Contracts and Derivativesand Contracts Price and DerivativePriceModelling. Energy Derivatives. Chapter 2of ManagingEnergy Risk Based on     Outline Weather Derivatives Hedging Put, , Covered , Put, Call, Options: Swaps and Futures Forwards, Schindlmayr , Wiley,2008. Chapter 9 of Electricity Markets . M. Burger, . M. Burger, B, . C. Harris. 2006. Graeber , G,

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metin /~ A bit on terminology:on A bit   forwardanotherderivative?Why ora generator) in advance Forwardsseller involveonly a To rearrange thebetween pricerisk buyer (          $1M extrato Term of the contract is from now until (the last)delivery. (Total) Contract million, $1 price is Luminant Cirro Cirro August wholesalerange overcan price Cirro $40/    Cirro MWh For thissort of certaintysupply/price Cirro Price is fixed at $100/ yields 500*6*31=93,000 Delivery 500 of replaces the price uncertainty of $80-4500/ forwardbuys froma needs is $9.3 million $9.3 delivery price is buys the forwardbuys the sells the the sells 93,000 Luminant Forwards forward forward MWh MWh . -buyer pair. duringAugust peak periodsof for Luminant MWh   or $10.75=1,000,000/93,000or per per hour during 6peak hours on each day of August MWh is is is short for each in the forward =93,000*100, or $100 per $100 =93,000*100, or , terms can be of the followingthe kindof can be , terms . Cirro $ in the forward 80/ MWh electricity retailer) and seller ( MWh MWh regardless of the delivery time. pays (side payment)pays (side − $4,500/ contract contract withcertaintyof price $100 by paying MWh Ercot MWh MWh , despite current price retail market . $1,000,000 Luminant , in total this this total in , to to electricity electricity Luminant of of

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metin /~  profits Buyer . . . Buyer pays:  Random Variable, price in the future  Parameter thatis safe to assume to be known    Contract parameters known now Conceptual Seller profits profits Seller Fair price: p obtain the K+ Continuous discounting exp Buyer’sbenefit in thefuture E( Buyer’s cost now p r T f K : Forward (delivery) priceper unit : Interestcontinuousrate,: compounding ( : Termof thecontract (T : Contractprice per unit T , , T exp(-rT T ) ) in thein future. K - K Who profits? if if = exp( commodity whoseis valuecommodity now and )f K K < exp( < > exp( -rT K exp(-rT ) E (p ) f -rT in theinto future -rT ) E ( p ) E Forward Example ) E ( p ) ( ) E (p ) (T - rT , p T ) ( (T )-f T (T (T , , T ) T , ) , T T )). - f )-f )-f ) ), or ), ) Contract parameters known now    Variable    Parameter thatis safe to assume to be known    Numerical 10 More on distributions when necessary. p r 1/ 10% annualinterest= discount by1/1.1 T f K and standarddeviation$10/ : $100/ : 0.1 per year so that : 120 days =1/3 years: 120days (T : 1,000,000/93,000=$10.75/ 1. . 75 , 1 T ) has “adistribution”) with mean $120/ = MWh - < 0. Who profits? 9 19 = , see swaps for swapsmultiple, see prices Aug . exp 34 ( −𝑟𝑟1) = 1/ exp 1. 1 or or = ( MWh − 𝑟𝑟 exp MWh 0. = Buyer profits Buyer 1/ ( − −0. 3) ln . ( 1) ( 0. 120 9) = − 0. 100 1 MWh

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metin /~   Market contractsForward : Futures Market: (CME, NYMEX) provides(CME, NYMEX) liquidity Sellersare producers (physical owners) thatdeliver the commodity (oil,electricity), or are typicallybetween private counterpartiesand cover longer term. Forward vs. Future Forward vs. Contract Sellers are oftenSellersarespeculate financialthatentities onthe price. financialinstitutionsor hedging/trading companies. Seller Seller price Obligation to deliverin the future K at the price f Obligationdeliver to at the price f Contract price K in the future (frequent transactions). Market set now set now Contract price K Buyer Buyer

