Commodity price modeling in EDF. Parameter estimation and ... · Commodity price modeling in EDF....

Commodity price modeling in EDF. Parameterestimation and calibration

Olivier FÉRON,1,2

1 Electricité de France (EDF)– Recherche & développements1 Avenue du Général de Gaulle, CLAMART, FRANCE

2 Laboratoire de Finance des marchés de l’énergieUniversité Paris Dauphine

Workshop on Electricity, Energy and Commodities Risk Managemen t - Fields Institute, Toronto, 2013 1 / 41

Agenda

1 IntroductionObjectives of an electricianConstraints in commodity price modeling

2 2-factor modelDescriptionCalibration methodsImpact on valuation

3 Strutural modelDescriptionStudy on forward prices reconstructionCalibration issues

4 Conclusion and perspectives

Commodity modeling in EDF: objectives

Short term (1 day → 2 weeks) prediction

Market comprehension

Mid term risk managementGross energy margin predictionRisk measurementHedging

PricingValuation of Production assetsValuation of flexibilities in supply contracts

Investment decision

EDF portfolios: some examples

Production unitsApproximation : Strip of European spread options of payoff CF (t)

CF (t) =(

Spowert − hSfuel

t − h′SCO2t − K

∀t ∈ [0 ; T ]

More realistic: dynamic constraintsStartup costs, limited number of startupsScheduled outage periods

Gas storage, Hydro damSwing option : lets the holder buy a flexible quantity Q ∈ [Qmin ; Qmax ]Additional constraints: minimal and maximal quantity per day, per month...

Supply contractsIndexed price: moving average opionsFlexibility in quantity: swing options

Spot market

Daily spot for fuels

(semi)Hourly spot price for power

0 4 8 12 16 20 24

Spot price Peak price Base price

Figure : French power spot prices on January 12th 2012

Future market

Power specificity: delivery period

Different maturities and different delivery periods

2011 2012

2010 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 2013

1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12

Future market

2011 2012

2010 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 2013

1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12

Time2011

Q1 Q2 Q3

1 2 3 4 5

15/12/2010

Future market

2011 2012

2010 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 2013

1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12

Time2011

Q1 Q2 Q3

1 2 3 4 5

15/12/2010

Q1 Q2 Q3

1 2 311 12

25/10/2011

Future market

2011 2012

2010 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 2013

1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12

Time2011

Q1 Q2 Q3

1 2 3 4 5

15/12/2010

Q1 Q2 Q3

1 2 311 12

25/10/2011 Zoom

October 2011 November 2011

W1 W2 W3 W4 W1 W2 W3 W4W1 W2 W3

25/10/2011

Contraints of modeling

Portfoltio expositionPower and fuels spot prices (or "unitary forward prices")Power and fuels forward productsLink between power and fuels

Available hedging productsforward products (with delivery periods for power)

Incomplete market: No derivatives (market information) on"unitary" (instantaneous) forward pricesany link (spread) between fuels and power

ConsequencesWe need to model spot prices and forward productsThe link is obtained from the forward curve

Calibration issues2 different sets of observed data (spot prices and forward products)Complex relationship between parameters and observed forward products

Electricity price modeling: two approaches

Forward curve of unitary future price as a starting point

dFt(T ) = µ(t ,T ,Ft(T ))dt + σ(t ,T ,Ft(T ))dWt

Spot price deduced as a limit St = limT→t Ft(T )Forward products deduced by no-arbitrage principles

Ft(T , θ) =

∫ θ

0w(u)Ft(T + u)du

Spot price as a starting point

dSt = µ(t ,St ,Xt)dt + σ(t ,St ,Xt)dWt

Some other stochastic processes Xt (fuel prices, demand...) may turn up.Forward prices deduced by no-arbitrage principles

Ft(T ) = EQt [ST ]

Two different works

Parameter estimation of the model currently used in EDF for r iskmanagement

2-factor model descriptionEstimation / Calibration on forward products and spot pricesImpact of calibration method on European option valuation

Structural model for commodity pricesModel descriptionStudy on the forward products reconstructionIntegration of parameters uncertainty in the forward products recontructionCalibration

Interests: performance of a model and its calibrationin representing market informationin introducing information

Two different works

Two-factor model description

FormulationdFt(T )

Ft(T )= σs(t)e

−α(T−t)dW (s)t + σl(t)dW (l)

with α ∈ R+∗ and σs(t) and σl(t) are positive integrable functions.

