Weather risk management and Black-Scholes option pricing model

Transcription

1 IJBFMR 5 07) 5-60 ISS Weaher risk managemen an Black-Scholes opion pricing moel Turgu ÖZKA Beyken Universiy, Faculy of Economics an Aminisraive Sciences, Ayazaga Maslak-Isanbul, Turkey. urguozkan@beyken.eu.r. Aricle Hisory Receive 9 Sepember, 07 Receive in revise form 6 Ocober, 07 Accepe 9 Ocober, 07 Keywors: Quano opion, Weaher Risk Managemen, Black-Scholes, Cooling egree ays. Aricle Type: Full Lengh Research Aricle ABSTRACT Weaher risk managemen can be escribe as all financial aciviies ha are geare owars keeping he income flow eviaions, which he weaher coniions will creae, in sabiliy ban by using prevenive financial ools. Like oher financial erivaives, weaher erivaives have special qualiy ypes ha consis of weaher relae securiies such as floor, cap, collar, swap, Quano opion besies convenional ypes like forwar, opion, fuures. In his suy, implemenaion of Black-Scholes opion pricing moel, which is use in pricing of opions, on weaher risk managemen an especially weaher risk opions is specifie. I is conclue ha, he moel can be use for effecive weaher risk managemen as well. 07 BluePen Journals L. All righs reserve ITRODUCTIO Among he noions ha are becoming increasingly imporan, affecing personal an economical life eeply, risk is he leaing one. Wih is broaes escripion, risk can be explaine as a conclusion which exiss in an even or a fac because of is naure an amages when i maerializes. Wihin his escripion, weaher risk, which is known as he oles risk noion, is of capial imporance. When he changes in weaher coniions are unforeseen an above seasonal normals, hey consiue unavoiable or sysemaic risk ype since hey are naural evens which can occur in every season. Global warming has mae he managemen of risks ha were creae by weaher risk by acceleraing he issenien from seasonal normals. Despie he meeorological forecas mehos ha are eveloping fas wih he help of echnological ai, here are sill en-ay of shor erms which inclue graually ecreasing forecas success. The possibiliy of a shor-erm preicion enhances he imporance of he weaher risk. Deviaions from seasonal normals ha are efine by long-erm local meeorological aa ofen cause 'geing caugh unprepare' by he risk an he exen of he amage o rise unexpecely. The enency of expecaion of global warming o coninue increasingly also means ha he risks which epen on weaher coniions will inensify. Thus, i shoul be expece ha he ineres an necessiy of financial ools for avoiing weaher risks will increase in parallel o his change in he fuure. The effec of weaher an climae change risk on numerous segmens an firms are much more han he changes in economic facors an price flucuaions of financial asses. Inee, he changes in weaher coniions an he volume of his change affec personal life, an changing one s habis accoringly, can affec he economic life irecly an eeply. Besies, accoring o he eviaion level of abnormal weaher coniions such as hurricane, exreme col, rain, snow, here can be caasrophic economic amages an loss of lives. These losses cause flucuaion of fun flow on he basis of iniviual, firm, segmen, naional economy an even global level. While an abnormally col winer perio can pu a fuel supplier or a firm ha sells

2 In. J. Bus. Financ. Manage. Res. 53 winer clohes on heir buge arges, i can cause elays in projecs of consrucion an conracing firms, hereby income losses an inemniies. Off-season anomalies such as heavy rain an snow or sorm, hurricane urge municipaliies o spen more han expece on operaing expeniures o eliminae he effecs of hese hiches. These coniions which are resource from he naure can cause he rise in lack of repaymen o crei insiuions, o push he limis of loss raio for inemniy paymen of insurance companies, enforcemen of reinsurance agreemens agains insurance companies an loss of profi. The risk of ski faciliies is inaequae snow eph, whereas he risk is no having ho weaher ha year for an air coniioner manufacurer. Moreover, i is known ha rain an air emperaure have a cerain effec on crop yiel in agriculure segmen. One hir of he Gross aional Prouc GP) of evelope counries are irecly