptions is more complex than tor tiie case ot swaps. Forillustration we consider a case in which the index distri-bution is norma! and is hence entirely specified by themean and the standard deviation. The mean can be cal-culated easily as tor swaps, [lut the calculation ot the stan-dard deviation is ditHcult. In jewson and C l̂aballero [2()03b|we describe three rather different methods that can beused to estimate the standard deviation:

• a simple but rather approximate metht)d based onhistorical meteorological data;

• a very complex bottom-up approach based on prob-abilistic forecasts of temperature and daily temper-ature simulation models (which we call "pruning");and

• a straightforward approach based on the use ofBrownian motion to model the expected index.

The third approach involves specifying a volatilitymodel tor the expected index. As an example of sucli amodel, the seasonal trapezium volatility" model that wedescribe in jewson [20()3h| captures both the overlappingeffects of weather forecasts at the start and ends of a con-tract and also the varying volatility' ot weather at ditlerenttimes ot year.

Market Based Pricing of Weather Options

We now consider the pricing ot weather options int!ie case in which there is a liquid market tor the weatherswap. In this case one can consider hedging the optionusing the swap, and under the simplest set of assumptions(that the market is balanced and the swap price is equalto the expected settlement index) the resulting model isa modified version of the Black [ 1976| model for pricingoptions on forwards (see jewson and Zervos [2003a]). Inthe Black model the underlying process is given by a geo-metric Brownian motion with drift while tor the weatherswap it is more appropriate to use an arithmetic Brownianmotion with no dritt (see ]ewson [2002] tor a detailedexplanation ot this). The arbitrage price one derives fromthis model is identical to the actuarially derived ("air priceand the risk neutral measure is the same as the naturalmeasure, unlike in the Black and Black-Scholes models.The basic weather version of the Black model can also beextended by including a dritt in the swap price to repre-

60 IMROIILHIIIO.N lO WtMHtR DtKlVAIiVL FALL 2004

sent an unbalanced market (see Jewson and Zervos|2()()3b|). In this case the arbitrage price is no longerexactly the actuarial fair price.

In reality, weather swap markets are not particularlyliquid and the abo\-e models are not particularly well jus-tified. depeudiiiL; as they do on assumptions of'iufiniteliquidity and zero transaction costs.

However, some hedging of" options witli swaps iscertainly possible. The optimum number of hedgesdepends on the liquidity of the swap market, the level oftransaction costs, and the risk aversion of the trader. Weha\-e explored a simple model of these effects, and theoption prices that result, injewson [2O()3i|.

When pricing options on an index for which theswap is commonly traded it is common practice to use theswap level instead of au actuarial estimate for the expectedindex. This can be considered as 1) actuarial pricing, withthe assumption that the swap is a good predictor for theexpected index, or 2) arbitrage pricing, where one is plan-ning to continuously hedge the option with the swap, or3) an approximation to the adjustment one should maketo the option price to take liitt) account the cost of hedgingwith just oue smgle swap transaction.

PORTFOLIO MANAGEMENT

The lack of correlation between different weathervariables, weather at different times, and weather in dif-ferent parts of the world allows the creation of very lowrisk portfolios of weather contracts, and there is consid-erable interest from market makers in trying to originatea wide variety of different trades for this reason.

Modeling a portfolio to estimate the distribution ofpossible outcomes can be performed by fitting marginaldistributions to each contract, estimating rank correla-tions between the indices, and simulating using the well-known simulation method of Iman and Conover |]'^J82].rile application of this model to weather derivatives wasfirst described by Jewson and Brix |2(KK)] but may havebeen in use m the weather derivative industry before then.

There are a luiniber of aspects to the managementof a weather portfolio, such as:

1. Jdentifyitiq and hed^in^ii the major sources of riskill the ciirreiil portfolio: A simple way to reduce the riskin a portfolit) is to hedge using the liquidly traded swapcontracts. The optimum (variance minimizing) size ofthe hedge is given by (minus one times) the regressioncoefficient (or beta) between the payofTs of the portfolio.\ud the index of the swap.

This can be calculated either using simulations or,for the normal distribution at least, analytically (see JewsonJ2()l)4a]). Since the more liquidly traded swap contractscan be traded at or close to fair price it may make senseto attempt beta neutrality with respect to these inciiceswith fairly frequent reliedging.

Other defniitions of optinium other than varianceminimizing can also be considered, and lead to differentsizes of hedge (seejewson |2()()4b|).

2, Pricing against the portfolio: If one cares aboutboth risk and return then all new contracts should ide-ally be priced in a way that rejects the impact tiiey willhave on both the risk and return of the portfolio. Therearc a number of f'rameworks that can be used for this suchas traditional mean-variance analysis (see Markowitz[ l'-^59|), risk-adjusted retui-n analysis (such as Sharpe ratiobased analysis), or stochastic dominance thet)ry (see Heyer[2001]). From an academic perspective one might alsoconsider utility theory, but this does not seem to be usedin practice.

