Commodities as Financial Assets

Commodities are different, because…

They are produced, consumed, They are produced, consumed, transported, and stored, so…transported, and stored, so… Market inventory swings wildlyMarket inventory swings wildly Owning a commodity at one place and time is a Owning a commodity at one place and time is a

completely different financial asset from owning it completely different financial asset from owning it at another. Enforcing arbitrage relationships at another. Enforcing arbitrage relationships between them is expensive or impossiblebetween them is expensive or impossible

Examples of Traded Commodities

EnergyEnergy crude oil, gasoline, heating oil, natural gas, electric power, etccrude oil, gasoline, heating oil, natural gas, electric power, etc

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.

The customers of commodity derivatives are The customers of commodity derivatives are industrial producers and consumers, and industrial producers and consumers, and sometimes governments who depend on the sometimes governments who depend on the revenue. revenue.

Particularly in energy, these customers are Particularly in energy, these customers are particularly risk averse, because of legal particularly risk averse, because of legal sanctions for failure to deliver. sanctions for failure to deliver.

The customers of commodity derivatives

The underlying assets for commodity derivatives are forwards and futures, not spot

This is a reflection of the statement that the This is a reflection of the statement that the same commodity at a different place or same 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, and because spot is much less liquid, and

it is impossible to short the spot it is impossible to short the spot commodity. commodity.

  

Forwards and Futures traded in the marketForwards and Futures traded in the market Physical forward delivers physical every day for a Physical forward delivers physical every day for a

month, 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 NYMEX Lookalike forwards, settles on NYMEX future NYMEX Lookalike forwards, settles on NYMEX future

price at expiryprice at expiry Publication forwards, e.g. Platt’s, settle on the Publication forwards, e.g. Platt’s, settle on the

monthly average of the Platt’s poll of closing spot monthly average of the Platt’s poll of closing spot pricesprices

Calendar Swap settles on monthly average of closing Calendar Swap settles on monthly average of closing NYMEX pricesNYMEX prices

Forwards and Futures on Commodities have special features

The first nearby is simply the forward contract The first nearby is simply the forward contract

closest to expiry. The second nearby is the closest to expiry. The second nearby is the second closest, etc. second closest, etc.

When a forward contract expires, it is said to When a forward contract expires, it is said to “roll off”. The second nearby becomes the “roll off”. The second nearby becomes the first, the third becomes the second, etc. first, the third becomes the second, etc.

Most exotic derivatives e.g. barriers and Most exotic derivatives e.g. barriers and average rates, are written on nearbys, rather average rates, are written on nearbys, rather than on particular forwards, so that they than on particular forwards, so that they actually refer to several different forwards. actually refer to several different forwards.

Forwards are referred to in terms of nearbys

There are no curve flattening arbitrages There are no curve flattening arbitrages available in commodities. available in commodities.

If the curve is upward sloping, then buy the If the curve is upward sloping, then buy the earlier forward and sell the later, but earlier forward and sell the later, but one has to take delivery, and store it. Can one has to take delivery, and store it. Can only make money if price difference is greater only make money if price difference is greater than storage costs, defines the “contango than storage costs, defines the “contango limit”. limit”.

If the curve is downward sloping, need to If the curve is downward sloping, need to short the spot commodity – impossible.short the spot commodity – impossible.

The Shape of the Forward Curve

Behavior of the Forward Curve Almost all commodities forward curve have Almost all commodities forward curve have

a stable long end, and a violent, whipping a stable long end, and a violent, whipping short end. short end.

