Commodity_Curve.pdf

7/28/2019 Commodity_Curve.pdf

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Commodities are different, because

They are produced, consumed,They are produced, consumed,transported, and stored, sotransported, and stored, so Market inventory swings wildlyMarket inventory swings wildly

Owning a commodity at one place and time is aOwning a commodity at one place and time is acompletely different financial asset from owning itcompletely different financial asset from owning itat another. Enforcing arbitrage relationshipsat another. Enforcing arbitrage relationships

between them is expensive or impossiblebetween them is expensive or impossible


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Examples of Traded Commodities

EnergyEnergy

crude oil, gasoline, heating oil, natural gas, electric power, ecrude oil, gasoline, heating oil, natural gas, electric power, etctc

Precious MetalsPrecious Metals

gold, silver, platinum, palladium etcgold, silver, platinum, palladium etc

Base MetalsBase Metals

aluminum, copper, nickel, zinc, etc.aluminum, copper, nickel, zinc, etc. AgriculturalAgricultural

grains, soy beans, coffee, pork bellies, etcgrains, soy beans, coffee, pork bellies, etc

OthersOthers

pulp, paper, weather, chemicals, etcpulp, paper, weather, chemicals, etc

Specifications need to be standardized to create trading volume.Specifications need to be standardized to create trading volume.


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The customers of commodity derivatives areThe customers of commodity derivatives areindustrial producers and consumers, andindustrial producers and consumers, andsometimes governments who depend on thesometimes governments who depend on therevenue.revenue.

Particularly in energy, these customers areParticularly in energy, these customers areparticularly risk averse, because of legalparticularly risk averse, because of legal

sanctions for failure to deliver.sanctions for failure to deliver.

The customers of commodity derivatives


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The underlying assets for commodity

derivatives are forwards and futures, notspot

This is a reflection of the statement that theThis is a reflection of the statement that the

same commodity at a different place orsame commodity at a different place or

time is a different financial asset.time is a different financial asset.

In addition, hedging with spot is impractical,In addition, hedging with spot is impractical,

because spot is much less liquid, andbecause spot is much less liquid, and

it is impossible to short the spotit is impossible to short the spot

commodity.commodity.

t


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orwar s an utures onCommodities have special features

Forwards and Futures traded in the marketForwards and Futures traded in the market

Physical forward delivers physical every day for aPhysical forward delivers physical every day for amonth, like an average of the spot pricemonth, like an average of the spot price

NYMEX futures, settles on physical forwardsNYMEX futures, settles on physical forwards

NYMEXNYMEX LookalikeLookalike forwards, settles on NYMEX futureforwards, settles on NYMEX futureprice at expiryprice at expiry

Publication forwards, e.g. Platts, settle on thePublication forwards, e.g. Platts, settle on the

monthly average of the Platts poll of closing spotmonthly average of the Platts poll of closing spotpricesprices

Calendar Swap settles on monthly average of closingCalendar Swap settles on monthly average of closing

NYMEX pricesNYMEX prices


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The first nearby is simply the forward contractThe first nearby is simply the forward contractclosest to expiry. The second nearby is theclosest to expiry. The second nearby is the

second closest, etc.second closest, etc.When a forward contract expires, it is said toWhen a forward contract expires, it is said to

roll off. The second nearby becomes theroll off. The second nearby becomes thefirst, the third becomes the second, etc.first, the third becomes the second, etc.

Most exotic derivatives e.g. barriers andMost exotic derivatives e.g. barriers and

average rates, are written onaverage rates, are written on nearbysnearbys, rather, ratherthan on particular forwards, so that theythan on particular forwards, so that theyactually refer to several different forwards.actually refer to several different forwards.

Forwards are referred to in terms of nearbys


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The Shape of the Forward Curve

There are no curve flattening arbitragesThere are no curve flattening arbitrages

available in commodities.available in commodities.