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metin /~  Background: 2. 1. profit?amake $1billion Question: PointState B  differencebillionof $1 your profit.is buyerfuturethe oftheentitled contract,billion. you are$2 receiveto The barrelsof oilthese to marketpayingbillion. you sellspotby $1 the When $50/ price2014,is the In Dec buyermillion with20 2014 tofutures aa contract unitsin Jun of In Jun 2014, theforward pricefor delivery Dec is $100/bbl. from c. b. a. shipment curtailments. p by taking - to Russia due increases rice the the Billion Oil.on Bloomberg, Jan issue. 21 Bit, S. Burton, K. Source: K. in in Long: no contractprice the buyer’s obligation to buy at the price f your obligationthepriceof at to deliver in Decoil the future atthecurrent high How short futuresoilyou use marketHow tocan to Capital, Capital, price price Jun’14 Oil own Profiting Profiting from OilPrice Drop: a contrarian’sa perspective drop. a fund, forwardbuy oil contracts to Case of price 𝐾𝐾 = Foxman2015. . 0 $ bbl . 107/ starts 2014 with$5.8starts so you buy 20 so canmillion barrels from Ukraine bbl price and Dec’14 oil price $50/bbl.priceand Dec’14 oil PointState Druckenmiller then as the market was was market the as then - Crimea issue and the associated associated the and issue Crimea . We We are are lower Alums at B long, in assets and profits profits and assetsin believe crude oil is goingoil crude is believe f – =$100/ PointState muchlower … I am sorry for you.I am You can sell sell can You expecting bbl Make $1 $1 Make Capital If $1 after assumingafteruncertainty price away you Sohn PointState Be long in forwards Be Tradingplaychilda is Investment Investment whenprices rise whenprices fall short in forwardsBe CEO Conf., NYC, Zach Schreiber Zach May 5,2014 May

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metin /~     Fair price exposure to fuelprice fluctuations. engage in contracts with fuel suppliers to fixthe price of the fuel and hence to eliminatetheirrisk theIn energy domain, power generators need continuous supply (e.g.,of fuel coal andgas) and can Amount exchanged in each period as well as the price f A buyer pays (receiver).sellerbetweenA swap buyerand a is (payer) a floatingat … pricefixed atBuyertimes commodity can buy themultiple   If periodicallyLibor pay - Outside energy domain,interest rateare swapscommon. In interestan rateswap,a company promises to Swap: K < of rather than its distribution.thanrather its So the Until right a 3-period swap: Swap: Repeated Forwards RepeatedSwap: 𝑝𝑝 Buyer Seller 𝑡𝑡 hand side,buyer profits. -hand now, now, − price Contract 1, 𝑡𝑡 all evaluationsall have required onlyexpected the pricevaluethe of − K based interest to another while it receivesa fixed rate interest from the other. 1 Energy electricity) time. over (coal,oil, sold is gas, , 𝑝𝑝 K Fixed 𝑡𝑡 = ,

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metin /~    Combinations: Combinations: Options (Distribution) Uncertainty Price Derivates of of derivatives of derivates of of

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metin /~ Multiplicative Evolution Forward p ..... (t price p p p p p -1, ( ( ( ( p p p ( T T t t T ( ( ( T + − T t t − − − , , ) 1 1 , T T 1 1 , 2 , T , T T , ) ) , T ) T T t ) ) -1 ) ) ) = = = = Exponential Brownian MotionExponential Brownian ε ε ε ε T t − 1 − 1 2 t , T = Evolution Forward Price ε T − ( t − Forward 1 ) p price (t , T ) from now Looking at future ln t

p p ( ( T T , , T T

) ) = = = = − ε .... ε ε ln 1 1 1 ε ε p p 2 2 ( ( .... t T p , ( T − ε T ) T 1 − − , = t T 2 p ∑ T i ) , = ( − T 1 t t , ln ) T ) ε i Each Each Price is Exponential Brownian Motion T ln ε i

p price Spot Normal. (T , T )

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metin /~     N. Meade. 2010. Oilprices Meade. N. Our evaluation of the two Gaussians was most successful at capturing mostsuccessfulwasthe jump Gaussians with[Normal]Gaussian a density function. … investigat [ term model becauseof time varying and the returns exhibit many jumps of a magnitude completely inconsistent plausible be to ceased models

p p ( Sep (Aug,Dec 4 normal distributionsto 4 fitlogarithms of epsilons. Good fit Forward prices Forward prices Forward prices for December from months Aug-Dec: ,