Long end sits near marginal cost of Long end sits near marginal cost of productionproduction

Short end governed by short term supply Short end governed by short term supply and demandand demand If short end is below long end we are in If short end is below long end we are in

glut == “contango”glut == “contango” If short end is above long end we are in If short end is above long end we are in

shortage == “backwardation”shortage == “backwardation”

Bias of the forward curve Most trading volume takes place at the long end of Most trading volume takes place at the long end of

the curve – industry buys well in advance.the curve – industry buys well in advance. Short end of the curve is used to cover unanticipated Short end of the curve is used to cover unanticipated

demanddemand Because industry in general, and utilities in particular Because industry in general, and utilities in particular

suffer out of proportion to the trading gain/loss if they suffer out of proportion to the trading gain/loss if they fail to deliver, the front of the forward curve is almost fail to deliver, the front of the forward curve is almost always bid up, i.e. backwardated. In financial terms, always bid up, i.e. backwardated. In financial terms, this translates to extreme risk-aversionthis translates to extreme risk-aversion

Investor indices such as GSCI have been invented to Investor indices such as GSCI have been invented to allow investors to enter this market, and ride up the allow investors to enter this market, and ride up the forward curveforward curve

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

What is special about commodities forward curves? BackwardationBackwardation

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What is special about commodities forward curves?

Forward curves display seasonality Intermediate points on commodities forward curves Intermediate points on commodities forward curves

tend to have humps at points of anticipated high tend to have humps at points of anticipated high demand, or supply constraint, and valleys where low demand, 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 capacity This is mitigated when there is storage capacity covering many more days than the length of the hump covering many more days than the length of the hump or valley. or valley.

Natural gas has a large hump in winter, a small one in Natural gas has a large hump in winter, a small one in summersummer

Gasoline has a large hump in the “summer driving Gasoline has a large hump in the “summer driving season”season”

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

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

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What is special about commodities forward curves?

Finer pointsFiner points

Shape of forward curve affected by available storage and transportationShape of forward curve affected by available storage and transportation

Sufficient short-term supply & transport implies short-term contangoSufficient short-term supply & transport implies short-term contango Aluminum marketAluminum market

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What is special about commodities forward curves?

Example: Storage and Seasonality in the US Natural Gas Market

US Natural gas production and consumption average US Natural gas production and consumption average 550bcf/mth.550bcf/mth.

We withdraw from November-April (“winter”) and store from We withdraw from November-April (“winter”) and store from April-November (“injection season”)April-November (“injection season”)

Total NG storage is 3.2 tcf, with a minimum of 500bcf. Total NG storage is 3.2 tcf, with a minimum of 500bcf. NG forwards are contango leading up to November, and NG forwards are contango leading up to November, and

contango afterwards. contango afterwards. Extreme volatility in the March contract, if it looks like we might Extreme volatility in the March contract, if it looks like we might

not have enoughnot have enough But April contract does not reflect this at all!But April contract does not reflect this at all! If it looks like storage tanks will fill completely before November, If it looks like storage tanks will fill completely before November,

can have downward spikes in supply, if it has nowhere to go. can have downward spikes in supply, if it has nowhere to go. Transport costs around $0.03/MMBTU, losses around 2%.Transport costs around $0.03/MMBTU, losses around 2%.

Regular demand/consumption patterns Regular demand/consumption patterns

reflected in the shape of the curve reflected in the shape of the curve Seasonality in natural gas, heating oilSeasonality in natural gas, heating oil

heating oil natgas

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What is special about commodities forward curves?

Example: Storage and Seasonality in the US Oil Markets

US consumes 22mm bbls/day, and produces about 5mm US consumes 22mm bbls/day, and produces about 5mm bbls/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/bbl mth - $0.30/bbl mth, total storage Storage costs $0.15/bbl mth - $0.30/bbl mth, total storage

capacity 350mm bbls, with a minimum of 265mm bbls. In capacity 350mm bbls, with a minimum of 265mm bbls. In addition, there is the US Strategic Petroleum Reserve, but this addition, there is the US Strategic Petroleum Reserve, but this is held out of the market most of the time. is held out of the market most of the time.

Transport costs are about $0.20/ bbl/ kmile.Transport costs are about $0.20/ bbl/ kmile. Little seasonality in crude oil, but there is seasonality in heating Little seasonality in crude oil, but there is seasonality in heating

oil, gasoline, etc. oil, gasoline, etc.

