energy markets iii: weather derivates - princeton university
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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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Exercise regions for N = 5 rights and finite maturity computed byMalliavin-Monte-Carlo.
Carmona Energy Markets
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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