Dec Aug ε ) 4 )=$100 = = Forward Price Evolution Example Evolution Price Forward ln ε ln 95 4 ε ε p 1 / 1 ( 100 = = for November for December Aug ln( ln( modelling 112 — p , 105 Dec (Sep,Dec)=$95 for longGeometric horizonsBrownianamotion ofthree monthsor less.not supportableis as Brownianmotionreversion?mean or / / 104 ) 101 Sep

p approaches …approaches ); ); ( from months Aug-Dec

Oct from monthsfrom Jul ln ln ε ε 2 , 2

Dec = = ln( ln( ε ) 3 104 101 = = p geometricBrownian motion - ε 92 diffusion process (Oct,Dec 3 / / p 90 92 / ( ing 95 -Nov Sep Oct ); );

] several alternative non alternative ] several

ln ln , , ε )=$92 Dec ε : using similar processusing data similar not shown explicitly: 3 3 = = Energy Economics, Vol. 32, )

p ln( ln( , providing, plausible density forecastsup to a year... ( T 90 92 = /

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⇒ Dec 93 95 p ( Nov (Nov,Dec exponential Brownianexponential motion. ); ); and ,

ln T ln

, ε ε = - Dec Nov mean reversion Gaussian densities 4 4 Dec = = ε ε 2 ) 1 )=$104 ln( ln( = = = ) 93 95 104 ε = 112 2 ε p / Iss / 1 100 98 ( / p / Oct . 6: 1485- . 104 92 ( ). T showed that ). − , p Dec a mixtureaof two 1 (Dec,Dec)=$112 = 1498: ) Nov Dec both , T =

Dec .

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metin /~

) Non These evaluations have E ( -discounted Value Callof = E max{ (call)commodity at fixed price f Ability Call f (optionwithout obligation) p p (t (t , , T T ) ) Options: Call and Put xpected     Random Variable, price in the future p Parameter thatis safe to assume to be known T r: Interestrate K: Price of option charged by the seller T f : Strike: price (to buy or sell, see below) call=0 Value of : Term: of the option of call Value at time T time at value) of a nonlinear term(max) and require the price distribution. to to p (T t purchase , T )-f, . 0} Non at time time at Abilityto(put) sell commodity at fixed price f -discountedmax{ = E Put Value of f T . p p (T (T (T , , T T , ) ) T ) T put=0 Value of f put of Value t f -p (T , T ), 0}

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metin /~     profits: Buyer Option Call Fair price: profits: Seller can be evaluated by usingcelebrated theBlack -Scholes formula. This isthe building block of buy the commodity at the ceilingprice f consumer owning optionscall canstill how high thepriceis,the market a consumer Calloption isan insurance the for K K K financial engineering.financialBlack, Scholes Economicsand MertonNobelgot1997.in Prize in ceiling = < > exp(-rT exp(-rT exp(-rT on theprice.No market matter If (of oil oil (of ) ) p ) E (max{ E (max{ (T E (max{ , T o ) isexponential) Brownian motion, non-discountedvalues option r electricity)to provide p p p (T (T E(max{ (T , , , T T T )-f, Option Profits Option )-f, )-f, 0 0}) 0 }) p }) (T , T )-f, . 0}) 0}) and E(max{ Fair price: profits: Seller     profits: Buyer Put Option commodity theat floor price f owning put optionsthesell canstill how low the marketprice is, the producer a producer Putoptionan insurance the is for K K K floor = < f-p > exp(-rT exp(-rT exp(-rT (T on the market price. No matter , T (of oilor toelectricity) provide ), 0}) 0}) ) ) ) E (max{ E (max{ E (max{ f f-p f -p -p (T (T (T , , , T T T ), ), ), . 0}) 0}) 0})

