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(1)

Energy Markets III:

Weather Derivates

Ren ´e Carmona Bendheim Center for Finance

Department of Operations Research & Financial Engineering Princeton University

(2)

Weather and Commodity

Stand-alone temperature precipitation wind In-Combination natural gas power heating oil propane

Agricultural risk (yield, revenue, input hedges and trading) Power outage - contingent power price options

(3)

Other Statistical Issues: Modelling Demand

For many contracts, delivery needs to match demand

Demand for energy highly correlated with temperature

Heating Season (winter) HDD Cooling Season (summer) CDD

Stylized Facts and First (naive) Models

Electricity demand = β * weather + α

Not true all the time

Time dependent β by filtering !

From the stack: Correlation (Gas,Power) = f(weather)

No significance, too unstable Could it be because of heavy tails?

Weather dynamics need to be included

(4)

Power Plant Risk Management

Hedging Volume Risk

Protection against the Weather Exposure

Temperature Optionson CDDs (Extreme Load)

Hedging Basis Risk

Protection against Gas & Electricity Price Spikes Gas purchase withSwing Options

(5)

Power Plant Risk Management

Hedging Volume Risk

Protection against the Weather Exposure

Temperature Optionson CDDs (Extreme Load)

Hedging Basis Risk

Protection against Gas & Electricity Price Spikes Gas purchase withSwing Options

(6)

Power Plant Risk Management

Hedging Volume Risk

Protection against the Weather Exposure

Temperature Optionson CDDs (Extreme Load)

Hedging Basis Risk

Protection against Gas & Electricity Price Spikes

(7)

Power Plant Risk Management

Hedging Volume Risk

Protection against the Weather Exposure

Temperature Optionson CDDs (Extreme Load)

Hedging Basis Risk

Protection against Gas & Electricity Price Spikes

(8)

Hedging Basis Risk

UseSwing Options

Multiple Rights to deviate (within bounds) from base load contract level

Pricing & Hedging quite involved! Tree/Forest Based Methods

Direct Backward Dynamic Programing Induction ( `a la Detemple-Jaillet-Ronn-Tompaidis) New Monte Carlo Methods

Nonparametric Regression ( `a la Longstaff-Schwarz) Backward Dynamic Programing Induction

(9)

Hedging Basis Risk

UseSwing Options

Multiple Rights to deviate (within bounds) from base load contract level

Pricing & Hedging quite involved!

Tree/Forest Based Methods

Direct Backward Dynamic Programing Induction ( `a la Detemple-Jaillet-Ronn-Tompaidis)

New Monte Carlo Methods

Nonparametric Regression ( `a la Longstaff-Schwarz) Backward Dynamic Programing Induction

(10)

Mathematics of Swing Contracts: a Crash Course

Review:Classical Optimal Stopping Problem: American Option

X0,X1,X2,· · · ,Xn, · · · rewards Right to ONE Exercise Mathematical Problem sup 0≤τ ≤T E{Xτ} Mathematical Solution Snell’s Envelop

Backward Dynamic Programming Induction in Markovian Case Standard, Well Understood

(11)

Mathematics of Swing Contracts: a Crash Course

Review:Classical Optimal Stopping Problem: American Option

X0,X1,X2,· · · ,Xn, · · · rewards Right to ONE Exercise Mathematical Problem sup 0≤τ ≤T E{Xτ} Mathematical Solution Snell’s Envelop

Backward Dynamic Programming Induction in Markovian Case

(12)

New Mathematical Challenges

In its simplest form the problem ofSwing/Recall option pricing is an

Optimal Multiple Stopping Problem

X0,X1,X2,· · · ,Xn,· · · rewards Right to N Exercises Mathematical Problem sup 0≤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

(13)

Instruments with Multiple American Exercises

Ubiquitous in Energy Sector

Swing / Recall contracts End user contracts (EDF)

Present in other contexts

Fixed income markets (e.g. chooser swaps) Executive option programs

Reload → Multiple exercise, Vesting → Refraction, · · · Fleet Purchase (airplanes, cars, · · · )

Challenges

Valuation

Optimal exercise policies Hedging

(14)

Instruments with Multiple American Exercises

Ubiquitous in Energy Sector

Swing / Recall contracts End user contracts (EDF)