If the curve is upward sloping, then buy theIf the curve is upward sloping, then buy theearlier forward and sell the later, butearlier forward and sell the later, but

one has to take delivery, and store it. Canone has to take delivery, and store it. Canonly make money if price difference is greateronly make money if price difference is greaterthan storage costs, defines thethan storage costs, defines the contangocontango

limitlimit.. If the curve is downward sloping, need toIf the curve is downward sloping, need to

short the spot commodityshort the spot commodity impossible.impossible.


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Behavior of the Forward Curve

Almost all commodities forward curveAlmost all commodities forward curvehave a stable long end, and a violent,have a stable long end, and a violent,

whipping short end.whipping short end. Long end sits near marginal cost ofLong end sits near marginal cost of

productionproduction

Short end governed by short termShort end governed by short termsupply and demandsupply and demand

If short end is below long end we areIf short end is below long end we arein glut ==in glut == contangocontango

If short end is above long end we areIf short end is above long end we arein shortage ==in shortage == backwardationbackwardation


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What is special about commodities forward curves?

BackwardationBackwardation

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Bias of the forward curve

Most trading volume takes place at the long end ofMost trading volume takes place at the long end ofthe curvethe curve industry buys well in advance.industry buys well in advance.

Short end of the curve is used to cover unanticipatedShort end of the curve is used to cover unanticipated

demanddemand

Because industry in general, and utilities in particularBecause industry in general, and utilities in particularsuffer out of proportion to the trading gain/loss if theysuffer out of proportion to the trading gain/loss if they

fail to deliver, the front of the forward curve is almostfail to deliver, the front of the forward curve is almostalways bid up, i.e. backwardated. In financial terms,always bid up, i.e. backwardated. In financial terms,this translates to extreme riskthis translates to extreme risk--aversionaversion

Investor indices such as GSCI have been invented toInvestor indices such as GSCI have been invented toallow investors to enter this market, and ride up theallow investors to enter this market, and ride up theforward curveforward curve

Recently, hedge funds have entered the market,Recently, hedge funds have entered the market,generating a large net speculative lengthgenerating a large net speculative length


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Forward curves display seasonality

Intermediate points on commodities forward curvesIntermediate points on commodities forward curvestend to have humps at points of anticipated hightend to have humps at points of anticipated highdemand, or supply constraint, and valleys where lowdemand, or supply constraint, and valleys where low

demand or high supply are anticipateddemand or high supply are anticipated This is mitigated when there is storage capacityThis is mitigated when there is storage capacity

covering many more days than the length of thecovering many more days than the length of the

hump or valley.hump or valley. Natural gas has a large hump in winter, a small oneNatural gas has a large hump in winter, a small one

in summerin summer

Gasoline has a large hump in theGasoline has a large hump in the summer drivingsummer drivingseasonseason

Electricity has yearly humps in summer and winter,Electricity has yearly humps in summer and winter,humps on weekdays, and humps during workinghumps on weekdays, and humps during working

hourshours


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The Build up to Gulf War IThe Build up to Gulf War I

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Finer pointsFiner pointsShape of forward curve affected by available storage and transpoShape of forward curve affected by available storage and transportationrtation

Sufficient shortSufficient short--term supply & transport implies shortterm supply & transport implies short--term contangoterm contango

Aluminum marketAluminum market

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Regular demand/consumption patternsRegular demand/consumption patternsreflected in the shape of the curvereflected in the shape of the curve

Seasonality in natural gas, heating oilSeasonality in natural gas, heating oil

heating oil natgas

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Example: Storage and Seasonality in the US Oil Markets

US consumes 22mm bbls/day, and produces about 5mmUS consumes 22mm bbls/day, and produces about 5mmbbls/daybbls/day

Extraction costs range from $2.50/bbl to $12/bblExtraction costs range from $2.50/bbl to $12/bbl

Storage costs $0.15/bblStorage costs $0.15/bbl mthmth -- $0.30/bbl$0.30/bbl mthmth, total storage, total storagecapacity 350mm bbls, with a minimum of 265mm bbls. Incapacity 350mm bbls, with a minimum of 265mm bbls. Inaddition, there is the US Strategic Petroleum Reserve, but thisaddition, there is the US Strategic Petroleum Reserve, but this isisheld out of the market most of the time.held out of the market most of the time.