affece by climae change. More han 80% of he worl economy is epenen on weaher coniions an he climae eiher irecly or inirecly. Every year, weaher risk-relae loss of he companies is abou 40 billion ollars in oal Global warming simulaes his unconrollable flucuaion more an increases he changes of weaher coniions an heir effecs. Accoring o he USA aional Weaher Service repors he average Earh emperaure was 3.6 C in March 05, which was he hoes March since 880 an 0.85 C above he 0 h cenury average. In he las en years, here has been an up-going ren of geing warmer in he Earh an he average Earh emperaure has accelerae from 5 o 7 C. A lo of segmens have been affece by his global warming ren such as waer resources, ry farming an generaion of hyro-elecric power. The expecaion ha global warming ren will accelerae means weaher coniion-relae risks o inensify as well. Thus, i shoul be expece ha he ineres an necessiy of financial ools for avoiing weaher risks will increase in parallel o his change in he fuure. All his example an such from real life srongly emphasizes he imporance of Weaher Risk Managemen noion in economic an financial lieraure. Weaher risk managemen is escribe as all financial aciviies ha are o keep he income flow eviaions, which he weaher coniions will creae, in sabiliy ban by using prevenive financial ools. In weaher risk managemen, besie using convenional financial ools such as insurance, weaher risk financial erivaives ha have evelope an iversifie in financial marke rapily, or weaher erivaives in shor, have he poenial o be a more effecive an common financial ool. The aims of his suy are o menion weaher risk opions which are one of he weaher risk by-proucs gaining a place in lieraure very recenly; however becoming more imporan; an o carry ou a mahemaical analysis of he usage of Black-Scholes opion pricing moel on weaher risk opions. For his purpose, firsly, weaher risk opions an a ype of hese opion conracs, Cooling Degree Days CDD), have been illusrae. Aferwars, Black- Scholes weaher risk opion pricing moel has been iscourse; hen he process of he moel in Quano opions has been especially emphasize, an in he las par of he suy, evaluaion an he conclusion have been inclue. WEATHER RISK OPTIOS Weaher erivaives aim o avoi he amages which can be creae by weaher anomalies ongoing above seasonal normals. Unforeseen ho an col weaher, rain, snow, fros, hurricane cause physical amage an his may cause naional an even global economic loss. Weaher coniions ha are above seasonal normals affec he firms sales, an cause exorbian expense an/or income loss. When he weaher coniions spoil he financial srucures, an cause unforeseen eviaions in fun flow an profis, i affecs almos every secor an especially becomes one of he main reasons of crop yiel ecline in agriculure secor. Weaher risk opions in financial marke are processe as sanarize weaher opion conracs epening on; Heaing egree ays HDD), CDD, rainfall, snowfall, growing egree ays GDD), fros freezing) egree ays FDD), meling egree ays MDD) an hurricane inices. Among hese opions, he process of CDD call opion has been illusrae. Figure shows how he opion ealer can avoi he weaher risk Özkan, 008: 07). Accoring o Figure, he righ sie shows he col, an he lef sie shows he ho emperaures. In reurn for he risk hey are uneraking, he opion ealer who pays for he opion risk premium in he amoun of DE o he oher opion ealer, acquires he righ o receive a cerain amoun from he opion ealer for each srike ha is above he hreshol in case he cumulaive col weaher emperaure is above he srike ha was sae in he conrac uring he conrac perio. A he en of he conrac, he opion ealer will muliply he emperaure srike ha is above hreshol above he emperaure ha was sae in he opion conrac-an he amoun ha was se per emperaure srike an make he paymen o he ealer on he expiry ae of he conrac. For a naural gas venor, when he winer is warmer han expece, i causes a sharp ecrease in sales revenue. If he firm wans o provie a hege agains such a risk, i purchases CDD call opion in reurn for a conrac premium as much as DE) an receive he loss unil he nex level ha was sae in he conrac from he