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metin /~ fair$27,145. contractpriceis (0.905)(30)The=(30) =thousands.0.1) 27.145 advance.The presentvalue of this amount is exp( - needthe present valueof this amount a yearin Total valuea year from now is $30 thousands. We The non- theIn comparisons of The non-discounted option value E (max{ is important. Finding out this value is called oil is $1 per barrel.$1 per oil is The non- wp wp wp wp wp K E (max{ evaluationan option of 15% 10% 20% 50% =(0.05)10+(0.10)5+(0.85)0=1. 5% and exp( and p discounted valueof the discountedoption valueof this crude on ( , max{ , , max{ , max{ , , max{ , , max{ , T , T p ) (T - f, p p p p p , -rT 0}) is a weightedaverage= T ( ( ( ( ( T T T T T )-f, Numerical Example , , , , , T T T T T ) E (max{ ) E ) ) ) ) ) 0}) - - - - - f, f, f, f, f, 0}= max{ 0}= 0}= max{ 0}= max{ 0}= 0}= max{ 0}= 0}= 0 0}= p call (T 110 100- 105 , 95 T option )-f, - - - 100, 100, 100, 100, 0}) 0}= 0 0}= 0}=10 0}= 5 0}= 0}= 0 0}= interestrateof r=0.1? pricewith1 year and ofa maturity continuous annual 30,000 barrels of crude oil, what is the fair contract If Southwest Airlines wants the option? this barrelof non-discounted value where distribution as perwhile barrel Southwest assesspricecrude the oil on the price of crude oil. The strikeprice issetat $100 in jet fuel prices, Southwest Airlines buys a withtheprices. crude oil To hedge againstincreasethe Southwest Airlinesuses fueljet whose prices increase $95 $75 wp wp wp 20%, $100 20%, 5%, $80wp 5%, stands for with probability. What is the per wp 15%, $105 15%, 10%, 10%, $85 call wp wp option above foroption 10%, 110 10%, 15%, 90 15%, call wp wp option 20% 5%,

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metin /~ The non-discounted option value is important. Finding out this value is $237,800. 0.2378 millions.fairThe contractpriceis amount is exp(- years) in advance.The present valueof this thepresent valueof this amount six month (1/2 Total valueon 1 isNov $0.25 million. need We The non- theIn comparisons of E (max{ crudeoil $2.5 peris barrel. The non- wp wp wp wp wp E (max{ K calledevaluation of anoption 50% 20% 15% 10% =(0.05)15+(0.10)10+(0.15)5+(0.70)0=2.5. 5% and exp( and f discounted valueof the discountedoption valueof this on - p , max{ , , max{ , max{ , max{ , max{ , ( T f , -p T ) 0.1/2) (0.25) = (0.951)(0.25)0.1/2)= (0.25) = , (T f f f f f 0}) is a weighted average= weighted a is 0}) -rT - - - - - p p p p p , ( ( ( ( ( T T T T T T PutOption NumericalExample ) E (max{ ) E ), , , , , , T T T T T Strikelower. pricewill be 0}) ) ) ) ) ) , , , , , 0}= max{90- 0}= 0 0}= 0}= max{90- 0}= max{90- 0}= max{90- f put -p (T option , 75, 90, 85, 80, T ), 0}=15 0}= 0 0}= 5 0}= 0}=10 0}) a put optiona put whose $100,000?priceisfair put companyexploration budgetedThe$100,000 the only for interestrateof r=0.1? what isthecontract fair price with continuous annual above on May100,000 1for production,itsbarrelsof explorationoilcompany the wantbuy the Ifto Texas as a collateralfor the loan. Thinking in thisline, Bank of obtainedoilthedrilling the program from can be thought The ofvalue program. drilling Bank Texasfund its toof from $40 millionexplorationcompany borrows oil An discounted value of this option? this barrelof non-discounted value where crude oil price distributionas theexplorationThe company assessNov 1. on barrel put $95 $75 option.Would be strikethepricehigherlower or for option for crude oilat the strikeprice of $90 per wp wp suggests/forces wp 20%, $100 stands for with probability. What is the per 5%, $80 wp wp the exploration company to buy a 15%, $105 10%, $85 wp wp 10%, 110 15%, 90 put wp wp option 20% 5%,

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metin /~      [Value of CallOption] – [Value What if strikepricesdiffer?What Buyingput a option of Put[Value Option] of CallOption]of Put = Contract]+[Value [Value Option].[Value ofForward getputyou cantothethe option, thatscheme use toaof a scheme find value If youhave contract.forward buying a Buying a call option and selling a put option = = = exp exp [Value of Forward Contract]Forward of [Value ( ( - - rT rT ) E [max{ E ) ) E [ E ) and f – - p Option Option [Value of Call Option]of Call[Value selling a call option option callselling a ( Call Call T , f T - p )] )] ( T , T [Value of Put Option][Value ) , 0} – - max{ Put = Forward = Put Combinations p at at ( T the same price strike same the = , T = = ) = − - exp(-rT f, exp(-rT ) E[ exp(-rT) [Value of Forward Contractof [Value 0}] = = 0}] at exp ) E [max{ ) ) E [max{ ) the same strike priceto fis equivalent ( - rT ) E [max{ E ) p(T f is equivalent to selling a forwardequivalenttois selling contract.a p p ,T : (T (T )-f , , T f T - p )-f, )-f, ] ( T ] 0} + 0}min{ , 0} 0} T ) , 0} + min{ +0} - - max{ -f f -p + p (T p ( (T T , , T T , ), T ) - +f, ), 0}] 0}] 0}]

