energy markets iii: weather derivates - princeton...
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Energy Markets III:Weather Derivates
Rene Carmona
Bendheim Center for FinanceDepartment of Operations Research & Financial Engineering
Princeton University
Banff, May 2007
Carmona Energy Markets
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Weather and Commodity
Stand-alonetemperatureprecipitationwind
In-Combinationnatural gaspowerheating oilpropane
Agricultural risk (yield, revenue, input hedges and trading)Power outage - contingent power price options
Carmona Energy Markets
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Other Statistical Issues: Modelling Demand
For many contracts, delivery needs to match demand
Demand for energy highly correlated with temperatureHeating Season (winter) HDDCooling Season (summer) CDD
Stylized Facts and First (naive) ModelsElectricity demand = β * weather + α
Not true all the timeTime dependent β by filtering !
From the stack: Correlation (Gas,Power) = f(weather)No significance, too unstableCould it be because of heavy tails?
Weather dynamics need to be includedAnother Source of Incompleteness
Carmona Energy Markets
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Power Plant Risk Management
Hedging Volume RiskProtection against the Weather ExposureTemperature Options on CDDs (Extreme Load)
Hedging Basis RiskProtection against Gas & Electricity Price SpikesGas purchase with Swing Options
Carmona Energy Markets
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Power Plant Risk Management
Hedging Volume RiskProtection against the Weather ExposureTemperature Options on CDDs (Extreme Load)
Hedging Basis RiskProtection against Gas & Electricity Price SpikesGas purchase with Swing Options
Carmona Energy Markets
![Page 6: Energy Markets III: Weather Derivates - Princeton Universityrcarmona/download/short_courses/Banff... · Energy Markets III: Weather Derivates ... (29 cities on CME) ... New (major)](https://reader034.vdocument.in/reader034/viewer/2022051800/5acc63657f8b9aa1518c4024/html5/thumbnails/6.jpg)
Power Plant Risk Management
Hedging Volume RiskProtection against the Weather ExposureTemperature Options on CDDs (Extreme Load)
Hedging Basis RiskProtection against Gas & Electricity Price SpikesGas purchase with Swing Options
Carmona Energy Markets
![Page 7: Energy Markets III: Weather Derivates - Princeton Universityrcarmona/download/short_courses/Banff... · Energy Markets III: Weather Derivates ... (29 cities on CME) ... New (major)](https://reader034.vdocument.in/reader034/viewer/2022051800/5acc63657f8b9aa1518c4024/html5/thumbnails/7.jpg)
Power Plant Risk Management
Hedging Volume RiskProtection against the Weather ExposureTemperature Options on CDDs (Extreme Load)
Hedging Basis RiskProtection against Gas & Electricity Price SpikesGas purchase with Swing Options
Carmona Energy Markets
![Page 8: Energy Markets III: Weather Derivates - Princeton Universityrcarmona/download/short_courses/Banff... · Energy Markets III: Weather Derivates ... (29 cities on CME) ... New (major)](https://reader034.vdocument.in/reader034/viewer/2022051800/5acc63657f8b9aa1518c4024/html5/thumbnails/8.jpg)
Hedging Basis Risk
Use Swing OptionsMultiple Rights to deviate (within bounds) from base loadcontract levelPricing & Hedging quite involved!
Tree/Forest Based MethodsDirect Backward Dynamic Programing Induction(a la Detemple-Jaillet-Ronn-Tompaidis)
New Monte Carlo MethodsNonparametric Regression (a la Longstaff-Schwarz) BackwardDynamic Programing Induction
Carmona Energy Markets
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Hedging Basis Risk
Use Swing OptionsMultiple Rights to deviate (within bounds) from base loadcontract levelPricing & Hedging quite involved!