3 Özkan 54 Figure. Cooling egree ays CDD) call opion. Source: Özkan 008). opion wrier. During he conrac perio, if he weaher in he winer) is above hreshol warmer ha foreseen) an ens o he lef sie, per each emperaure ha has exceee hreshol, i gives he righ o receive he loss unil he nex level ha was sae in he conrac from he opion wrier. The ifference beween he maximum paymen level of he inemniy an he conrac premium will consiue he ne inemniy sum of he pary ha has he opion. Moreover, a consrucion an conracing firm can purchase a CDD pu) opion agains he risk of much coler weaher han foreseen in he winer. Therefore, i can hege he loss which is he resul of he slow work ue o oo col weaher or he amage ha can be creae by oo col weaher on he consrucion. Accoringly, i is possible o escribe CDD as below: Call opion CDD) = max0, Tm 65 F) coler han expece) Pu opion CDD) = max0, 65 F Tm) hoer han expece) Benh an Benh, 03:, 8; Rier e al., 00): WEATHER RISK OPTIOS AD BLACK-SCHOLES OPTIO PRICIG MODEL There are various mehos o calculae he pricing of byproucs an o eermine he marke price or forwar prices. Among hese mehos ha complicae formulas form, -espie all he criicisms- he mos frequenly use one is Black-Scholes opion pricing moel. Black-Scholes opion pricing moel Black-Scholes formulas are use o calculae he vali cash opion values. Therefore, i provies vali price an he marke price comparison bu i is no effecive in guessing he prices in he fuure. Hence, i canno es he implemenaion price of he fuure). Despie similar inaequacies, i is sill a useful meho in calculaing aily ransacions. Black-Scholes opion pricing moel is use as below in orer o calculae he marke price or curren value of a Europe call C) opion:,0] CDD, ) isc con max8t ),0) isc max[ T ) c,0] C S, ) = S ) Ke -rt-) ) max8t ),0) S ln ) r ) T ) K T

4 In. J. Bus. Financ. Manage. Res. 55 Consiine, 009): S ln ) r ) T ) K T T E M CDD P CDD) Q CDD) CDD) 0 As for Pu P) opion, he formula below is use o calculae is value: P S, ) = Ke -rt-) S + C S,) or; P S, ) = Ke -rt-) ) S ) The relaion beween he call an pu opion prices of a European ype which has he same forwar an implemenaion price is calle pu-call pariy. In case he lack of he pariy or he relaion showe below, here can be arbirage beween he call an pu opions. An invesor can obain a risk-free arbirage by purchasing call an pu opions, which are base on he same financial asse an wih equal erms an implemenaion prices, simulaneously. P = C S + Ke -r P + S 0 = C + PV K) Informula: Q: Spo price of heunerlyingshares, K: Srike price of opion, σ: Yiel variabiliy of he unerlying asse sanar eviaion), r: Yearly coninuous compouning value of he risk-free ıneres rae T - : Mauriy perio / year's ays, e-r T-): coninuousiscouningfacor e =.788). n ): cumulaive isribuion funcion of hehe sanar normal isribuion or cumulaive normal isribuion funcion. Black-Scholes opion pricing moel in evaluaing weaher risk opions Accoring o Davis 00) an Geyser 004), Black- Scholes opion pricing moel will be able o be implemene on he weaher risk by-proucs uner cerain conjecures. These conjecures are; all aa showing Brownian moion, maximizaion of he expece profi, naural growing raio of egree ays, naural enhancemen of cash prices such as perol, which are relae o weaher risk an naural expansion of firm profis. The paymen amoun of a CDD weaher risk byprouc is a funcion of he srike, he probabiliy isribuion of CDD an he paymen amoun per CDD conrac. I can be showe wih he pariy below Inequaliy: E: Expece paymen amoun of CDD opion, M: Amoun of ollars per CDD in conrac, P CDD): Probabiliy isribuion of CDD sanar eviaion), Q CDD): Opion paymen amoun of per CDD uni, CDD): Differenial of CDD are sae. The price of a weaher risk opion is effece by hree elemens ha are sae below Consiine, 009): ) Sanar eviaion of ime series ), ) Disancefrom srike hreshol o mean μ) of series, 3) Money amoun of conracper egree ays. I is a well-known fac ha, seasonal aum are no sable. This means he normals expece) values of ime series ha are forme of seasonal inicaors an sanar eviaions are also erivaive. Taking he seasonal aum for 0, 0 an even 50 oes no change his feaure of climae aum. Unusual