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metin /~     Collar Profits depending on future price future depending on Profits f f C P Combinations Other party exercises call exercises party Other Typically collarsare costless, i.e., are set such thatthe fair contractprice iszero, Then i.e., K=0. Value Collarof = [Value of Put] option:AnotherCall can buy at party Put option: We can atsell least Weput p p    (T (T is a combinationa is of buyingput a , , T T If If If loss is loss ) ) E Fora priceprocessaroundissymmetric that meanits (e.g., Brownian), f f p max P P (T < < p , p T f { C ) ≤ f ≤ ) (T (T 𝑝𝑝 ≤ ( , , 𝑇𝑇 T p T P (T ) ≤ ) , )-f < 𝑇𝑇 , C ) f T f C . T C − ), we do not exercise put but the other party exercises calland our incurred sold call sold Loss from , we exercise put to earn b Gain from , neither we nor the other party exercises put or call ought put ought 𝑓𝑓 Collar 𝐶𝐶 , : Collarto Bound Market 0} p (T Do not buy Do 𝑓𝑓 = sell above , 𝑃𝑃 – T . Cannot ) [Value of Call] =[Value Call] of E option and below max f C = from us where { 𝑓𝑓 𝑃𝑃 Put Put p − (T sellinga call 𝑝𝑝 , f ( T f P f C 𝑇𝑇 P ) -p , − E max{ 𝑇𝑇 (T Collaran implieseffective market price ) , 𝑓𝑓 , 0} T 𝐶𝐶 Call ), ), . > estricted to theto[ interval restricted the otherpartythe does not exercisecall. option f P -p 𝑓𝑓 𝑃𝑃 (T . , T What if if What ), 0 } - E . f P max{ = K=0 f Prices C ? p (T implies implies f P , , T f C )-f ]. C , 0}. f C = f P .

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metin /~    ProtectivePut Contract is acombination of buying a forward and buying a put Profits depending on future price future depending on Profits Non Put option: We Forward contract: We will buy at Combinations = ≥    = E p = f E P -discountedCovered Call Valueof max{ If If If - (T f F f f p . , F F T (T < < p )- f P , T ,p f f P ) ≤ f ≤ ) F (T (T ≤ +E max{ buy at buy at , , p T T (T F ) ≤ ) )} < , -f T f f P ), ), F P , at at , then f then we then wethen exerciseput the option andearn P f -p P where (T : ProtectivePut we exercise f p P , (T T -f ), F do exercisenot , T 0 f ) } f P F (Value) Earning . > = Protected f F . the Forward Forward f F put option and earn the putthe option and f P + Contracts p (T , T f Put f P P ) - - earn f f F F . . p (T , T ) - f F .

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metin /~    CoveredContract Call Profits depending on future price future depending on Profits Non option:AnotherCall can buy at party Forwardbuy atwill Wecontract: ≤    = f E C -discountedCovered Call Valueof p If If If - (T Combinations f F f f p . , F F T (T < < p )- , T f f C Covered Call ) ≤ f ≤ ) F (T ≤ -E max{ , T p F (T ) ≤ ) isa combination of buying a forward and sellinga call < , T f f C C ), ), , thenotherparty, the does notoption exercisethe calland we earn , p then the other party exercises thenotherparty the does not exercise (T , f T C )-f p -f -f (T F F C , , 0} = T (Value) Earning f : CoveredCall ) F . f E C from us where p = (T f , F T )+E min{ )+E Forward Forward f C Covered the calloption f -p C (T > p , f T F (T . )+f the call optionand we earncall the , T − ) C Contracts , 0} -f Call and we earnand we F = E min{= E f C f ,p C - (T f F , T . )} p p -f (T (T F , , T T ) ) - - f f