Transport costs are about $0.20/ bbl/Transport costs are about $0.20/ bbl/ kmilekmile..

Little seasonality in crude oil, but there is seasonality in heaLittle seasonality in crude oil, but there is seasonality in heatingtingoil, gasoline, etc.oil, gasoline, etc.


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Example: Storage and Seasonality in the US Power Markets

Power is segmented into separate markets by time of day.Power is segmented into separate markets by time of day.

You can buy either On Peak, or Off Peak, there is a smallerYou can buy either On Peak, or Off Peak, there is a smallermarket in individual hours.market in individual hours.

These different times of day have such different properties andThese different times of day have such different properties andpricing that they are regarded as different assets.pricing that they are regarded as different assets.

Seasonalities are intraSeasonalities are intra--day, intraday, intra--week, and intraweek, and intra--year.year.

Power supply is generated by plants with varying efficienciesPower supply is generated by plants with varying efficiencies

and start up times, arranged in a generation stack. The mostand start up times, arranged in a generation stack. The mostefficient longest startup time plants are at the bottom, and theefficient longest startup time plants are at the bottom, and theothers are arranged in descending order of efficiency, in aothers are arranged in descending order of efficiency, in ageneration stackgeneration stack

Power price jumps with demand as we move up the generationPower price jumps with demand as we move up the generationstack.stack.

It is also possible to transport, if there is spare capacity, buIt is also possible to transport, if there is spare capacity, butttransport between neighboring markets costs 1transport between neighboring markets costs 1--5$/MW5$/MW--hr, out ofhr, out of$35/MW$35/MW--hr for a typical plant. Also 3% is lost in transmissionhr for a typical plant. Also 3% is lost in transmissionwires.wires.


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Monthly, weekly, dailyMonthly, weekly, daily seasonalityseasonality for powerfor power

peak offpeak weekends+holidays

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Example: Storage and Seasonality in London Base Metals

Storage for base metals is cheap, and plentifulStorage for base metals is cheap, and plentiful

Transport costs around $0.05/lbTransport costs around $0.05/lb -- $0.08/lb$0.08/lb

Certain metals have seasonality of demand, but thisCertain metals have seasonality of demand, but thisdoes not show up in forward curve, possibly becausedoes not show up in forward curve, possibly becauseof plentiful storage.of plentiful storage.

Aluminum is demanded in summer, by beverageAluminum is demanded in summer, by beverage

makersmakers Lead is demanded in winter, by battery makersLead is demanded in winter, by battery makers


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Forward Curves are Frequency-limited by

trading from storage operators.

In order to move the markets for a single futures contract, itIn order to move the markets for a single futures contract, it

takes trading in a volume of size with the same magnitude astakes trading in a volume of size with the same magnitude asthe daily usage.the daily usage. To stamp out a peak of width T in the forward curve, we needTo stamp out a peak of width T in the forward curve, we need

around T days usage in storage (roughly).around T days usage in storage (roughly). To stamp out a trough of width T in the forward curve, we needTo stamp out a trough of width T in the forward curve, we need

around T days usage of storage capacity (roughly).around T days usage of storage capacity (roughly). Thus, we expect to see details in the forward curve no smallerThus, we expect to see details in the forward curve no smaller

than the number of days usage in storage, in normal situations.than the number of days usage in storage, in normal situations. When close to the lower limit of storage, we can see finer detaiWhen close to the lower limit of storage, we can see finer detailsls

in peaks, when close to the upper limit of storage, we can seein peaks, when close to the upper limit of storage, we can seefiner details in troughs.finer details in troughs.