an infrequen climae evens such as hurricane, sorm, an hail increase he erivaives of he series of a specific meeorological area an eflec from he normal isribuion. o only long-erm series are prouce in meeorological aa source fiel, bu also he resuls are parly accurae an hey provie low-qualiy analyses. However, when he seasonal perios are aken ino consieraion, calculaing mean an sanar eviaions of egree ays wih Gaussian moels urn he specifie saisical conceps ino very well esimaors. Therefore, he erms of weaher risk by-proucs are ominanly limie by seasons. A imes, purchasing a series of conrac ha carries he season o shorer erms is more effecive han purchasing seasonal conrac in weaher risk by-proucs. This siuaion, which seems agains he common belief of he feaure of classic opion conracs, happens because here is lack of iner-perio correlaion an he seasonal aum occur ranomly. Insea of buying a whole single perio s pars separaely, when a single conrac is bough in balk incluing a few sub-perios; here is sanar eviaion risk) lower han oal of sub-perios. However, since he sanar eviaion of he conrac series increase in oal; when compare o shor-erm ones, long-erm conracs are reae wih more expensive opion premium. I is possible o exemplify he subjec as below wih he help of aum in Table Consiine, 009). If we assume ha implemenaion or hreshol srike for an opion

5 Özkan 56 Table. Exercise hreshol of opion conrac. Mean µ) Sanar eviaion ϭ) January HDD February HDD 3 64 C,, K,, ) e E [max S,, ) K,0) S ] r ) In Q In Pu opion value: P,, K,, ) e E [max K S,, ),0) S ] r ) In Q In conrac in sock or Over-he-Couner OTC) markes is foun by aing he 50% of he sanar eviaion o he average value; he implemenaion hreshol of a whole single conrac incluing January, February an boh will be calculae as below: K JAUARY = [ )] = = 45 K FEBRUARY = 3 + [ )] = = 354 K J+F = [ ) 0.50] = = 779 r ) r ) C,, K,, ) e S,,, ) ) e K ) This means ha, if he emperaure r ) goes above he r ) C,, K,, ) e S,,, ) ) e K ) calculae implemenaion hreshol, call opion will be pracice. The sanar eviaion of boh monhs; r ) r ) P,, K,, ) e K ) e S,,, ) P r ) ) v r, v v P, v, K,, ) e K ) e S,,, ) ) will be calculae as below by using his formula. JAUARY FEBRUARY JAUARYFEBRUARY ,9 JAUARY JAUARY FEBRUARY FEBRUARY O; Ş As i is seen, he sanar eviaion of a whole single conrac incluing boh monhs is o be more han he eviaions of he monhs i inclues. For his reason, he possibiliy of long erm conracs o be more profiable han shor-erm ones provies hem o be reae wih more expensive opion premium. This suy The pricing of opions an corporae liabiliies by Black an Scholes 973), who applie Brownian moion meho, which is use o guess ranom movemens ha will occur from he collusion of liqui molecules for example, us, pollen) wih smaller paricles for example, gas, liqui) wih sochasic mehos, o guessing opion prices successfully, is calle Black-Scholes opion pricing moel. Wih he help of Black-Scholes opion pricing moel, i is possible o escribe he calculaion meho of call C) an pu P) opion values or prices of weaher Risk managemen opions for HDD, CDD an CAT) as below Benh an Benh, 03: 68; Cao an Wei, 004): Call opion value: SCAT,, ) K,,, K) CAT,,, ) If we efine he call an pu opions as for CDD, HDD Heaing Degree Days) or CAT Caasrophe Risk), in orer o calculae he call an pu opion values, efines cumulaive normal isribuion funcion, an we can wrie: ln S,, ) ln K,, ) s s s,, ) s ,9 ln S,, ) ln K,, ) s s s,, ) s Call an pu values subsaniae he posiions lack of arbirage. In he formulas, In efines he inex value of weaher changes ha were calculae in τ, τ ) perios, showing CAT, CDD or HDD values. As for,, K,, ), i C In is he call or pu opion price a he ime of 0, which is in opion exercise erm from K exercise price τ. Moreover, i is τ τ uring he calculaion of he opion an S: is he Spo Price of he Base Sock. In his case, we can efine he pu-call pariy as below: C,, K,, ) P,, K,, ) e S,, ) e r ) In In In ) ),,,, ),,,, ) r,, ) r In In In C K P K e S e K Pu-call pariy is efine as he pariy ha shows he relaion beween he European ype of call an pu opion price base on he same erm an implemenaion price.