F

. F

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metin /~    When K is explicit, is K When will decrease the (floor) strike price. K, When buying these options, Buyer ofthe  call option seller will theincrease (ceiling)strike price and similarlyput option seller theice.com/ Call time totime figure out priceswhich change fromone hour to another. exercise or cancel cancel or exercise Buyerto decides severaloption before sellers buying an option, Cargilltry the swap swap the Swaption : Option + : Option Swaption electricity.jhtml what value should itvalue take?what has the option to cancel the swap before thethe first beforeto option swap cancel thehas K can be explicitor implicitprices caninstrikeK be the , BP , Buyer Selle price price Contract , ShellNegotiateoil.for prices. You for not have mattoo much r K

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p t Floating price Floating , CMEGroup.com, Fixed swap becomes due. Period

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t +1) +1, ( t t p +1 Floating price Floating

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metin /~    an arithmetic meancollected ofan arithmetic pricesfromnow maturity.until Asian American European    Evaluation of optionsAsian is challenging cannotdoneand analytically. be Averaging prices over perioda makesan option contracts can include terms on averageprices during that period. Asian options are relevant in energy becauseoil, gas, electricity transactions happen over a time period and optionsexercised are onlythe payoffbutmaturitybut maturityatthejust typically atnot price is . . Options options canbe exercisednow from any time Use existing numerical approximations. You developcan your own numericalmethod, or . . American and Asian Options and Asian American Search onlineSearchcalledis forthethatmanual theout First check the package “ package the (discussed above) can be exercisedmaturity only at foptions ”. Rmetrics packageof open source software R: path - dependent foptions.pdf to maturity, and significantly evaluation.its complicates , when this linethis written. is when, i.e., T o . ver https://www.rmetrics.org the interval[0, T ]. . especially especially .

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metin /~        SwaptionOption: + Question: What is Protective Put Covered Call Protective Put Question: What isForward + Collar,allat the same strikeprice Collar= Put ForwardCall = – Arithmetic - Arithmetic Derivative Undermatching strike prices, – = = Call, differentstrikeprices – Forward Combinations makesbalancing harder; Forward Put,pricesame strike Swap Derivativeof Derivatives Be carefulBe and humble! − + Call Put Protective Put – Covered Call? = Put – – Covered Call = Collar Call Sarcastically Call Call

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metin /~       OilCompanies vs. OilServices Companies What should oil service companiesserviceshould What oildo? Oil companies are hedged betteragainst price drops than oilservice companies. July Oil service companies are more exposed to oilprice fluctuationsthan oilcompanies. price of the oildrops, oilservices companies suffermore than oilcompanies. options.WhenOilput thesamecannottheyso buy the Services sellto companies do have notoil and the revenue of the oilproducing company. spot price of oildrops below thatcertainprice. This strategyhedges the risk of dropping oilprice Oil     (producing) 1 Change in the Oil Companies Index (XOI): 1,687.12 Change in the Oil Service Companies Index (OSX): 310.72 Change violently, Land - Dec 31, drillers drillers in the Oil price:100.78 as theas owners theof expensive, depreciatingthe genesiscapitalofat new activity. companies buy 2014 [oil service[oilcompanies] are - , Credit Suisse analyst James WicklundSuisse analyst in quoted put options totheir selloilat acertain floor price even when the Hedging of  Buy airline . Buy airline 53.27: the end of theend of the 47% drop  Barron’s. Jan 7, 2014. 2014. 7, Jan Barron’s. whip. and mostThey move first 1,348.13:  210.86: 20% drop 32% drop

