Introduction to Weather Pricing

STEPHEN JEWSON

STEPHEN JEWSON ver the last seven ye;irs weather For those who trade and invest in weatber is Director. Weather Risk, derivatives have emerged as an derivatives the methods used for pricing and at Kisk Maiiai^L-iiieiK attractive new asset class, iincor- risk management are very important and this Solutions ill London. related with almost al! other kinds article gives an overview of the pricing and risk X @stephenjewson.i'om oOt iiivestniL-nt. and .1 number otlnsurancc com- tnanagement methods that are used in industry. panies, reinsurance companies, banks, Also at tbe end of the article we briefly men- funds, and energy companies have set up tion some pricing methods that bave been sug- weather trading desks. The weather derivative gested in the academic literature. portfolios they hold might contain a few hun- We start witb the simplest case which is dred contracts, and contract sizes (in terms ot^ tbe pricing ot weatber swaps. We then discuss the maximum payout) typically range from a tbe pricing of weather options, and argue that tew hundred thousand to tens of millions of both actuarial and arbitrage pricing can be rel- US. dollars. evant. We describe briefly the most important Trading strategies vary from company to aspects of portfolio modeling and risk man- company, and weather derivatives can be used agement. Finally we mention some other to create profitable investment portfolios in a important topics sucb as d.iily modeling of nuiiibcr ot wnys. For example: temperature and seasonal forecasts.

• A diversified portfolio of wcatlicr deriv- PRICING OF WEATHER SWAPS atives can give gt><>d return for very low risk because ot the many different and We start our discussion of tbe pricing of uncorrclatcd weather nidices on wbich W'-eatber deri\atives with tbe simplest case, weather derivatives are based. which is the pricing ot weather swaps. Sucb • A portfolio of weather derivative and contracts are, technically speaking, either for- commodity trades can be high return but ward or future contracts. They are traded low risk because of the correlations without a premium and have a payoff that is between weather and commodiry prices. linearly dependent on some weather index. • A portfolio of more standard investments Prices are quoted in terms of the strike level (such as stocks and bonds) that contains of the index. Swaps as forwards (which are a small number of weather derivatives can mostly capped, and so are not strictly linear) give a lower risk than a portfolio of stan- are traded OT(' for a very wide range of loca- dard investments alone because of the lack tions and indices. Swaps as futures (which don't of correlation between the weather deriv- have caps) are traded 011 the Chicago Mer- atives and tbc wider fuiaucial markets. cantile Exchange for monthly and seasonal