Present in other contexts

Fixed income markets (e.g. chooser swaps) Executive option programs

Reload → Multiple exercise, Vesting → Refraction, · · · Fleet Purchase (airplanes, cars, · · · )

Challenges

Valuation

Optimal exercise policies Hedging

(15)

Instruments with Multiple American Exercises

Ubiquitous in Energy Sector

Swing / Recall contracts End user contracts (EDF)

Present in other contexts

Fixed income markets (e.g. chooser swaps) Executive option programs

Reload → Multiple exercise, Vesting → Refraction, · · · Fleet Purchase (airplanes, cars, · · · )

Challenges

Valuation

Optimal exercise policies Hedging

(16)

Exercise regions for N = 5 rights andfinitematurity computed by Malliavin-Monte-Carlo.

(17)

First Faculty Meeting of New PU President

Princeton University Electricity Budget

2.8 M $ over (PU is small)

The University has its own Power Plant Gas Turbine for Electricity & Steam Major Exposures

Hot Summer (air conditioning) Spikes in Demand, Gas & Electricity Prices

(18)

Risk Management Solution

Never Again such a Short Fall !!! Student (Greg Larkin) Thesis

Hedging Volume Risk

Protection against the Weather Exposure

Temperature Optionson CDDs (Extreme Load)

Hedging Basis Risk

Protection against Gas & Electricity Price Spikes

(19)
(20)

The Need for Temperature Options

Rigorous Analysis of the Dependence between the Shortfall and the Temperature in Princeton

Use of Historical Data (sparse) & Definition of a Temperature Protection

Period of the Coverage Form of the Coverage

Search for the Nearest Stations with HDD/CDD Trades

La Guardia Airport (LGA) Philadelphia (PHL)

Define a Portfolio of LGA & PHL forward / option Contracts Construct a LGA / PHL basket

(21)

Pricing: How Much is it Worth to PU?

Actuarial / Historical Approach

Burn Analysis

Temperature Modeling & Monte Carlo VaR Computations Not Enough Reliable Load Data

Expected (Exponential) Utility Maximization (A. Danilova)

Use Gas & Power Contracts Hedging in Incomplete Models Indifference Pricing

(22)
(23)
(24)

Weather Derivatives

OTC & Exchange traded (29 cities on CME)

Still ExtremelyIlliquid Markets (except for front month) Misconception: Weather Derivative = Insurance Contract

No secondary market

Mark-to-Market (or Model) does not change

Not Until Meteorologykicks in(10-15 days before maturity) Mark-to-Market (or Model)changesevery day

Contracts change hands

That’s when major losses occur and money is made Thishotperiod is not considered in academic studies

Need forupdates: new information coming in (temperatures, forecasts, ....)

(25)

Weather Derivatives

OTC & Exchange traded (29 cities on CME)

Still ExtremelyIlliquid Markets (except for front month)

Misconception: Weather Derivative = Insurance Contract No secondary market

Mark-to-Market (or Model) does not change

Not Until Meteorologykicks in(10-15 days before maturity) Mark-to-Market (or Model)changesevery day

Contracts change hands

That’s when major losses occur and money is made Thishotperiod is not considered in academic studies

Need forupdates: new information coming in (temperatures, forecasts, ....)

(26)

Weather Derivatives

OTC & Exchange traded (29 cities on CME)

Still ExtremelyIlliquid Markets (except for front month) Misconception: Weather Derivative = Insurance Contract

No secondary market

Mark-to-Market (or Model) does not change

Not Until Meteorologykicks in(10-15 days before maturity) Mark-to-Market (or Model)changesevery day

Contracts change hands

That’s when major losses occur and money is made Thishotperiod is not considered in academic studies

Need forupdates: new information coming in (temperatures, forecasts, ....)

(27)

Weather Derivatives

OTC & Exchange traded (29 cities on CME)

Still ExtremelyIlliquid Markets (except for front month) Misconception: Weather Derivative = Insurance Contract

No secondary market

Mark-to-Market (or Model) does not change

Not Until Meteorologykicks in(10-15 days before maturity)

Mark-to-Market (or Model)changesevery day

Contracts change hands

That’s when major losses occur and money is made

Thishotperiod is not considered in academic studies Need forupdates: new information coming in (temperatures, forecasts, ....)