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The Volatility Surface Constituents

The volatility surface is made up of options onThe volatility surface is made up of options onfutures, one option maturity for each futuresfutures, one option maturity for each futurescontract, maturing within a few days (up to acontract, maturing within a few days (up to a

week or two) of the futures maturity. In mostweek or two) of the futures maturity. In mostmarkets, the liquid options can range inmarkets, the liquid options can range inmoneynessmoneyness from 0.5 to 2, and possibly more.from 0.5 to 2, and possibly more.

Because these futures are really differentBecause these futures are really differentassets, this is not a volatility surface in theassets, this is not a volatility surface in theusual senseusual sense


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Volatilities in commodities markets are

almost always backwardated

Long end moves with long termLong end moves with long termdemand, determined by weather,demand, determined by weather, gdpgdp

growth. Very slow, little volatility, 2growth. Very slow, little volatility, 2--10%10%

instantaneous volatilityinstantaneous volatility Short end whips around with short termShort end whips around with short term

supply and demand (200%supply and demand (200%--300%)300%) Reversion occurs over a few weeks.Reversion occurs over a few weeks.

H l ili i l d h


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How volatility term-structure is related to the

demand & consumptionMeanMean--reverting nature of the market is reflected inreverting nature of the market is reflected in

the termthe term--structure of volatilitiesstructure of volatilities

Supply/demand imbalancesSupply/demand imbalances excessive whippiness of the front end of the forward curveexcessive whippiness of the front end of the forward curve

high volatility of shorthigh volatility of short--dated optionsdated options

Backwardated vol curveBackwardated vol curve

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1 M a y 0 2 1 2 S e p 0 51 J a n 0 3 1 J u l 0 3 1 J a n 0 4 1 J u l 0 4 1 J a n 0 5

In crises volatility can become contango


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In crises, volatility can become contango

And variance can backwardate!!!

On occasion there is a supply crunch whichOn occasion there is a supply crunch which

affects one month, and not the succeeding one.affects one month, and not the succeeding one.Implied Volatil ities explode for the affectedImplied Volatil ities explode for the affectedmonth, but then drop back down for themonth, but then drop back down for thesucceeding month. This can even go to thesucceeding month. This can even go to theextent of backwardating the variances. Becauseextent of backwardating the variances. Becauseone cannot short spot, this cannot be arbitraged.one cannot short spot, this cannot be arbitraged.

In March 2003, this happened in the US NaturalIn March 2003, this happened in the US Natural

Gas markets, because it was a cold winter and weGas markets, because it was a cold winter and weran out of Natural gas in Texas.ran out of Natural gas in Texas.


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The volatility skew is primarily determined

by inventory effects

Most market participants are industrial, extremely riskMost market participants are industrial, extremely risk

averse, hedging exposure.averse, hedging exposure. Producers want OTM puts, Consumers want OTMProducers want OTM puts, Consumers want OTM

calls.calls. Market is rarely in balance, and in some cases it isMarket is rarely in balance, and in some cases it is

extreme.extreme. Electricity hedging is only done by producers, volElectricity hedging is only done by producers, vol

surface is a diagonal line. ITM puts can be bought atsurface is a diagonal line. ITM puts can be bought ator close to intrinsic value, because dealers are so fullor close to intrinsic value, because dealers are so fullof them, they cannot bear further risk.of them, they cannot bear further risk.

Nat Gas hedging is only done by consumers. SkewNat Gas hedging is only done by consumers. Skewis very heavy the other way, because the market is allis very heavy the other way, because the market is all

one way.one way.


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What is volatility skew and how is it related


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What is volatility skew, and how is it related

to who dominates the marketScenario 2.Scenario 2.