FAN 2(KI4 I.W ALTi;it.NATiVE INVESTMENTS 57 contracts on 21 locations: 15 in the U.S., 5 in Europe, tage that it is simple, but the predictions it gives are biased and 1 in Japan. All the exchange traded swaps are based (because of the trend) and estimates ot probabilities in the on daily tern pe rat tires, btit the daily temperatures are con- tail of the distribution are very poor. The latter method verted into the monthly or seasonal settlement index in has the advantage that the bias is lower and the tails ot different ways in different regions. In the U.S.. daily winter the distribution are better estimated but the disadvantage temperatures are first converted into daily heating degree that the variance of errors is typically greater because of days (a measure of cold based on the size of temperature parameter uncertainty on the parameters in the trend excursions below a baseline) and daily summer temper- model. These issues, along with some empirical com- atures are converted into daily cooling degree days (a mea- parisons of different detrending methods applied to U.S. sure of heat based on the size of temperature excursions temperature data, are described in detail in Jewson and above a baseline). Both heating and cooling degree days Brix |20O4| and Jewson [20(Mh|. are then summed across the period of the weather con- The main goal of actuarial pricing tor swaps is to tract to create the settlement index. In Europe, daily winter establish the expectation ot the settle me nt index, which temperatures are converted into heating degree days in is often called the tair strike. It a contract is traded many the same way while daily summer temperatures are times at the fair strike then neither part\' will wm or lose summed across the period of the contract directly. In Japan on average. One can estimate this tair strike simply as the the monthly or seasonal index is formed as the average average of the (possibly detrended) historical values for temperature across the period of the contract. the settlement index. Because of the trends and the lim- ited number of years of data there is significant uncer- Actuarial Pricing tainty about these estimates: simple methods for estimating the size of pricing uncertainty for both swaps and options How should contracts be priced, i.e., how arc given in Jewson [2003t]. should the strike level be determined? We first consider actuarial pricing. For many OTC weather contracts this Market Pricing is the only possible method tor pricing since observable markets don't exist. We ncnv consider cases in which there is an observ- Actuarial pricing consists ot considering appropriate able market tor the weather swap. For certain locations, historical meteorological data and meteorological forecasts particularly London, Chicago, and New York, one might and using this data to derive a prediction for the distribu- consider that the weather swap market trades frequently tion of outcomes for the settlement index. The tirst stage enough (currently several trades per day) that this market of such an analysis is that the historical data has to be cleaned is likely to be reasonably ctticicnt as a mechanism tor tair and corrected because of the presence of gaps {due to fail- strike discovery. For valuation of currently held positions ures in measuring equipment and data transmission sys- these market strikes can then be used instead ot results ot tems) and discontinuities (due to station changes): we don't aT) actuarial analysis. One can also compare these market discuss these cleaning and correction processes in detail, prices with the results of actuarial analysis: in most cases but refer the reader to Boissonnade et al. [2()()2|. one tinds that the market prices lie within the range ot Even after such cleaning and correction meteoro- possible values given by the actuarial analysis and only logical data is generally not stationary but contains trends occasionally do the prices tor commonly traded contracts due to climate change and the urbanization ot weather sta- move out of this range, presumably because of supply and tions over time. These etTects mean that the weather over demand imbalance. the last 50 years is not a good mdication of the future As a weather swap contract progresses toward expiry, weather and the distribution of likely settlement levels ot market price inevitably converges onto tundaniental price a weather contract. This problem can be tackled in two and the two are equal at expiry. ways: either one can use only a short period of recent data, such as 10 years, and assume that over this short The Role of Forecasts period the non-stationarit\' is small enough to ignore, or one can use a longer period ot data, such as 30 or 50 Immediately bctorc and during the measurement years, and attempt to model the trend. Both these methods period tor a swap contract weather torccasts play an impor- are commoiily used in practice: the former has the advan- 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 traded PRICING OF WEATHER OPTIONS contracts arc almost entirely driven by changes in weather forecasts. Weather forecasts can be incorporated into esti- We now move on to discuss the pricing of weather mates of the fair strike of most weather swap contracts in (3ptit]ns, which are contracts with a non-linear ratbcr than a very simple fashion. The starting point is to ensure tliat linear payofF, and whicb are exchanged tor a premium. the forecast being used represents tbe expcctatioii of future Tbe hrst case we consider is that of a weather on tcmperatttres. Tins is not always the case for commercially a location for which there is no weather swap market. available forecasts and methtxls to cnstire tbat a particular Tbis would be the case for most OTC weather options weather forecast really does represent an expectation are structured for end users witb weather risk. In tbis case given in Jew^son [2002]. Then, for a contract based on the the only pricing method available is to perform an actu- sum of temperatures over the contract period, or a con- arial analysis based on historical meteorological data. The tract on degree diys for whicli there is no cbance of crossing analysis is slightly more complicated than that for swap the baseline, tbe results from tbe weatber forecast can contracts since one is now interested in the distribution simply be added to an actuarial analysis for the remainder ot possible values for the settlement index, rather than of the contract. Other cases, such as contracts based on just the expected value. Tbe concept of primary interest degree days in situations where there is some cbance ot the is the expected payoff of the option (with expectations cal- temperature crossing the baseline, are more complex and culated under tbe objective measure), and this is often should be priced using probabilistic forecasts of future tem- referred to as tbe fair price of tbe option. An option con- peratures. Methods for making such forecasts from the tract traded many times witb the premium set at this fan single and ensemble meteorological forecasts available com- price will neither make nor lose money on average. mercially are discussed in Jewson [2004c|. How proba- The simplest way to estimate this expected payoff bilistic forecasts should be used in pricing is discussed in uses so-called burn analysis in which we calculate how tbe jewson and Caballero |2(H)3b|. option would bave performed in previous years. A slightly more complex approach involves fitting a distribution to