Tree/Forest Based MethodsDirect Backward Dynamic Programing Induction(a la Detemple-Jaillet-Ronn-Tompaidis)
New Monte Carlo MethodsNonparametric Regression (a la Longstaff-Schwarz) BackwardDynamic Programing Induction
Carmona Energy Markets
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Mathematics of Swing Contracts: a Crash Course
Review: Classical Optimal Stopping Problem: American OptionX0,X1,X2,· · · ,Xn, · · · rewardsRight to ONE ExerciseMathematical Problem
sup0≤τ≤T
E{Xτ}
Mathematical SolutionSnell’s EnvelopBackward Dynamic Programming Induction in Markovian Case
Standard, Well Understood
Carmona Energy Markets
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Mathematics of Swing Contracts: a Crash Course
Review: Classical Optimal Stopping Problem: American OptionX0,X1,X2,· · · ,Xn, · · · rewardsRight to ONE ExerciseMathematical Problem
sup0≤τ≤T
E{Xτ}
Mathematical SolutionSnell’s EnvelopBackward Dynamic Programming Induction in Markovian Case
Standard, Well Understood
Carmona Energy Markets
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New Mathematical Challenges
In its simplest form the problem of Swing/Recall option pricing is an
Optimal Multiple Stopping Problem
X0,X1,X2,· · · ,Xn,· · · rewardsRight to N ExercisesMathematical Problem
sup0≤τ1<τ2<···<τN≤T
E{Xτ1 + Xτ2 + · · ·+ XτN}
Refraction period θ
τ1 + θ < τ2 < τ2 + θ < τ3 < · · · < τN−1 + θ < τN
Part of recall contracts & crucial for continuous time models
Carmona Energy Markets
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Instruments with Multiple American Exercises
Ubiquitous in Energy SectorSwing / Recall contractsEnd user contracts (EDF)
Present in other contextsFixed income markets (e.g. chooser swaps)Executive option programsReload→ Multiple exercise, Vesting→ Refraction, · · ·Fleet Purchase (airplanes, cars, · · · )
ChallengesValuationOptimal exercise policiesHedging
Carmona Energy Markets
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Instruments with Multiple American Exercises
Ubiquitous in Energy SectorSwing / Recall contractsEnd user contracts (EDF)
Present in other contextsFixed income markets (e.g. chooser swaps)Executive option programsReload→ Multiple exercise, Vesting→ Refraction, · · ·Fleet Purchase (airplanes, cars, · · · )
ChallengesValuationOptimal exercise policiesHedging
Carmona Energy Markets
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Instruments with Multiple American Exercises
Ubiquitous in Energy SectorSwing / Recall contractsEnd user contracts (EDF)
Present in other contextsFixed income markets (e.g. chooser swaps)Executive option programsReload→ Multiple exercise, Vesting→ Refraction, · · ·Fleet Purchase (airplanes, cars, · · · )
ChallengesValuationOptimal exercise policiesHedging
Carmona Energy Markets
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Exercise regions for N = 5 rights and finite maturity computed byMalliavin-Monte-Carlo.
Carmona Energy Markets
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First Faculty Meeting of New PU President
Princeton University Electricity Budget
2.8 M $ over (PU is small)
The University has its own Power PlantGas Turbine for Electricity & Steam
Major ExposuresHot Summer (air conditioning) Spikes in Demand, Gas & ElectricityPricesCold Winter (heating) Spikes in Gas Prices
Carmona Energy Markets
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Risk Management Solution
Never Again such a Short Fall !!!Student (Greg Larkin) ThesisHedging Volume Risk
Protection against the Weather ExposureTemperature Options on CDDs (Extreme Load)
Hedging Basis RiskProtection against Gas & Electricity Price SpikesGas purchase with Swing Options
Carmona Energy Markets
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Average Daily Load against Average Daily Temperature (PJM data).
Carmona Energy Markets
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The Need for Temperature Options
Rigorous Analysis of the Dependence between the Shortfall andthe Temperature in PrincetonUse of Historical Data (sparse) & Definition of a TemperatureProtection
Period of the CoverageForm of the Coverage
Search for the Nearest Stations with HDD/CDD TradesLa Guardia Airport (LGA)Philadelphia (PHL)
Define a Portfolio of LGA & PHL forward / option ContractsConstruct a LGA / PHL basket
Carmona Energy Markets
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Pricing: How Much is it Worth to PU?