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 smaller market in You can buy either On Peak, or Off Peak, there is a smaller market in

individual hours. individual hours. These different times of day have such different properties and These different times of day have such different properties and

pricing that they are regarded as different assets. pricing that they are regarded as different assets. Seasonalities are intra-day, intra-week, and intra-year. Seasonalities are intra-day, intra-week, and intra-year. Power supply is generated by plants with varying efficiencies and Power supply is generated by plants with varying efficiencies and

start up times, arranged in a generation stack. The most efficient start up times, arranged in a generation stack. The most efficient longest startup time plants are at the bottom, and the others are longest startup time plants are at the bottom, and the others are arranged in descending order of efficiency, in a “generation stack”arranged in descending order of efficiency, in a “generation stack”

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

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

Monthly, weekly, daily “seasonality” for power Monthly, weekly, daily “seasonality” for power

peak offpeak weekends+holidays

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What is special about commodities forward curves?

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/lb - $0.08/lbTransport costs around $0.05/lb - $0.08/lb Certain metals have seasonality of demand, but this Certain metals have seasonality of demand, but this

does not show up in forward curve, possibly because does not show up in forward curve, possibly because of plentiful storage. of plentiful storage.

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

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

The Volatility Surface Constituents

The volatility surface is made up of options on The volatility surface is made up of options on futures, one option maturity for each futures futures, one option maturity for each futures contract, maturing within a few days (up to a contract, maturing within a few days (up to a week or two) of the futures maturity. In most week or two) of the futures maturity. In most markets, the liquid options can range in markets, the liquid options can range in moneyness from 0.5 to 2, and possibly more.moneyness from 0.5 to 2, and possibly more.

Because these futures are really different Because these futures are really different assets, this is not a volatility surface in the assets, this is not a volatility surface in the usual senseusual sense

Volatilities in commodities markets are almost always backwardated

Long end moves with long term Long end moves with long term demand, determined by weather, gdp demand, determined by weather, gdp growth. Very slow, little volatility, 2-10% growth. Very slow, little volatility, 2-10% instantaneous volatilityinstantaneous volatility

Short end whips around with short term Short end whips around with short term supply and demand (200%-300%)supply and demand (200%-300%)

Reversion occurs over a few weeks. Reversion occurs over a few weeks.

How volatility term-structure is related to the demand & consumptionMean-reverting nature of the market is reflected in Mean-reverting nature of the market is reflected in

the term-structure of volatilitiesthe term-structure of volatilitiesSupply/demand imbalances Supply/demand imbalances excessive “whippiness” of the front end of the forward curve excessive “whippiness” of the front end of the forward curve high volatility of short-dated optionshigh volatility of short-dated options Backwardated vol curveBackwardated vol curve

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On occasion there is a supply crunch which On occasion there is a supply crunch which affects one month, and not the succeeding one. affects one month, and not the succeeding one. Implied Volatilities explode for the affected Implied Volatilities explode for the affected month, but then drop back down for the month, but then drop back down for the succeeding month. This can even go to the succeeding month. This can even go to the extent of backwardating the variances. Because extent of backwardating the variances. Because one cannot short spot, this cannot be arbitraged. one cannot short spot, this cannot be arbitraged.

In March 2003, this happened in the US Natural In March 2003, this happened in the US Natural Gas markets, because it was a cold winter and we Gas markets, because it was a cold winter and we ran out of Natural gas in Texas. ran out of Natural gas in Texas.

In crises, volatility can become contangoAnd variance can backwardate!!!

The volatility skew is primarily determined by inventory effects

Most market participants are industrial, extremely risk Most market participants are industrial, extremely risk averse, hedging exposure. averse, hedging exposure.

Producers want OTM puts, Consumers want OTM Producers want OTM puts, Consumers want OTM calls. calls.

Market is rarely in balance, and in some cases it is Market is rarely in balance, and in some cases it is extreme. extreme.

Electricity hedging is only done by producers, vol Electricity hedging is only done by producers, vol surface is a diagonal line. ITM puts can be bought at surface is a diagonal line. ITM puts can be bought at or close to intrinsic value, because dealers are so full or close to intrinsic value, because dealers are so full of them, they cannot bear further risk. of them, they cannot bear further risk.

Nat Gas hedging is only done by consumers. Skew Nat Gas hedging is only done by consumers. Skew is very heavy the other way, because the market is all is very heavy the other way, because the market is all one way. one way.