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THF.JOL'RNAL Oh Al IhKNM IVL INVESTMENTS 59 the historica] mdex v;ilues ;ind c;ilculating thte expected Use of Weather Forecasts payoff from this distribLitit)n. Botli of these methods ]iave been in use since the begnimngs of the weather marlcet, The use ot weather torecasts in the pricing of weather although the earliest reference we have been ab]e to fnid options is more complex than tor tiie case ot swaps. For is Goldman Sachs [1999], f-ittmg a distribution may pos- illustration we consider a case in which the index distri- sibly give slightly more accurate estimates of the fair pre- bution is norma! and is hence entirely specified by the mium in some cases, and ma\' well give more accurate mean and the standard deviation. The mean can be cal- estimates of the and the risk of extreme events {see culated easily as tor swaps, [lut the calculation ot the stan- Jevvson [2003e] for a quantitative comparison between dard deviation is ditHcult. In jewson and C^laballero [2()03b| the burn and index modehng approaches), hi the case of we describe three rather different methods that can be the normal and kernel distributions, a num[ier ot closed- used to estimate the standard deviation: torm expressions have been puhhshod (see Mclntyre [1999] and Henderson [2(.)()2[ for inciividual examples, • a simple but rather approximate metht)d based on and Jewson |2()()3a, 2(H)3b, 2(K)3d| for a comprehensive historical meteorological data; list of results tor all common contract types). Normal dis- • a very complex bottom-up approach based on prob- tributions are generally appropriate tor seasonal contracts, abilistic forecasts of temperature and daily temper- but not necessarily lor standard monthly contracts, and cer- ature simulation models (which we call "pruning"); tainly not tor exotic contracts based on the number ot and extreme weather events such as tlie number ot treczing • a straightforward approach based on the use of days overwinter In jewson [2()l)4t] we give an ana]ysis Brownian motion to model the expected index. ot tlie etiectiveness ot t]ie norma] distribution tor seasona] and mont[i]\- indices based on U.S, temperatures. For dis- The third approach involves specifying a tributions otiier tlun tlie norma] or kernel closed-torm model tor the expected index. As an example of sucli a solutions may also be possible, and tor all distributions model, the seasonal trapezium volatility" model that we Monte Carlo simulations or numerical integration can be describe in jewson [20()3h| captures both the overlapping used to calculate tlie fair price and other diagnostics. effects of weather forecasts at the start and ends of a con- The amount of liistorical data available does not pin tract and also the varying volatility' ot weather at ditlerent down the distribution ot the settlement index very pre- times ot year. cisely, and tiiere are often several distributions that cannot be rejected using statistical testing. For options with strikes Market Based Pricing of Weather Options near the mean and when calculating the expected payotl^ one can choose any of these distributions tor pricing since We now consider the pricing ot weather options in the choice of distribution does not have a major etfect on t!ie case in which there is a liquid market tor the weather the results: the choice of number of years or trend is much swap. In this case one can consider hedging the option more important. For options with strikes tar trom the using the swap, and under the simplest set of assumptions mean, the choice of distribution becomes more important, (that the market is balanced and the swap price is equal however (see jewson |2(K)4e| fora detailed comparison ot to the expected settlement index) the resulting model is the relative importance ot trends and distributions). a modified version of the Black [ 1976| model for pricing Having calculated the expected payotl ot an option. options on forwards (see jewson and Zervos [2003a]). In the seller may wish to add a premium to compensate for the the underlying process is given by a geo- the unhedgeable risk being taken on. Such a premium metric Brownian motion with drift while tor the weather would usually be calcu]ated to be proportiona] to some swap it is more appropriate to use an arithmetic Brownian measure of risk; e,g,, one might add 30% of the standard motion with no dritt (see ]ewson [2002] tor a detailed deviation ot the payotfs. Closed-torm expressions tor the explanation ot this). The arbitrage price one derives from standard deviation ot the payotTs of options on normal this model is identical to the actuarially derived ("air price and kernel distributions are given in jewson [2003cJ. The and the risk neutral measure is the same as the natural standard deviation of payoffs for other distributions can measure, unlike in the Black and Black-Scholes models. be calculated using Monte Carlo methods. The basic weather version of the Black model can also be extended 