6 In. J. Bus. Financ. Manage. Res. 57 This pariy means, he value of a porfolio consising of a long posiion an a shor posiion opion will be equal o he value of a forwar conrac wih he same implemenaion price an forwar conrac. If here is no equaliy or relaion showe wih formulas, here is arbirage possibiliy beween he call an pu opions. The by-proucs o hege he weaher risk can be forwar, fuures or opion conracs. If he weaher risk is forwar conrac, he paries conrac an engagemen o swap he prouc an he cash base or anchore upon HDD, CDD or anoher weaher inex value no maer wha he weaher inex value is in a specifie erm in OTC marke. Therefore, by guaraneeing a specific weaher inex, wih a weaher risk forwar conrac; HDD, ) max8t ),0) in T in erm, inex or conclusion, for insance HDD value is fixe. If we also ake forwar premium ino consieraion, i is possible o urn he pariy ino he form below Cao an Wei, 004): cov[ max8 T ),0) HDD, ) max8 T ),0] E ) The firs erm in he pariy shows he expece cash value of HDD in he fuure, an he secon erm shows he forwar premium. I is clear ha he same formula can be use for CDD forwar conrac oo. In his pariy, if he marke is in normal backwaraion posiion for forwar conrac, negaive risk premium provies he balance, if i is in conango posiion, posiive risk premium provies i. We can examine he pricing of weaher by-proucs accoring o Black-Scholes opion pricing moel wih an example: Le s assume ha srike price K), which is base upon he meeorological weaher aum of Chicago O Hare Airpor, of a call opion base on February cumulaive HDD inex of a specific year is 700 an he paymen sep per egree ay is $ If he aum wih lognormal isribuion feaure aes an he foreseen average or expece mean: μ) value is 70, an he sanar eviaion of HDD inex value ha is calculae by aking is naural logarihm is equal o ϭ) 0.07 Hull, 0: 760; Company e al., 007); we can wrie: C = [70 ) 700 )] If an show he cumulaive normal isribuion values: 70 0, 07) ln ) 700 0, 376 0, , 07) ln ) 700 0,676 0,07 is calculae. Accoringly, from Cumulaive ormal Disribuion Table, i is 0.376) = an 0.676) = When we replace hese values in he firs pariy, he paymen amoun ha he call opion is o provie a he en of expiry ae is; C = [70 ) 700 )] = [ ) )] C = [5,9] = $ 5.90,00 If he risk-free ineres rae is 3%, he value of he opion in he beginning of he erm he previous year) presen value: PV) e =.788) is calculae as shown; PV C = 5.90,00 e -0.03) = 5.90, ) = $ , If he expiry ae HDD inex value recees from 70 o 705 warmer winer han foreseen on 0.704% level) because of global warming or warmer winer han foreseen, i is; 705 0,07) ln ) 700 0, ,07) ln ) 700 0,07 0,367 0,0, ) = ve ) = An he expece value of he call opion for expiry ae is calculae as below; C = [ ) )] = [.3655] = $ 3.65,50 The value of he opion in he beginning of he erm he previous year) base on he 3% risk-free ineres rae is; PV C = 3.65,50 e -0,03) = 3.65, ) = $ 7.04,60 Therefore, in his case, when HDD inex is warm in 0.704% = % HDD) raio, i causes a ecline in expiry

7 Özkan 58 ae of he weaher opion in 0.96% [% C = 3.65, ,00) / 5.90,00] raio an in he same raio as [0.96% = 7.04, ,)/ , = % PV C ] a he beginning of he erm. As i is seen, he changes in inex value an in opion value are no in a linear, bu convex relaion. I means i is; HDD < C an HDD < PV C Such a convex relaion creaes arbirage possibiliy beween weaher by-proucs wih ifferen erms an an acive porfolio managemen policy. The change in ineres rae is an imporan elemen affecing r) opion. If he ineres rae increases from 3 o 4% r = + %), i recees o he value below: PV C = 5.90,00 e -0.04) = 5.90, ) = $ 4.340,7 This means, in given circumsances, an increase by % leas o a ecrease by $.45,5 C = , 4.340,7) or in 0.995% raio per erm value of he weaher opion. In case of a rop for example, from 3 o % raio an r = %) % PV C increases PV C = $46.6,, PV C = $.449,89 an % PV C = %). There is also a convex relaion insea of linear relaion here, an he correlaion beween % HDD, % PV C an r can be more or less han. In shor; he negaive relaion principle beween he ineres an he financial asse is also vali for weaher by-proucs. Quano opions The abbreviaion of Quaniy-Ajusing opion; Quano, Quano opions or Cross-Currency erivaives are byproucs, arranging he opion epening on an unerlying asse from a fixe rae wih anoher currency while unerlying asse is efine by a specifie currency. As is, Quano opions are somehow a currency forwar wih erivaive amoun. This opion ype, enabling wo ifferen currencies o inves irecly or inirecly an minimizing he exchange risk; is a suiable by-prouc for he invesors who wan o inves in financial asses ha are processe by foreign bill an speculaors. Thus, an effecive proecion is provie agains currency flucuaions. For insance, a Turkish invesor buys German Commerzbank or American General Elecric Co. on Swiss Sock Exchange SIX) wih Sweish Franc CHF), hey also buy risk agains he changes in CHF / TRY currencies. By purchasing Quano opion wih erivaive amoun forwar, he invesor can exchange o omesic currency from a fixe currency in erm an can proec hemselves agains he currency flucuaions. From his aspec, compare o he proecion wih swap conrac agains a risky posiion wih he same qualificaion, amoun opions inclue a broaer risk an a lower cos avanage. Quano opions are use in weaher risk managemen or energy marke. However, Quano opion an energy Quano opion are ifferen by-proucs in erms of process ynamics hey have. In response o heavily-use amoun opions in currency markes, energy-amoun opions are hinner markes. Furhermore, since he unerlying asse canno be calle or pu, he liquiiy egree of energy Quano opions is more limie. In organize or OTC markes, especially in energy Quano opions ha are weaher-coniione an energy by-prouc feaure, he paymen epens on he marke price of a commoiy for example, elecriciy) being processe in he marke. In case he emperaure, rain or oher weaher coniions excee a fixe exercise price srike price) below or above, aking he righ of opion paymen has a financial value. For insance, in case he weaher is warmer coler) han expece, elecric consumers urn on heir air coniioners heaers), an he increase of eman agains fixe elecriciy offer causes a ramaic increase in prices. In his case, he expenses of reailer elecriciy consumers increase as well. When an amoun opion is purchase agains he risk of his increasing expenses, he opion is exercise in case he emperaure excees or says below a fixe hreshol. Depening on he hreshol value, per each negaive eviaion or ick in emperaure, he righ of collecing inemniy is use. Thus, unexpece elecriciy expenses resourcing from emperaure eviaions can be hege. As anoher example we can give a naural gas or heaing fuel reailer. Afer he winer passes warmer han expece, he energy isribuor firm will face a ecline in price in boh eman an price; conrary o oher economic price-eman feaure. Complicae sraegies ha are implemene by means of sanar by-proucs such as fuures, opion or common ones, will no be able o proec agains he risks in which cash flow volailiy is affece by a number of facors price an eman) an will urge o ake expensive an risky posiions. As in he examples, Quano opions are relae o a hir price sanar which is calle Quano facor as well as he emperaure weaher coniions) an elecriciy commoiy) volume relaion. So i can hege energy Quano opion agains hree risks ha can occur because of price an emperaure an volumeric risks. In relaion o his, in Quano opions in energy markes, here are hree facors enabling o exercise he opion: ) Temperaure or HDD as a echnicalprovisions of emperaure) ) Volume an 3) Theprice of energy.