Market dominated byMarket dominated by consumersconsumers ..

positivepositive call skew,call skew, negativenegative put skewput skew..

uick Delta

NGJ03 EXCHANGE Vol Skew [Graph #9]

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What is olatilit ske and ho is it related to


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What is volatility skew, and how is it related to

who dominates the marketScenario 3.Scenario 3.Market dominated by neitherMarket dominated by neither producersproducers ,,

nornor consumersconsumersSkew tend to be fairly symmetric and positive for the calls andSkew tend to be fairly symmetric and positive for the calls and putsputs..

u i c k D e l t a

W T IU 9 7 E XC H A N G E V o l S k e w [ G r ap h # 15 ]

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Kurtosis appears immediately , and lasts a long time

Jumpy behavior visible in observation of futuresJumpy behavior visible in observation of futures

trading, and in option prices close to expiry.trading, and in option prices close to expiry. Kurtosis is jumpKurtosis is jump--like, in that it appears immediately,like, in that it appears immediately,

does not build up.does not build up.

Kurtosis is also StochasticKurtosis is also Stochastic--volvol--like, in that it lasts alike, in that it lasts along time (more than a year).long time (more than a year).

Spikes are present, but do not affect vanilla optionSpikes are present, but do not affect vanilla optionvalue that much.value that much.


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Comes about when a stored supply isComes about when a stored supply isexhausted, or when demand outrunsexhausted, or when demand outrunsproduction capacityproduction capacity

Behavior is difficult to model withBehavior is difficult to model withMarkov models, requires regimeMarkov models, requires regime--switching, or extreme mean reversionswitching, or extreme mean reversion

Does not really influence value ofDoes not really influence value ofvanillas, but very important for barriers.vanillas, but very important for barriers.

Non-Black-Scholes Behavior: Spiking


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Non-Black-Scholes Behavior: Negative Prices

Happens in the power markets,Happens in the power markets,because there is no storage, andbecause there is no storage, and

because it costs a lot of money to shutbecause it costs a lot of money to shut

down and start up certain kinds ofdown and start up certain kinds ofplants (nuclear, coal).plants (nuclear, coal).

Happens in natural gas markets, butHappens in natural gas markets, butvery rarely.very rarely.


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Common Commodity Exotics: Transport Options

A simple option on the differenceA simple option on the differencebetween prices in two locations.between prices in two locations.

Sold as a strip.Sold as a strip. Incorporates a loss rateIncorporates a loss rate

Can be tricky to model, as correlation isCan be tricky to model, as correlation isclose to 1, yet poorly known, mostclose to 1, yet poorly known, mostmodels are singular atmodels are singular at =1=1


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Common Commodity Exotics: Load Serving Deals

Power Utilities would like to hedge not just the power price, buPower Utilities would like to hedge not just the power price, butt

the demand as well, because they cannot refuse to serve.the demand as well, because they cannot refuse to serve. The load is also highly correlated with the power price, as wellThe load is also highly correlated with the power price, as well asas

with weather, and with long term economic growth.with weather, and with long term economic growth.

There is no market in load, so crude models are marked toThere is no market in load, so crude models are marked tohistoric datahistoric data

There are no satisfactory models of load, andThere are no satisfactory models of load, and andand almost noalmost nowork has been done to model it, even though it is critical to mawork has been done to model it, even though it is critical to manynypeople.people.


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Common Commodity Exotics:

Crack Spread Options

Payoff is the difference between OilPayoff is the difference between OilProduct (Heating Oil, Fuel Oil) andProduct (Heating Oil, Fuel Oil) and

Crude, minus strike.Crude, minus strike.

The natural hedge for a refinery.The natural hedge for a refinery.


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Common Commod Exotics:

Spark Spread Options

The natural hedge for a gas burning powerThe natural hedge for a gas burning power

plant, the payoff is Payoff = max( Pplant, the payoff is Payoff = max( P

H *H *G,0)G,0)

Heat rate H represents efficiency of theHeat rate H represents efficiency of theplant, and varies from deal to deal, and fromplant, and varies from deal to deal, and fromplant to plant.plant to plant.

For less efficient plants, higher up theFor less efficient plants, higher up thegeneration stack, a strike is sometimesgeneration stack, a strike is sometimesincluded.included.