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 This can be calculated either using simulations or, |2()()3b|). In this case the arbitrage price is no longer for the normal distribution at least, analytically (see Jewson exactly the actuarial fair price. J2()l)4a]). Since the more liquidly traded swap contracts In reality, weather swap markets are not particularly can be traded at or close to fair price it may make sense liquid and the abo\-e models are not particularly well jus- to attempt beta neutrality with respect to these inciices tified. depeudiiiL; as they do on assumptions of'iufinite with fairly frequent reliedging. liquidity and zero transaction costs. Other defniitions of optinium other than variance However, some hedging of" options witli swaps is minimizing can also be considered, and lead to different certainly possible. The optimum number of hedges sizes of hedge (seejewson |2()()4b|). depends on the liquidity of the swap market, the level of 2, Pricing against the portfolio: If one cares about transaction costs, and the risk aversion of the trader. We both risk and return then all new contracts should ide- ha\-e explored a simple model of these effects, and the ally be priced in a way that rejects the impact tiiey will option prices that result, injewson [2O()3i|. have on both the risk and return of the portfolio. There When pricing options on an index for which the arc a number of f'rameworks that can be used for this such swap is commonly traded it is common practice to use the as traditional mean-variance analysis (see Markowitz swap level instead of au actuarial estimate for the expected [ l'-^59|), risk-adjusted retui-n analysis (such as Sharpe ratio index. This can be considered as 1) actuarial pricing, with based analysis), or stochastic dominance thet)ry (see Heyer the assumption that the swap is a good predictor for the [2001]). From an academic perspective one might also expected index, or 2) arbitrage pricing, where one is plan- consider utility theory, but this does not seem to be used ning to continuously hedge the option with the swap, or in practice. 3) an approximation to the adjustment one should make to the option price to take liitt) account the cost of hedging RISK MANAGEMENT with just oue smgle swap transaction. There are a number of sources of risk in a weather PORTFOLIO MANAGEMENT portfolio. The most fundamental is the weather itself: a typical portfolio will have a range of possible outcomes The lack of correlation between different weather that depends on the weather, and if the weather mo\'es variables, weather at different times, and weather in dif- against the trader the trader can lose money. This risk is ferent parts of the world allows the creation of very low typically measured in two ways. The first, and most usefiil, risk portfolios of weather contracts, and there is consid- is the so-called expiry VaR in which one considers holding erable interest from market makers in trying to originate all the contracts to expiry and calculating the lower quan- a wide variety of different trades for this reason. tiles of the distribution of profit and loss. These quanti- Modeling a portfolio to estimate the distribution of ties can be estimated using the simulation methods for possible outcomes can be performed by fitting marginal portfolios that have been described above. The second is distributions to each contract, estimating rank correla- the so-called horizon VaR, which is more similar to the tions between the indices, and simulating using the well- standard definition of VaR since it measures the risk of known simulation method of Iman and Conover |]'^J82]. loss over a specific time horizon. Horizon VaR for weather rile application of this model to weather derivatives was portfolios can be calculated on an actLiarial basis or a first described by Jewson and Brix |2(KK)] but may have market price basis. When considered on an actuarial basis been in use m the weather derivative industry before then. horizon VaR is defined as one of the lower quantiles of There are a luiniber of aspects to the management the distribution of changes in the expected payoff from of a weather portfolio, such as: the portfolio over a fixed time Iiorizon. There are a 1. Jdentifyitiq and hed^in^ii the major sources of risk number of ways this can be calculated: a fully detailed ill the ciirreiil portfolio: A simple way to reduce the risk calculation is extremely complex, but a simplified calcu- in a portfolit) is to hedge using the liquidly traded swap lation based on the martingale nattire of expected weather contracts. The optimum (variance minimizing) size of indices is much simpler (seejewson [2003g|). When con- the hedge is given by (minus one times) the regression sidered on a market basis horizon VaR is defined as the coefficient (or beta) between the payofTs of the portfolio possible change in market valtie of a weather contract or .\ud the index of the swap. portfolio ot contracts o\-er a specified time horizon. This