Actuarial / Historical ApproachBurn AnalysisTemperature Modeling & Monte Carlo VaR ComputationsNot Enough Reliable Load Data
Expected (Exponential) Utility Maximization (A. Danilova)Use Gas & Power ContractsHedging in Incomplete ModelsIndifference PricingVery Difficult Numerics (whether PDE’s or Monte Carlo)
Carmona Energy Markets
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Carmona Energy Markets
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Carmona Energy Markets
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Weather Derivatives
OTC & Exchange traded (29 cities on CME)Still Extremely Illiquid Markets (except for front month)Misconception: Weather Derivative = Insurance Contract
No secondary marketMark-to-Market (or Model) does not change
Not Until Meteorology kicks in (10-15 days before maturity)Mark-to-Market (or Model) changes every dayContracts change handsThat’s when major losses occur and money is made
This hot period is not considered in academic studiesNeed for updates: new information coming in (temperatures,forecasts, ....)Filtering is (again) the solution
Carmona Energy Markets
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Weather Derivatives
OTC & Exchange traded (29 cities on CME)Still Extremely Illiquid Markets (except for front month)Misconception: Weather Derivative = Insurance Contract
No secondary marketMark-to-Market (or Model) does not change
Not Until Meteorology kicks in (10-15 days before maturity)Mark-to-Market (or Model) changes every dayContracts change handsThat’s when major losses occur and money is made
This hot period is not considered in academic studiesNeed for updates: new information coming in (temperatures,forecasts, ....)Filtering is (again) the solution
Carmona Energy Markets
![Page 26: Energy Markets III: Weather Derivates - Princeton Universityrcarmona/download/short_courses/Banff... · Energy Markets III: Weather Derivates ... (29 cities on CME) ... New (major)](https://reader034.vdocument.in/reader034/viewer/2022051800/5acc63657f8b9aa1518c4024/html5/thumbnails/26.jpg)
Weather Derivatives
OTC & Exchange traded (29 cities on CME)Still Extremely Illiquid Markets (except for front month)Misconception: Weather Derivative = Insurance Contract
No secondary marketMark-to-Market (or Model) does not change
Not Until Meteorology kicks in (10-15 days before maturity)Mark-to-Market (or Model) changes every dayContracts change handsThat’s when major losses occur and money is made
This hot period is not considered in academic studiesNeed for updates: new information coming in (temperatures,forecasts, ....)Filtering is (again) the solution
Carmona Energy Markets
![Page 27: Energy Markets III: Weather Derivates - Princeton Universityrcarmona/download/short_courses/Banff... · Energy Markets III: Weather Derivates ... (29 cities on CME) ... New (major)](https://reader034.vdocument.in/reader034/viewer/2022051800/5acc63657f8b9aa1518c4024/html5/thumbnails/27.jpg)
Weather Derivatives
OTC & Exchange traded (29 cities on CME)Still Extremely Illiquid Markets (except for front month)Misconception: Weather Derivative = Insurance Contract
No secondary marketMark-to-Market (or Model) does not change
Not Until Meteorology kicks in (10-15 days before maturity)Mark-to-Market (or Model) changes every dayContracts change handsThat’s when major losses occur and money is made
This hot period is not considered in academic studiesNeed for updates: new information coming in (temperatures,forecasts, ....)Filtering is (again) the solution
Carmona Energy Markets
![Page 28: Energy Markets III: Weather Derivates - Princeton Universityrcarmona/download/short_courses/Banff... · Energy Markets III: Weather Derivates ... (29 cities on CME) ... New (major)](https://reader034.vdocument.in/reader034/viewer/2022051800/5acc63657f8b9aa1518c4024/html5/thumbnails/28.jpg)
Weather Derivatives
OTC & Exchange traded (29 cities on CME)Still Extremely Illiquid Markets (except for front month)Misconception: Weather Derivative = Insurance Contract
No secondary marketMark-to-Market (or Model) does not change
Not Until Meteorology kicks in (10-15 days before maturity)Mark-to-Market (or Model) changes every dayContracts change handsThat’s when major losses occur and money is made
This hot period is not considered in academic studiesNeed for updates: new information coming in (temperatures,forecasts, ....)Filtering is (again) the solution
Carmona Energy Markets
![Page 29: Energy Markets III: Weather Derivates - Princeton Universityrcarmona/download/short_courses/Banff... · Energy Markets III: Weather Derivates ... (29 cities on CME) ... New (major)](https://reader034.vdocument.in/reader034/viewer/2022051800/5acc63657f8b9aa1518c4024/html5/thumbnails/29.jpg)
Daily Average Temperature at La Guardia.
Carmona Energy Markets
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Prediction on 6/1/2001 of daily temperature over the next four months.