8 In. J. Bus. Financ. Manage. Res. 59 If we show he HDD inex an inex values of implemenaion values in he amoun opion conrac of average gas price P) in cash marke wih K HDD an K P, an if he egrees are above he implemenaion values sae in he conrac because of a coler winer han expece; which means benefie from Benh e al., 03) if i is; HDD > K HDD ve P > K P The invesor who has he opion will exercise he call opion, C [[max HDD K,0)][max P K,0)] Mikar] HDD will receive inemniy as much from he opion. We can calculae HDD an P inex values as below: HDD u g T ) an P S u Informulas; S u : Spo marke energy prices, T : Temperaures on Time as a funcion of g [g) = max0, 8 C g)] g: Daily average emperaure). If he egrees are below he implemenaion values sae in he conrac because of a warmer winer han expece, which means; HDD<K HDD an P < K P he invesor who has he opion will exercise he call opion, C [[max K HDD,0)][max K P,0)] Mikar] HDD will receive cash flow as much from he opion. As i is seen, in Quano opions, if boh he emperaure an he price is below or above) he conrac implemenaion values, he opion is exercise. This srucure explains why Quano opions are a beer alernaives compare o oher sanar by-proucs. This feaure of Quano opions enable Quano opions o use a grea amoun in energy marke. Because Quano opions are suiable an effecive by-proucs in erms of managemen of risks boh resource from price an volume in energy marke. For his reason, oher han Quano opions; opions such as Quano fuures, Quano swap, Quano barrier, Quano forwar, Quano corrior are also being use effecively in he markes as of now. u P P I is possible o calculae he value of a Quano forwar puing conrac base on Black-Scholes opion pricing moel wih he help of formulas below Wysup, 008; Haugh, 00; Benh e al., 03; González-Gaxiola an Chávez, 06): Applicaion coniion of a Quano opion = Q [ØS T K)] + C r T X, ) Qe Q S0 e T S0 ln T K T Correcion value: r r f, f ) K f an Informula: Q: Quano facor, Ø: Shor-long posiion inicaor, S 0 : Marke price of he unerlying asse presen value) T: Opion mauriy K: Exercise price srike price), : Opion ime-o-mauriy, rt Q e : Coninuous iscoun facor for Quano facor e Coefficien =.788), T e : Coninuous iscoun facor for unerlying asse, ): Cumulaive normal isribuion funcion, ln: Logarihm, σ : Yiel variabiliy of he unerlying asse, σ : Variance of he unerlying asse, μ: Correcion coefficien, ρ,f : Correlaion coefficien beween omesican foreign currencies, r : Yearly coninuous compouning value of he omesic risk-free ineres rae, r f : Yearly coninuous compouning value of he foreign risk-free ineres rae. Conclusion A sysemaic risk ype an having sochasic feaures, a weaher risk is very imporan in erms of following elemens, ) preeermining he amages, ) measuring he risk, 3) specifying he measures an aking hege posiions, 4) urning of fun flow flucuaions ino a sable ren, 5) price sabilizaion. Accoringly, weaher risk managemen can be escribe as all financial aciviies ha are o keep he income flow eviaions, which he weaher coniions will creae, in sabiliy ban by using prevenive financial ools. Weaher risk is possible o hege wih classic mehos such as insurance, in 996, a financial proucion which is a much more effecive, accurae an wih ispue