2tK)4 Ol ALIkRNAMVh ilMVESTMENTS 61 is generally very hard to estimate given the limited amount Seasonal Forecasts and El Nino of weather market price data available. If contracts are being traded in and out, rather than El Nino and La Niiia have significant effects on the just held to expiry, then there is the potential of market U.S. climate, and since they can be predicted a few months risk: even if the weather is favorable, one may not be able in advance it is important to consider their impact on the to trade out of a contract at a reasonable price. Again this fair price of options and swaps (see Dischel [1998b] and can be difficult to evaluate since it involves understanding Dischel |2l)00|). However, litde work has been published the variations in liquidity in the market for which little on how to model the efTects of El Nino on the distribu- data is available. tion of temperatures at individual stations. Our own inves- Other sources of risk that one may consider are li- tigations into this subject are described injewson [20(l4dj. quidity risk (the risk that a portfolio that is performing well may suffer a short-term liquidity crisis because of Other Variables the staggered timing of the payments from different con- tracts), and credit risk (the risk that a counterparty may Our discussion thus far has focused entirely on tem- go bankrupt). perature. However, a small but growing proportion ot the weather market is based on precipitation, wind, and other variables. Most of the methods described above apply FURTHER TOPICS equally well to these other variables, with appropriate Daily Modeling modifications. Discussions of the pricing of contracts based on precipitation have been given in Moreno [20011. There has been considerable academic and some practitioner interest in the idea that actuarial pricing of Pricing Weather-Commodity weather options could be performed using simulation Dual Trigger Contracts models for daily temperatures (see for instance Dischel [l99Ha], Cao and Wei |200()|, Dormer and Querel [2()00|, In addition to weather based contracts and com- Moreno [2000], Moreno and Roustant, [2002), Torro et modity price based cotitracts a number ot contracts with al. [2001], Davis [2001], Alaton et al. [2002], Caballero a dual strike based on weather and commodity' prices have et al. 12002]. Brody et al. [2002]. and Jewson and Caballero been traded in both the U.S. and Europe. The only article [2003a]). we know of that addresses the question of how to price Modeling daily temperatures could, in principle, such deals is Carmona and Villani [2003]. lead to more accurate pricing, especially for short con- tracts (as shown injewson [2004g|), but is a difficult sta- Alternative Pricing Paradigms tistical problem. The difTiculty arises because observed temperatures often sliovv seasonalit\' in all of the mean, A number of alternative pricing paradigms for weather variance, distribution, and autocorrelations, and long derivatives have been suggested in the academic literature. memory in the autocorrelations. The risk with daily mod- Examples are the use of CAPM-style equilibrium models eling is that small mis-specifications in the models can (Cao and Wei [2000]) and using relations between the lead to extreme mispricing of contracts. Many of the pub- weather and gas or electricity prices (see Geman [1999a] lished articles on the subject, for instance, consider very and L^avis [2001 ]). As far as the author is aware none of these simple models for daily temperature that give prices that ideas have been used in practice at this point. are much less accurate than the more sti'aightfonvard burn and index modeling methods described above. A typical CONCLUSIONS problem is that daily models underestimate the autocor- relation of temperature fluctuations about the seasonal We have reviewed the methods used by practitioners cycle and hence underestimate the standard deviation of for the pricing of weather derivatives, in summary, weather the settlement index. derivative pricing is mostly based on an actuarial analysis of past weather data. Weather forecasts also play a role immediately before and during contracts, and seasonal forecasts 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 seascjnal Dischel. R. •• lilack-Scholcs Won"t Do."" Hi/iTi,')' Power and Risk temperature contracts on London. Chicago, and New- Ma!ia\;viiieiit Heather Risk Special Report, 10 (1998a). York, enough of a market exists that swaps can be valued simply by observing the market price and one can attempt . "Seasonal Forecasts ;md ihc Weather Risk Market."" to value options using arbitrage pricing based models. Applied Derivatives Tradrn'^. 11 (1998b). . "Seasonal Weather Forecasts and Dcriwitive Valua- FURTHER READING tion." Energy Power and Ri>k Manai>enient Weather Risk' Special Report, 8 (2000). CitKid sources of articles on weather derivatives are the Artemis website at http://www.artemis.bm and the Dornicr, F, and M Qucrcl. "Caution to the Wind." Eneri;y Social Science Research Network at http://www.ssrn.org. Power and Risk \hnh\{;en!ent llhither Risk Special Report, 8 (2ll(K)). There are also several books that cover general aspects of pp. 30-32. the market and have short sections on pricing, written in English (Geman [1999b]. Element Re [2002], and Dis- Element Re. Heather Risk Management. l'algrave. 2(J02. chel [2O02[), Japanese (Hirose [2003], Hijikata [1999], Geman, H. "The Bermuda Triangle: Weather, Electricity and and Hijikata [2OO3|). and French (Marteau et al. |2004]). Insurance Derivatives." In Insurance and Weather Derivaiives. chap. The author has written a book that covers pricing issues 2!. Risk Books, 1999a, pp. 197-204. in more detail (Jewson et al. [2004]). . cd. Insurance and llearher Derii'atii'es. Risk Books. 1999b. REFERENCES Goldman Sachs. Kc-lvin Ltd. Otfcrini; circular. 1999. Al.itoii, P.. IJ. DjL-hiclic, and D. Stillbergcr. "On Modeling and I'ricing Weather ncriv.itives." Applied Mathematical Finance, Vol. Henderson. R. "Pricing Weather in Risk." In li'eaiher Risk '). No. 1 (2002). pp. 1-3(1. Manai;einent. l'algrave. 2Oli2. pp. 167-198.