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Common Commodity Exotics: Swing Options

This is an option to hedge out the flexibility that aThis is an option to hedge out the flexibility that a

customer has in buying natural gas.customer has in buying natural gas. A customer contracts to buy a certain quantity ofA customer contracts to buy a certain quantity of

natural gas over a series of periods. He has thenatural gas over a series of periods. He has theoption to take a certain amount each day, at theoption to take a certain amount each day, at the

floating rate. He must buy at least a minimumfloating rate. He must buy at least a minimumamount within the period, or there are penalties.amount within the period, or there are penalties.There is rebating in the next period if he buys moreThere is rebating in the next period if he buys morethan the maximum in a period.than the maximum in a period.

This has a lot of optionality, and is very timeThis has a lot of optionality, and is very time--consuming to evaluate, even in a simple model.consuming to evaluate, even in a simple model. This is another interesting problem for academics.This is another interesting problem for academics.


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Commodities Models: Basic Features

Spot Price ModelsSpot Price Models

Evaluate futures asEvaluate futures as F_tTF_tT = E( S_T |= E( S_T | S_tS_t ), almost), almost

always a smooth function (Cant have discontinuousalways a smooth function (Cant have discontinuousforward curve!)forward curve!)

Almost always have mean reversionAlmost always have mean reversion

parametrizeparametrize forward curve with convenience yield yforward curve with convenience yield yF_tTF_tT == S_tS_t exp( (r + uexp( (r + u y)(Ty)(T--tt) ), u = storage rate.) ), u = storage rate.

Spot models are limited, cant have negative forwardSpot models are limited, cant have negative forwardvariance in futures. Hard to put in sharply varyingvariance in futures. Hard to put in sharply varyingforward curves.forward curves.

But Spot models are much more tractable, with fewerBut Spot models are much more tractable, with fewerfactors.factors.


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Commodity Models: Basic Features

Models of whole curve (i.e. 1 factor forModels of whole curve (i.e. 1 factor foreach futures maturity) are capable ofeach futures maturity) are capable of

encompassing most observedencompassing most observed

phenomena, but have many morephenomena, but have many morefactors, and so are hard to evaluate.factors, and so are hard to evaluate.

BGMBGM--like Factor models are a kind oflike Factor models are a kind ofcompromise.compromise.


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Should have some form of mean reversionShould have some form of mean reversion

Should beShould be generalizablegeneralizable to a multito a multi--commodity model,commodity model,or multior multi--location modellocation model

A model capturing the vol smile should beA model capturing the vol smile should be

calibratablecalibratable to oddto odd--shaped vol surfaces, distorted byshaped vol surfaces, distorted byinventory effects.inventory effects.

A model capturing the vol smile should probablyA model capturing the vol smile should probablycontain jumps.contain jumps.

A model of storage, transport, or refineryA model of storage, transport, or refinery--capacitycapacityconstrained commodities should include theseconstrained commodities should include thesevariables, and their relation to their limits.variables, and their relation to their limits.


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Market Specific: Natural Gas models mayMarket Specific: Natural Gas models may

want to use the storage limits, and currentwant to use the storage limits, and currentvalue of storage as a parameter, controllingvalue of storage as a parameter, controllingjumpiness, now that there is a forward marketjumpiness, now that there is a forward marketin storage numbers.in storage numbers.

Market Specific: Electricity markets shouldMarket Specific: Electricity markets should

separate different parts of the curve intoseparate different parts of the curve intodifferent assets, hour, daydifferent assets, hour, day--ofof--week, seasonweek, season


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Some example models: Gibson-Schwartz Model

A spot model for electricity, withA spot model for electricity, withstochastic convenience yield.stochastic convenience yield.

Cannot accommodate sharply varyingCannot accommodate sharply varying

forward curves, kurtosis, skew, negativeforward curves, kurtosis, skew, negativeforward variance.forward variance.

Does not meanDoes not mean--revert, so variancerevert, so variancegrows too fast at long times.grows too fast at long times.