RISK MANAGEMENT

There are a number of sources of risk in a weatherportfolio. The most fundamental is the weather itself: atypical portfolio will have a range of possible outcomesthat depends on the weather, and if the weather mo\'esagainst the trader the trader can lose money. This risk istypically measured in two ways. The first, and most usefiil,is the so-called expiry VaR in which one considers holdingall the contracts to expiry and calculating the lower quan-tiles of the distribution of profit and loss. These quanti-ties can be estimated using the simulation methods forportfolios that have been described above. The second isthe so-called horizon VaR, which is more similar to thestandard definition of VaR since it measures the risk ofloss over a specific time horizon. Horizon VaR for weatherportfolios can be calculated on an actLiarial basis or amarket price basis. When considered on an actuarial basishorizon VaR is defined as one of the lower quantiles ofthe distribution of changes in the expected payoff fromthe portfolio over a fixed time Iiorizon. There are anumber of ways this can be calculated: a fully detailedcalculation is extremely complex, but a simplified calcu-lation based on the martingale nattire of expected weatherindices is much simpler (seejewson [2003g|). When con-sidered on a market basis horizon VaR is defined as thepossible change in market valtie of a weather contract orportfolio ot contracts o\-er a specified time horizon. This

2tK)4 Ol ALIkRNAMVh ilMVESTMENTS 6 1

is generally very hard to estimate given the limited amountof weather market price data available.

If contracts are being traded in and out, rather thanjust held to expiry, then there is the potential of marketrisk: even if the weather is favorable, one may not be ableto trade out of a contract at a reasonable price. Again thiscan be difficult to evaluate since it involves understandingthe variations in liquidity in the market for which littledata is available.

Other sources of risk that one may consider are li-quidity risk (the risk that a portfolio that is performingwell may suffer a short-term liquidity crisis because ofthe staggered timing of the payments from different con-tracts), and credit risk (the risk that a counterparty maygo bankrupt).

FURTHER TOPICS

Daily Modeling

There has been considerable academic and somepractitioner interest in the idea that actuarial pricing ofweather options could be performed using simulationmodels for daily temperatures (see for instance Dischel[l99Ha], Cao and Wei |200()|, Dormer and Querel [2()00|,Moreno [2000], Moreno and Roustant, [2002), Torro etal. [2001], Davis [2001], Alaton et al. [2002], Caballeroet al. 12002]. Brody et al. [2002]. and Jewson and Caballero[2003a]).

Modeling daily temperatures could, in principle,lead to more accurate pricing, especially for short con-tracts (as shown injewson [2004g|), but is a difficult sta-tistical problem. The difTiculty arises because observedtemperatures often sliovv seasonalit\' in all of the mean,variance, distribution, and autocorrelations, and longmemory in the autocorrelations. The risk with daily mod-eling is that small mis-specifications in the models canlead to extreme mispricing of contracts. Many of the pub-lished articles on the subject, for instance, consider verysimple models for daily temperature that give prices thatare much less accurate than the more sti'aightfonvard burnand index modeling methods described above. A typicalproblem is that daily models underestimate the autocor-relation of temperature fluctuations about the seasonalcycle and hence underestimate the standard deviation ofthe settlement index.

Seasonal Forecasts and El Nino

El Nino and La Niiia have significant effects on theU.S. climate, and since they can be predicted a few monthsin advance it is important to consider their impact on thefair price of options and swaps (see Dischel [1998b] andDischel |2l)00|). However, litde work has been publishedon how to model the efTects of El Nino on the distribu-tion of temperatures at individual stations. Our own inves-tigations into this subject are described injewson [20(l4dj.

Other Variables

Our discussion thus far has focused entirely on tem-perature. However, a small but growing proportion ot theweather market is based on precipitation, wind, and othervariables. Most of the methods described above applyequally well to these other variables, with appropriatemodifications. Discussions of the pricing of contracts basedon precipitation have been given in Moreno [20011.

Pricing Weather-CommodityDual Trigger Contracts

In addition to weather based contracts and com-modity price based cotitracts a number ot contracts witha dual strike based on weather and commodity' prices havebeen traded in both the U.S. and Europe. The only articlewe know of that addresses the question of how to pricesuch deals is Carmona and Villani [2003].

Alternative Pricing Paradigms

A number of alternative pricing paradigms for weatherderivatives have been suggested in the academic literature.Examples are the use of CAPM-style equilibrium models(Cao and Wei [2000]) and using relations between theweather and gas or electricity prices (see Geman [1999a]and L̂ avis [2001 ]). As far as the author is aware none of theseideas have been used in practice at this point.

CONCLUSIONS

We have reviewed the methods used by practitionersfor the pricing of weather derivatives, in summary, weatherderivative pricing is mostly based on an actuarial analysisof past weather data. Weather forecasts also play a roleimmediately before and during contracts, and seasonalforecasts are important in the U.S. For some of the more

62 INTRODUCTION TO WEATHER DEiuvATivt PRICING FALL 2UII4

liquidly traded contracts, especially monthly and seascjnaltemperature contracts on London. Chicago, and New-York, enough of a market exists that swaps can be valuedsimply by observing the market price and one can attemptto value options using arbitrage pricing based models.

FURTHER READING

CitKid sources of articles on weather derivatives arethe Artemis website at http://www.artemis.bm and theSocial Science Research Network at http://www.ssrn.org.There are also several books that cover general aspects ofthe market and have short sections on pricing, written inEnglish (Geman [1999b]. Element Re [2002], and Dis-chel [2O02[), Japanese (Hirose [2003], Hijikata [1999],and Hijikata [2OO3|). and French (Marteau et al. |2004]).The author has written a book that covers pricing issuesin more detail (Jewson et al. [2004]).

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64 iNTRonucjrON TO WL.ATHER DFRIVATIVL PKICING FALL 2()I)4