BLick. F "The I'ricinij; of Cloinniodity Contracts." Joiinhv/ of HL'yer. D. "Stochastic Dominance: A Tool for Evaluating Rein- l-iuiiiicidl luoiiouiicf. 3 (1976), pp. U>7-!79. surance Alternatives." C-asualty Actuarial Society Eorum. Summer 2lH)|, Boissonnadc, A.. L. Heitkenipcr. and D. Whitehead. "Weather Oata: CkMiiiTig and Enhancement."' In Climate Risk' nnd the Hijikata. K., ed. Tenkoo Derivatives No Siihcte (All about weather niMhcr Market, chap. 5. Risk Books, (2002), pp. 73-98. derivatives). Sigma Base Capital, 1999.

Bnidy. H.. J. Syroka. and M. Zervos. "Dynamical Pricing ot . Sooron Tenkoo Derivatives (All theories about weather Weather Derivatives." Quaniiiatii'c Finance, 2 (2002), pp. 189- derivatives). Sigma Base Capital, 2003.

Hirosf, N.. cd. Everythin^^ About Weather Derivatives. Tokyo Elec- C\ib.illLTo. li,.. S. Jewson, and A. lirix. "Long Memory in Sur- tricity University. 2(103. \.\cc Air Temperacure: Detection. Modeling and Application CO Wcatlu'r Derivative Valuation." CJiiiiatc Rcsdin'h, 21 (201)2), Iman. R.. and W. Conover. "A Distribution-Free Approach to pp. )27-14(). hidticing Rank Correlation Among Input Variables."' Coinnm- in'cations in Statistics, 11 (1982). pp. 31 1-334. C:ao, M., .Hid j. Wei. "PricniL: the Weather/" RisL; Vol. 1 3. No. 5. (20(10). Jewson, S. "Weather Derivative Pricing and Risk Manai^cment: Volatility- and Value .it Risk." http://ssrn.com/abstract=4(>58(0, Cirmona, R., and D. Villani. "Monte C~arlo Helps with 2002. Pricing." Etivironmental Finance, 6 (2003). . "Closed-Form Expressions for the Pricing ot Weather Davis, M. "Pricing Weather Derivatives by Marginal Value.'" Derivatives: Part 1—The Expected Payoff."" http://ssrn.com/ Qiuwtitative Finance, Vol. 1, Nos. t-4 (2001). abstract=436262, 2()()3a.

Dischel. IJ.. cd. Cliniati- Risk iVid the WealluT Miirhet. Risk Books, 2002.

lALL 21X14 Tub JOUKNAI Oh Al.TFRNATIVF. INVESTMENTS 63 . "C~losed-Forni Expressions for the Pricing; of Weather . "Weather Derivative _ am! the Year Ahead Fore- Derivatives: Part 2—The Greeks." http://ssrn.coTn/abstraet= casting ot Temperature Part 2—Theory." http://ssrn.coni/ 436263. 2n(l3b. abstract-535143, 2O()4h.