(28)

Weather Derivatives

OTC & Exchange traded (29 cities on CME)

Still ExtremelyIlliquid Markets (except for front month) Misconception: Weather Derivative = Insurance Contract

No secondary market

Mark-to-Market (or Model) does not change

Not Until Meteorologykicks in(10-15 days before maturity)

Mark-to-Market (or Model)changesevery day

Contracts change hands

That’s when major losses occur and money is made

Thishotperiod is not considered in academic studies

Need forupdates: new information coming in (temperatures, forecasts, ....)

(29)
(30)
(31)

The Future of the Weather Markets

Social functionof the weather market

Existence of a Market of Professionals (for weather risk transfer)

Under attackfrom (Re-)Insurance industry

Utilities (trying to pass weather risk to end-customer) EDF program in France

Weather Normalization Agreements in US Cross Commodity Products

Gas & Power contracts withweather triggers/contingencies New (major) players:Hedge Funds provide liquidity

World Bank

(32)

The Future of the Weather Markets

Social functionof the weather market

Existence of a Market of Professionals (for weather risk transfer)

Under attackfrom

(Re-)Insurance industry

Utilities (trying to pass weather risk to end-customer)

EDF program in France

Weather Normalization Agreements in US

Cross Commodity Products

Gas & Power contracts withweather triggers/contingencies New (major) players:Hedge Funds provide liquidity

World Bank

(33)

The Future of the Weather Markets

Social functionof the weather market

Existence of a Market of Professionals (for weather risk transfer)

Under attackfrom

(Re-)Insurance industry

Utilities (trying to pass weather risk to end-customer)

EDF program in France

Weather Normalization Agreements in US Cross Commodity Products

Gas & Power contracts withweather triggers/contingencies

New (major) players:Hedge Funds provide liquidity

World Bank

(34)

The Future of the Weather Markets

Social functionof the weather market

Existence of a Market of Professionals (for weather risk transfer)

Under attackfrom

(Re-)Insurance industry

Utilities (trying to pass weather risk to end-customer)

EDF program in France

Weather Normalization Agreements in US Cross Commodity Products

Gas & Power contracts withweather triggers/contingencies

New (major) players:Hedge Funds provide liquidity

World Bank

(35)

The Future of the Weather Markets

Social functionof the weather market

Existence of a Market of Professionals (for weather risk transfer)

Under attackfrom

(Re-)Insurance industry

Utilities (trying to pass weather risk to end-customer)

EDF program in France

Weather Normalization Agreements in US Cross Commodity Products

Gas & Power contracts withweather triggers/contingencies

New (major) players:Hedge Funds provide liquidity

World Bank

(36)

Incomplete Market Model & Indifference Pricing

Temperature Options: Actuarial/Statistical Approach Temperature Options: Diffusion Models (Danilova) Precipitation Options: Markov Models (Diko)

Problem: Pricing in an Incomplete Market Solution: Indifference Pricing `a la Davis

d θt = 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)dWt(S)] θt non-tradable St tradable

(37)

Mathematical Models for Temperature Options

Example:Exponential Utility Function

˜ pt =E{ ˜ φ(YT)e− RT t V (s,Ys)ds} E{e− RT 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 the

temperature ˜

pt =e−γ(1−ρ 2)p

t

where pt is price of the option at time t

Yt is the diffusion: dYt = [g(t, Yt) − µ(t, Yt) −r σ(t, Yt) h(t, Yt)]dt + h(t, Yt)d ˜Wt starting from Y0=y

(38)

The Weather Market Today

Insurance companies: Swiss Re, XL, Munich Re, Ren Re Financial Houses: Goldman Sachs, Deutsche Bank, Merrill Lynch, ABN AMRO

Hedge funds: D. E. Shaw, Tudor, Susquehanna, Centaurus, Wolverine

Trading

OTC

Exchange: CME (Chicago Mercantile Exchange) 29 cites globally traded, monthly / seasonal contracts

Strong end-user demand within the energy sector Northeast and Midwest LDCs most prevalent in US

(39)

Marking-to-Market

Only a subset of locations are traded on a daily basis Exchange settlement prices depart from OTC market prices (viewed by traders)

Denoting by µ the mean of the swaps delivering in a given season, by Σ their covariance matrix:

inf µ, Γµ=π(µ − µ t simΣ −1(µ − µ sim)

where Γ defines the set of observable trades and π is the vector of market prices.

References

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