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Some example models: Schwartz-Smith Model

A spot model for electricity, modeling spot as a lowA spot model for electricity, modeling spot as a low--

vol long term rate, plus a rapidly varying difference,vol long term rate, plus a rapidly varying difference,mean reverting to zero.mean reverting to zero.

dd == -- kk dtdt ++ __ dZdZ__

dd == __ dtdt ++ __ dZdZ__Spot =Spot = ++

Cannot accommodate sharply varying forwardCannot accommodate sharply varying forwardcurves, kurtosis, skew, negative forward variance.curves, kurtosis, skew, negative forward variance.

Has some deHas some de--correlation of futures, for time spreadcorrelation of futures, for time spreadoptionsoptions


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Some example models: Gabillon Model

A spot model for energy, modeling spot as aA spot model for energy, modeling spot as a

single factor Gaussian process that meansingle factor Gaussian process that meanreverts to a lognormal long term ratereverts to a lognormal long term rate dSdS/S/S == (( lnln LL lnln S )S ) dtdt ++ _S_S dZ_SdZ_S

dLdL/L =/L = _L_L dtdt ++ _L_L dZ_LdZ_L Cannot accommodate sharply varyingCannot accommodate sharply varying

forward curves, kurtosis, skew, negativeforward curves, kurtosis, skew, negative

forward variance.forward variance. Has some deHas some de--correlation of futures, for timecorrelation of futures, for time

spread optionsspread options


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Bibliography Platts Gas Daily, Platts Electricity DailyPlatts Gas Daily, Platts Electricity Daily

EydelandEydeland, A. and Wolyniec, K. Energy and Power, A. and Wolyniec, K. Energy and PowerRisk Management, Wiley Finance, 2003Risk Management, Wiley Finance, 2003 GemanGeman && VasicekVasicek Plugging into Electricity, Risk,Plugging into Electricity, Risk,

Aug. 2001.Aug. 2001.

ShijieShijie Deng, Stochastic Models of EnergyDeng, Stochastic Models of EnergyCommodity Prices and Their Applications: MeanCommodity Prices and Their Applications: Mean--reversion with Jumps and Spikesreversion with Jumps and Spikes

CavusCavus, Mustafa and, Mustafa and PaxsonPaxson, Dean A. The Valuation, Dean A. The Valuationand Effectiveness of Long Term Forward Contracts.and Effectiveness of Long Term Forward Contracts.

AudetAudet, N.,, N., HeiskanenHeiskanen, P.,, P., KeppoKeppo, J. and, J. and VehvilainenVehvilainen,,I I ModellingModelling of Electricity forward curve dynamics.of Electricity forward curve dynamics.


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Acknowledgments: I would like to express my gratitude to the followingI would like to express my gratitude to the following

people for their willing and able help. Jamie Coxpeople for their willing and able help. Jamie Cox--Jones, Ben Freeman, MichaelJones, Ben Freeman, Michael KirchKirch, Ilya Ustilovsky,, Ilya Ustilovsky,DanDan SharfmanSharfman, Elisha Wiesel, Alex, Elisha Wiesel, Alex LesinLesin, Roberto, RobertoCaccia, Derek Yi, SofiaCaccia, Derek Yi, Sofia CheidvasserCheidvasser, Karhan, Karhan

Akcoglu, Bill Cowieson, Jeremy Glick,Akcoglu, Bill Cowieson, Jeremy Glick, LavanyaLavanyaViswanathanViswanathan, Alan Yamamura and Pavel Langer for, Alan Yamamura and Pavel Langer fortheir advice and criticisms. I would also like to thanktheir advice and criticisms. I would also like to thank

Peter Carr, Marco Avellaneda, Bob Kohn, EmanuelPeter Carr, Marco Avellaneda, Bob Kohn, EmanuelDerman, as well as Valerie Perugini.Derman, as well as Valerie Perugini.

Documents

Commodity_Curve.pdf