Carmona Energy Markets
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The Future of the Weather Markets
Social function of the weather marketExistence of a Market of Professionals (for weather risk transfer)
Under attack from(Re-)Insurance industryUtilities (trying to pass weather risk to end-customer)
EDF program in FranceWeather Normalization Agreements in US
Cross Commodity ProductsGas & Power contracts with weather triggers/contingenciesNew (major) players: Hedge Funds provide liquidity
World BankUse weather derivatives instead of insurance contracts
Carmona Energy Markets
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The Future of the Weather Markets
Social function of the weather marketExistence of a Market of Professionals (for weather risk transfer)
Under attack from(Re-)Insurance industryUtilities (trying to pass weather risk to end-customer)
EDF program in FranceWeather Normalization Agreements in US
Cross Commodity ProductsGas & Power contracts with weather triggers/contingenciesNew (major) players: Hedge Funds provide liquidity
World BankUse weather derivatives instead of insurance contracts
Carmona Energy Markets
![Page 33: Energy Markets III: Weather Derivates - Princeton Universityrcarmona/download/short_courses/Banff... · Energy Markets III: Weather Derivates ... (29 cities on CME) ... New (major)](https://reader034.vdocument.in/reader034/viewer/2022051800/5acc63657f8b9aa1518c4024/html5/thumbnails/33.jpg)
The Future of the Weather Markets
Social function of the weather marketExistence of a Market of Professionals (for weather risk transfer)
Under attack from(Re-)Insurance industryUtilities (trying to pass weather risk to end-customer)
EDF program in FranceWeather Normalization Agreements in US
Cross Commodity ProductsGas & Power contracts with weather triggers/contingenciesNew (major) players: Hedge Funds provide liquidity
World BankUse weather derivatives instead of insurance contracts
Carmona Energy Markets
![Page 34: Energy Markets III: Weather Derivates - Princeton Universityrcarmona/download/short_courses/Banff... · Energy Markets III: Weather Derivates ... (29 cities on CME) ... New (major)](https://reader034.vdocument.in/reader034/viewer/2022051800/5acc63657f8b9aa1518c4024/html5/thumbnails/34.jpg)
The Future of the Weather Markets
Social function of the weather marketExistence of a Market of Professionals (for weather risk transfer)
Under attack from(Re-)Insurance industryUtilities (trying to pass weather risk to end-customer)
EDF program in FranceWeather Normalization Agreements in US
Cross Commodity ProductsGas & Power contracts with weather triggers/contingenciesNew (major) players: Hedge Funds provide liquidity
World BankUse weather derivatives instead of insurance contracts
Carmona Energy Markets
![Page 35: Energy Markets III: Weather Derivates - Princeton Universityrcarmona/download/short_courses/Banff... · Energy Markets III: Weather Derivates ... (29 cities on CME) ... New (major)](https://reader034.vdocument.in/reader034/viewer/2022051800/5acc63657f8b9aa1518c4024/html5/thumbnails/35.jpg)
The Future of the Weather Markets
Social function of the weather marketExistence of a Market of Professionals (for weather risk transfer)
Under attack from(Re-)Insurance industryUtilities (trying to pass weather risk to end-customer)
EDF program in FranceWeather Normalization Agreements in US
Cross Commodity ProductsGas & Power contracts with weather triggers/contingenciesNew (major) players: Hedge Funds provide liquidity
World BankUse weather derivatives instead of insurance contracts
Carmona Energy Markets
![Page 36: Energy Markets III: Weather Derivates - Princeton Universityrcarmona/download/short_courses/Banff... · Energy Markets III: Weather Derivates ... (29 cities on CME) ... New (major)](https://reader034.vdocument.in/reader034/viewer/2022051800/5acc63657f8b9aa1518c4024/html5/thumbnails/36.jpg)
Incomplete Market Model & Indifference Pricing
Temperature Options: Actuarial/Statistical ApproachTemperature Options: Diffusion Models (Danilova)Precipitation Options: Markov Models (Diko)
Problem: Pricing in an Incomplete MarketSolution: Indifference Pricing a la Davis
dθt = p(t , θ)dt + q(t , θ)dW (θ)t + r(t , θ)dQ(θ)
t
dSt = St [µ(t , θ)dt + σ(t , θ)dW (S)t ]
θt non-tradableSt tradable
Carmona Energy Markets
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Mathematical Models for Temperature OptionsExample: Exponential Utility Function
pt =E{φ(YT )e−
∫ Tt V (s,Ys)ds}
E{e−∫ T
t V (s,Ys)ds}
where
φ = e−γ(1−ρ2)f
where f (θT ) is the pay-off function of the European call on thetemperature
pt = e−γ(1−ρ2)pt
where pt is price of the option at time t
Yt is the diffusion:
dYt = [g(t ,Yt)−µ(t ,Yt)− rσ(t ,Yt)
h(t ,Yt)]dt + h(t ,Yt)dWt
starting from Y0 = y
V is the time dependent potential function:
V (t , y) = −1− ρ2
2(µ(t , y)− r)2
σ(t , y)2
Carmona Energy Markets
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The Weather Market Today
Insurance companies: Swiss Re, XL, Munich Re, Ren ReFinancial Houses: Goldman Sachs, Deutsche Bank, MerrillLynch, ABN AMROHedge funds: D. E. Shaw, Tudor, Susquehanna, Centaurus,Wolverine
TradingOTCExchange: CME (Chicago Mercantile Exchange) 29 citesglobally traded, monthly / seasonal contractsStrong end-user demand within the energy sector Northeast andMidwest LDCs most prevalent in US
Carmona Energy Markets
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Marking-to-Market
Only a subset of locations are traded on a daily basisExchange settlement prices depart from OTC market prices(viewed by traders)Denoting by µ the mean of the swaps delivering in a givenseason, by Σ their covariance matrix:
infµ, Γµ=π
(µ− µtsimΣ−1(µ− µsim)
where Γ defines the set of observable trades and π is the vectorof market prices.
Carmona Energy Markets