What is volatility skew, and how is it related to who dominates the marketScenario 1.Scenario 1.

Market dominated by “producers”.Market dominated by “producers”.““positive” put skew, “negative” call skewpositive” put skew, “negative” call skew..

Quick Delta

WTIF03 EXCHANGE Vol Skew [Graph #10]

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What is volatility skew, and how is it related to who dominates the marketScenario 2.Scenario 2.

Market dominated by “consumers”.Market dominated by “consumers”.

““positive” call skew, “negative” put skewpositive” call skew, “negative” put skew..

Quick Delta

NGJ03 EXCHANGE Vol Skew [Graph #9]

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Scenario 3.Scenario 3.

Market dominated by neither “producers”, Market dominated by neither “producers”, nor “consumers”nor “consumers”

Skew tend to be fairly symmetric and positive for the calls and putsSkew tend to be fairly symmetric and positive for the calls and puts..

Quick Delta

WTIU97 EXCHANGE Vol Skew [Graph #15]

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Kurtosis appears immediately , and lasts a long time

Jumpy behavior visible in observation of futures Jumpy behavior visible in observation of futures trading, and in option prices close to expiry. trading, and in option prices close to expiry.

Kurtosis is jump-like, in that it appears immediately, Kurtosis is jump-like, in that it appears immediately, does not build up. does not build up.

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

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

Comes about when a stored supply is Comes about when a stored supply is exhausted, or when demand outruns exhausted, or when demand outruns production capacityproduction capacity

Behavior is difficult to model with Behavior is difficult to model with Markov models, requires regime-Markov models, requires regime-switching, or extreme mean reversionswitching, or extreme mean reversion

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

Non-Black-Scholes Behavior: Spiking

Non-Black-Scholes Behavior: Negative Prices

Happens in the power markets, Happens in the power markets, because there is no storage, and because there is no storage, and because it costs a lot of money to shut because it costs a lot of money to shut down and start up certain kinds of down and start up certain kinds of plants (nuclear, coal).plants (nuclear, coal).

Happens in natural gas markets, but Happens in natural gas markets, but very rarely. very rarely.

Common Commodity Exotics: Transport Options

A simple option on the difference A simple option on the difference between 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 is Can be tricky to model, as correlation is

close to 1, yet poorly known, most close to 1, yet poorly known, most models are singular at models are singular at ρρ=1=1

Common Commodity Exotics: Load Serving Deals

Power Utilities would like to hedge not just the power price, but Power Utilities would like to hedge not just the power price, but 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 well as The load is also highly correlated with the power price, as well as 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 to There is no market in load, so crude models are marked to historic datahistoric data

There are no satisfactory models of load, and and almost no There are no satisfactory models of load, and and almost no work has been done to model it, even though it is critical to many work has been done to model it, even though it is critical to many people. people.

Payoff is the difference between Oil Payoff is the difference between Oil Product (Heating Oil, Fuel Oil) and Product (Heating Oil, Fuel Oil) and Crude, minus strike. Crude, minus strike.

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

Common Commodity Exotics: Crack Spread Options

Common Commod Exotics: Spark Spread Options

The natural hedge for a gas burning power The natural hedge for a gas burning power plant, the payoff is Payoff = max( P – H * plant, the payoff is Payoff = max( P – H * G,0) G,0) 

Heat rate H represents efficiency of the Heat rate H represents efficiency of the plant, and varies from deal to deal. plant, and varies from deal to deal.

For less efficient plants, higher up the For less efficient plants, higher up the generation stack, a strike is sometimes generation stack, a strike is sometimes included. included.

Common Commodity Exotics: Swing Options

This is an option to hedge out the flexibility that a This 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 of A customer contracts to buy a certain quantity of natural gas over a series of periods. He has the natural gas over a series of periods. He has the option to take a certain amount each day, at the option to take a certain amount each day, at the floating rate. He must buy at least a minimum floating rate. He must buy at least a minimum amount 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 more There is rebating in the next period if he buys more than the maximum in a period. than the maximum in a period.