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metin /~         $20: see CME offersUS Monthly WeatherHDD Future contractswhere degreeeach unit (1 1 day)for worths 40,000 hot or cold): 40,000or hot what the HDD contract offersisnot an increase in profitsbut a stability of profits(40,000 and Although Flag’sSixwashearningson average can (40,000losses away = its ProfitfromHDD units+ Change in Ticket revenue WeatherMarchHot in ColdWeather March in MarchoutIf turnsbe hot to MarchoutIf turnsbe cold to paying $80,000by (=80*50*20).unitsasset.ConsiderbuysHDD 80 Supposean Flags Six as index Suppose the Dallas HDD index for March is50. – – – – – – – – – – – http://www.cmegroup.com/trading/weather/temperature/us Net profits with contract= 40,000 and 40,000 from ticket salesas well as the contract. profits without= contract Net − units=$80,000.fromLoss HDD willunitsSix Flag’s worth(=80* HDD0 $0 Profitfrom units=$160,000. HDD willunitsSix Flag’s worth HDD $240,000 (=80*150*20). {65- max 31Jan [over days sum Jan: in HDD Dallas in temperature daily {average max = day particular days) on a degree CDD (cooling Similarly, {65- max = day HDDparticular on a 160,000 TradingDerivative WeatherExample 80,000 − + 30*4,000 30*4,000 Weather derivatives decrease riskexposure = = = HDD lowis so Six Flags ticketrevenue increases by$4,000 per day. = HDD is high so Six Flags ticket revenue drops by $4,000 per day $4,000by perrevenue drops ticketFlagsSix sohigh = HDD is 40,000is if March 40,000is if March and HDDand becomes and HDD becomesand − average daily temperature} 120,000 and 120,000120,000 only and fromticketsales cold hot *20). average daily temperature} 0 - 150 , - monthly (as heating is required), weather - heating.html ] . - 65} 80,000)/2), -80,000)/2), (160,000

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metin /~   from { from are obtained with or without the contract but the contractreduces still the variability of the payoff If the example,theappearsIn last it thatcontract always deliversnet“more”than profits otherwise. – – – Contractreducedthevariance down to zeroin this specific example. ( ( thingsconcrete,makeTo morevarianceof payoffs:think ofthe Thisincorrect is onaverageif probability for This is correct on averageif ( ( 3/4 1/3 0.5 cold 1/3 -120,000,120,000}40,000. to TradingDerivative WeatherExample )40,000+( )40,000+( )(40,000- )40,000+( March and 40,000) 0.5 1/4 2/3 )40,000=40,000 )40,000 = 40,000)40,000 = Average PayoffAverage and Risk )40,000 = 40,000)40,000 = hot 2 +( 2/3 March have probabilities1/3of and 2/3,expected same (average) payoffs )(40,000 cold and > < = 0 = 0 - 40,000) 60,000 = (3/4 60,000 = ( hot 2/3 0.5 March are are March )(120,000)+( (120,000)+ 2 =0 < =0 cold (38.4/3)*10 March is less than )(120,000)+( equally likely 0.5 1/3 ( )( - 120,000). - 120,000) and 9 = ( 2/3 1/4 )(120,000- )( as as - 120,000). 1/3 , say 1/4, as1/4, say , 40,000) 2 +( 1/3 )( - 120,000- 40,000

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metin /~ Summary     Weather Derivatives Hedging Collar, Put, Call, Options: Put, CoveredCall, Covered Swaps and Futures Forwards,

For

𝑞𝑞𝑓𝑓 ≤ 𝑞𝑞𝑝𝑝 ≤ 𝑞𝑞𝑐𝑐 Swaption

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metin /~  Summary Summary of theCourse  Resources  1 2. Oil Oil 2. 3 4 Technology . Hydrocarbon . .  Gas Coal 5 9. Hydro et al. Hydro9. et 8. 7. 6. Trans . Enhanced Oil. 13. Electric 12. 11. Electricity 10. Wind Solar Nuclear E&P Markets 16. 15. Market Pipelines 14. Refineries (-formation & Derivatives Markets Geology Grid and Risk and Generation Recovery Models -portation ) O ur ur My teaching is over is teaching My Do not give up ever up give not Do learning is forever is learning Towards better Towards

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metin /~ Aside: Continuous Compounding Aside:      What if I earninterestan of (r/infinity)per (12/infinity) months? compounding when m approaches infinity. Thinkcontinuous of compounding as the special casediscrete of If I earn an interest of (r/m) per (12/m) months, what is my ofend the year? If I earn an interest of r/2 per six months, what is my interest+ investment at the interest+investment If my $1investment earns an interest of r per year, whatis my Answer [Present valuefuture of value V :

m lim → ∞    1 + m at the endthe of year? r    m Answer: (1+r/2) (1+r) Answer: Answer: (1+r/m) Answer: = e r where

e = 2 m at time ∑ n ∞ = 0 n! 1 = T] = exp( 1 1 + 1 1 + 1 2 -rT + ? interest+investment )V. 1 6 + - time time 24 1 + 120 1

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metin /~