—•—. "Closed-Form Expressions for the Pricing of Weather Jewson, S., and A. Brix. "Sunny Outlook for Weather Investors." Derivatives: Part 3—The Payoff Variance." http://ssrn.com/ Environmental Finance. 11 (200(1). ahstract-4819(12, 2()03c. . "Weather Derivative Pricing and the Year Ahead Fore- . "Closed-Form Expressions for the Pricing of Weather casting of Temperature Part 1—Empirical Results." http:// Derivatives: Part 4—The Kernel Densit)'." http://ssrn.coni/ ssrn.coni/abstract-535142, 2004. abstract=486422. 2lU)3d. lewson, S.. A. Brix, and C. Ziehniann. Weather Derivative Val- . "Comparing the Potential Accuracy of Burn and Index uation. Cambridge University Press, 2004. In press. Modeling for Weather Option V.iluation.'" http://ssrn.coin/ ahstract=486342, 2()03e. Jewson, S., and R.Caballero. "Seasonality in the Dytiamics of Surface Air Temperature and the Pricing of Weather Deriva- . "Estimation of Uncertainty in the Pricing of Weather tives." Journal of .Applied Mcteoroloj^y, 11 (2003a). Options." http://ssrn.com/abstract=^4412H6, 2(IO3f. . "The use of Weather Forecasts in the Pricing of Weather "Horizon Value at Risk for Weather Derivatives Part Derivatives." jouwai of Applied .Meleorolo^>y, 1 1 (20()3b). 2—Portfolios." http://ssrn.com/abstract=478(bl, 2003g. Jewson, S., and M. Zervos, "The lJlack-Scholes Equation for . "Sitnple Models for the Daily Volatiht\- of Weather Deriv- Weather Derivatives." http://ssrn.coni/abstract=436282, 20()3a. ative Underlyings." http://ssrn.coni/abstract-477163, 20(l3!i. . "No Arbitrage Pricing of Weather Derivatives ni the •. "Weather Option Pricing with Transaction C'osts." Uncri^y- Presence of a Liquid Swap Market." Submitted to The hiteyna- Power iVid Rish Maiia\;eiiiciil. 2()O3i. tioiial journal of Tliccretical and Applied i'inamc. 2003b.

. "Closed-Form E.xprcssions for the Beta of a Weatlier Markowitz, H., ed. Portfolio Selection: Hfjicieiil Diversification of Derivative Portfolio." http://ssrn.com/abstract^486442, 2U()4a. iin-estmenis. Wiley, 1959.

. "Four Methods for the Static Hedging of Weather Deriv- Marteau. D..J. Carle, S. Fourneaux, R. Holz, and M. Moreno. ative Portfolios." http://ssrn.com/abstract=486302, 2(.H.)4b. Lit Gcslioii dii Risque Cliiimtiqiie. Econonuca. 2004.

. "Improving Probabilistic Weather Forecasts Using Sea- Mclntyre, R. "Black-Scholes Will Do." Ener'^y Power •iiid Risk sonally Varying Calibration Parameters." arxiv:physics/04(J2026, Management, II (1999). 20()4c. Moreno, M. "l^iding the Temp." futures and Options World. \ I . "A Preliminary Assessment of the Utility of Seasonal (2000). Forecasts for the Pricing of U.S. Temperature Based Weather Derivatives." http://ssrn.com/abstract=531062. 2O(l4d. . "Rainfall Derivatives." Derii'atii'es iVeek. September 200].

. "The Relative Importance of Trends, Distributions and Moreno. M., and O. Rtiustant. "Modelisation de la Tempera- the Number of Years of Data in the Pricing of Weather ture: Application Aux Derives C'limatiqnes." In Lti Rcasfinaiice. Options." http://ssrn.coni/abstract=516503. 2(IO4e. Approclic 'Rrliniijiie, chap. 29. Economica, 2002.

. "Weather Derivative Pricing and the Norniahty of Stan- Torro, H., V. Meiieu.. and E. Valor. "Single Factor Stochastic dard US Temperature Itidices." http://ssrii.com/abstract- Models with Seasonality Applied to Underlying Weather Lieriv- 535982, 20O4f. atives Variables." Technical Report 60. European Financial Association, Februarv 20(11. . "Weather Derivative Pricing and the Potential Accuracy of Daily Temperature Modeling." http://ssrn.com/ abstract= 535122, 2OO4g, To order reprints of this article, please contact Ajani Malik at anicilik(a),iijourjia!s.com or 212-224-3205.

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