This has a lot of optionality, and is very time-This has a lot of optionality, and is very time-consuming to evalute, even in a simple model. consuming to evalute, even in a simple model.

This is another interesting problem for academics.This is another interesting problem for academics.

Common Commodity Exotics: Storage Options

Very similar to Swing optionsVery similar to Swing options A user is rented a storage tank. He has the option each day to A user is rented a storage tank. He has the option each day to

buy natgas and inject into the tank, or withdraw and sell natgas buy natgas and inject into the tank, or withdraw and sell natgas from the tank, or do nothing.from the tank, or do nothing.

He pays operating fees to inject or withdraw. He pays operating fees to inject or withdraw. He must return the tank at some level of fill.He must return the tank at some level of fill. He has a daily injection limit and a daily withdrawal limit.He has a daily injection limit and a daily withdrawal limit. This option has a lot of optionality, is difficult and time-This option has a lot of optionality, is difficult and time-

consuming to evaluate, even in a simple modelconsuming to evaluate, even in a simple model This is another place where academics can make a real This is another place where academics can make a real

contribution to the business.contribution to the business.

Commodities Models: Basic Features Spot Price ModelsSpot Price Models

Evaluate futures as F_tT = E( S_T | S_t ), almost Evaluate futures as F_tT = E( S_T | S_t ), almost always a smooth function (Can’t have discontinuous always a smooth function (Can’t have discontinuous forward curve!)forward curve!)

Almost always have mean reversionAlmost always have mean reversion parametrize forward curve with convenience yield yparametrize forward curve with convenience yield y

F_tT = S_t exp( (r + u – y)(T-t) ), u = storage F_tT = S_t exp( (r + u – y)(T-t) ), u = storage rate.rate.

Spot models are limited, can’t have negative forward Spot models are limited, can’t have negative forward variance in futures. Hard to put in sharply varying variance in futures. Hard to put in sharply varying forward curves.forward curves.

But Spot models are much more tractable, with fewer But Spot models are much more tractable, with fewer factors.factors.

Commodity Models: Basic Features

Models of whole curve (i.e. 1 factor for Models of whole curve (i.e. 1 factor for each futures maturity) are capable of each futures maturity) are capable of encompassing most observed encompassing most observed phenomena, but have many more phenomena, but have many more factors, and so are hard to evaluate. factors, and so are hard to evaluate.

BGM-like Factor models are a kind of BGM-like Factor models are a kind of compromise. compromise.

Commodities Models: Basic Features

Should have some form of mean reversionShould have some form of mean reversion Should be generalizable to a multi-commodity Should be generalizable to a multi-commodity

model, or multi-location modelmodel, or multi-location model A model capturing the vol smile should be A model capturing the vol smile should be

calibratable to odd-shaped vol surfaces, calibratable to odd-shaped vol surfaces, distorted by inventory effects. distorted by inventory effects.

A model capturing the vol smile should A model capturing the vol smile should probably contain jumps.probably contain jumps.

Commodities Models: Basic Features

Market Specific: Natural Gas models may Market Specific: Natural Gas models may want to use the storage limits, and current want to use the storage limits, and current value of storage as a parameter, controlling value of storage as a parameter, controlling jumpiness, now that there is a forward market jumpiness, now that there is a forward market in storage numbers.in storage numbers.

Market Specific: Electricity markets should Market Specific: Electricity markets should separate different parts of the curve into separate different parts of the curve into different assets, hour, day-of-week, seasondifferent assets, hour, day-of-week, season

Some example models: Gibson-Schwartz Model

A spot model for electricity, with A spot model for electricity, with stochastic convenience yield. stochastic convenience yield.

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

Does not mean-revert, so variance Does not mean-revert, so variance grows too fast at long times.grows too fast at long times.