9 Özkan 60 prevenive conclusions beween he paries is rae in Chicago Mercanile Exchange CME). This financial proucion, processe in organize an OTC markes an in all global sock is calle weaher erivaives an ue o he serious avanages i creaes, is process iversiy an volume are increasing in a fas pace in he worl. Black-Scholes opion pricing moel is sill an effecive moel in pricing opions ha are base on commoiy an oher financial proucs an aking suiable posiions o hege probable risks. I is unersoo ha, especially in pricing he opions o hege weaher risks, uner specific circumsances, Black-Scholes moel can be use effecively in weaher risk managemen. However; regaring he use of Black-Scholes opion pricing moel in weaher risk opion conracs, here are some limiaions compare o oher financial erivaives. While he moel gives effecive resuls in conracs abou agriculural proucs, he iea of being an effecive preicion ool in caasrophic risks like hurricane wihou ransacion volume which he number of conracs creae meaningful, saisical resuls, is suspicious. Besies, he moel is a goo preicor especially in seasonal conracs in price preicions regaring weaher opions abou agriculural proucs. While he moel gives more accurae resuls in shorer perios han season, causes o an increase in he oal premium cos of he conrac porfolio which consiss of shor perios. The assumpions which Davis 00) an Geyser 004) sae abou applicabiliy of he moel are also vali for oher opion ypes. Whereas as Consiine 009) sae, unlike oher opions, long-erm average meeorological hreshols abou weaher coniions for price preicions of he weaher opions have criical imporance. In aiion, global warming causes increasingly negaive impacs on he sabiliy of he climaic ime series, raise he sanar eviaion variabiliy problem an causes a convex ren in climae inex values in accorance wih Cao an Wei 004) approaches. Therefore, he moel also provies effecive resuls in eermining he arbirage opporuniy beween weaher opions, mainly Quano opions. REFERECES Benh F. E. & Benh J. S. 03). Moeling an pricing in financial markes for weaher erivaives. Worl Scienific Publishing Co., Avance series on saisical science an applie probabiliy. 7: 4. Benh F. E., Lange. & Myklebus T. A. 03). Pricing an Heging Quano opions in energy markes. e ). Black F. & Scholes M. 973). The pricing of opions an corporae liabiliies. The Journal of Poliical Economy. 83): Cao M. & Wei J. 004). Weaher erivaives valuaion an marke price of weaher risk. The Journal of Fuures Markes. 4): Company R., Jóar L., Rubio G. & Villanueva R. J. 007). Explici soluion of Black Scholes opion pricing mahemaical moels wih an impulsive payoff funcion. Mah. Compu. Moel. 45:80-9. Consiine G. 009). Inroucion o weaher erivaives, weaher erivaives group, Aquila Energy. Davis M. 00). Pricing weaher erivaives by marginal value. Quan. Financ. : Geyser J. M. 004). Weaher Derivaives: Concep an applicaion for heir use in Souh Africa, Agrekon. 43:4. González-Gaxiola O. & Chávez J. R. 06). A nonlinear opion pricing moel hrough he Aomian ecomposiion meho. In. J. Appl. Compu. Mah. : Haugh M. 00). Foreign exchange an Quanos. e ). Hull J. C. 0). Opions, fuures an oher erivaives, Eigh Eiion, Pearson Prenice Hall, ew Jersey. Özkan T. 008). Financial risk managemen by erivaives cause from weaher coniions: Is applicabiliy for Türkiye, risk managemen an value: Valuaion an asse pricing, worl scienific priners. Worl Scienific in Inernaional Economics. 3: 644. Rier M., Muβhoff O. & Oening M. 00). Meeorological Forecass an he Pricing of Weaher Derivaives, Weaher Derivaives an Risk CRC 646 Conference, Berlin. aional Weaher Service). Wysup U. 008). Foreign exchange Quano opions. hp:// e ).