Some example models: Schwartz-Smith Model

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

ddχχ = - k = - k χχ dt + dt + σσ__χχ dZ_ dZ_χχddζζ = = μμ__ζζ dt + dt + σσ__ζζ dZ_ dZ_ζζSpot = Spot = ζζ++χχ

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

Has some de-correlation of futures, for time spread Has some de-correlation of futures, for time spread optionsoptions

Some example models: Gabillon Model

A spot model for energy, modeling spot as a A spot model for energy, modeling spot as a single factor Gaussian process that mean single factor Gaussian process that mean reverts to a lognormal long term ratereverts to a lognormal long term rate

dS/SdS/S = = ββ( ln L – ln S ) dt + ( ln L – ln S ) dt + σσ_S dZ_S_S dZ_SdL/L = dL/L = μμ_L dt + _L dt + σσ_L dZ_L_L dZ_L

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

Has some de-correlation of futures, for time Has some de-correlation of futures, for time spread optionsspread options

Some example models: Deng Model

A spot model for electricity and natural gas together, modeling A spot model for electricity and natural gas together, modeling them as 2 mean-reverting models with jumps, and either them as 2 mean-reverting models with jumps, and either stochastic vol, or regime switchingstochastic vol, or regime switching

dX = K ( θ – X ) dt + M dW + ΔZ^1_t+ ΔZ^2_t, dX = K ( θ – X ) dt + M dW + ΔZ^1_t+ ΔZ^2_t, where X is a 2-vector containing prices, M is Cholesky decomp, where X is a 2-vector containing prices, M is Cholesky decomp, ΔZ^i are two R^2 Poisson processes, one for up and one for ΔZ^i are two R^2 Poisson processes, one for up and one for down. down.

Has kurtosis, skew, spikes!Has kurtosis, skew, spikes! Cannot accommodate sharply varying forward curves, negative Cannot accommodate sharply varying forward curves, negative

forward variance.forward variance. Has some de-correlation of futures, for time spread optionsHas some de-correlation of futures, for time spread options A heavy model to evaluate.A heavy model to evaluate.

Some example models: Model of Audet, Heiskanen, Keppo and Vehvilainen

An HJM-like curve model for electricity, in which each An HJM-like curve model for electricity, in which each forward is a mean-reverting lognormal process. forward is a mean-reverting lognormal process.

dF_{tT}/F_{tT}=exp(-α(T-t))σ(T)dB_{T}(t)dF_{tT}/F_{tT}=exp(-α(T-t))σ(T)dB_{T}(t)with dB_T(t) dB_T’(t) = exp(-with dB_T(t) dB_T’(t) = exp(-*|T’-T|) dt *|T’-T|) dt

Accomodates singular forward curves, and negative Accomodates singular forward curves, and negative forward variance,forward variance,

Easy to generalize to multi-commodityEasy to generalize to multi-commodity Forwards nicely de-correlatedForwards nicely de-correlated Easy to solveEasy to solve Has no skew, kurtosis, jumps, spikes.Has no skew, kurtosis, jumps, spikes.

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Acknowledgments:

I would like to express my gratitude to the following I would like to express my gratitude to the following people for their willing and able help. Jamie Cox-people for their willing and able help. Jamie Cox-Jones, Ben Freeman, Michael Kirch, Ilya Ustilovsky, Jones, Ben Freeman, Michael Kirch, Ilya Ustilovsky, Dan Sharfman, Elisha Wiesel, Alex Lesin, Roberto Dan Sharfman, Elisha Wiesel, Alex Lesin, Roberto Caccia, Derek Yi, Sofia Cheidvasser, Karhan Caccia, Derek Yi, Sofia Cheidvasser, Karhan Akcoglu, Bill Cowieson, Jeremy Glick, Lavanya Akcoglu, Bill Cowieson, Jeremy Glick, Lavanya Viswanathan, Alan Yamamura and Pavel Langer for Viswanathan, Alan Yamamura and Pavel Langer for their advice and criticisms. I would also like to thank their advice and criticisms. I would also like to thank Peter Carr, Marco Avellaneda, Bob Kohn, as well as Peter Carr, Marco Avellaneda, Bob Kohn, as well as Valerie Perugini, Gabrielle Maloney and Lillibeth Valerie Perugini, Gabrielle Maloney and Lillibeth Gecale for setting up this event so helpfully. Gecale for setting up this event so helpfully.