"Discretized" forward product formulation Ft(T , θ) = 1θ

∑θ−1i=0 Ft(T + i)

Forward product process

dFt(T , θ) =1θ

θ−1∑

σs(t)e−α(T+i−t)dW (s)

t + σl(t)dW (l)t

Ft(T + i)

The forward product process is not Markovian [Benth & Koekebakker (2008)]

shaping factors λt,T ,θ

i = Ft (T+i)Ft (T ,θ)

and Ψ(α, t ,T , θ) =∑θ

i=0 λt,T ,θ

i e−αi

dFt(T , θ)

Ft(T , θ)= σs(t)e

−α(T−t)Ψ(α, t ,T , θ)dW (s)t + σl(t)dW (l)

Approximation on shaping factors, Markovian process

Strong approximation for the calibrationConstant shaping factors λ

t,T ,θ

i ≡ 1Constant volatility functions σs(t) ≡ σs and σl(t) ≡ σl [Kiesel et al. (2007)]

Consequences: Ψ(α, t ,T , θ) ≡ Ψ(α, θ) is determistic and the dynamicson forward products becomes Markovian

Two different methods of parameters estimationEstimation on Marginal volatilities (average of log-returns variance overthe cotation period)

Express the theoretical marginal volatility of a forward product as a function ofmodel parametersFit the theoretical marginal volatility to the empirical (or implied) marginalvolatility computed from observed cotations

Estimation from spot pricesLong term volatility estimated from long term forward productsShort term parameters estimated from spot prices

Estimation on marginal volatilities

N observations (Fn(T , θ))n=1,...,N at dates (tn)n=1,...,N , with tn+1 − tn = δt

Model on geometric returns

Rn(T , θ) = ln(

Fn+1(T , θ)

Fn(T , θ)

= σse−α(T−tn)Ψ(α, θ)√

ν(δt)εsn + σl

√δtεl

with ν(δt) = e2αδt−1

Marginal volatility: average of instantaneous return variance

MV 2(T , θ, α, σs, σl) =1N

Var [Rn(T , θ)]

Historical (or implied) marginal volatility M̂V2(T , θ) from observed forward

returns (or options)

Estimation: minimization of the squared difference

(α̂, σ̂s, σ̂l) = arg min(α,σs,σl )

(T ,θ)

M̂V2(T , θ)− MV 2(T , θ, α, σs, σl)

Two-factor model: Estimation on spot prices

dFt(T )

Ft(T )= σse−α(T−t)dW (s)

t + σldW (l)t

Spot price formulation:

ln St = ln F0(t)−12

1 − e−2αt

2α+ σ

0σse−α(t−u)dW (s)

︸︷︷︸

X (s)t

0σldW (l)

︸︷︷︸

X (l)t

Deseasonalization step

Long term volatility σl estimated from long term forward products

Short term parameters estimated from a state-space model

State equation: two hidden factors X (s)t and X (l)

tσs and α estimated by Likelihood maximization on deseasonalized spotprices, by using a Kalman filter

Estimation results

Date of estimation: 12-Mar-2013

1 year of historical data

Two different estimation methodsEstimation on (empirical) marginal volatilitiesEstimation on long term forward products (for the long term factor) andspot prices (for the short term factor)

Study on resulting parameter values: strip of European options onmonthly forward products (Apr-2013 → Mar-2015)

(F (t ,T , θ)− K )+, K = F (t0,T , θ)

Comparison in European option pricing"Benchmark" value defined from a multi-factor model fitted on empirical

volatilities M̂V2(T , θ)

Results: estimation on UK Power

1WAH 1MAH 1QAH 1SAH 6SAH0.08

Products

Empirical Vol2F calibrated on MV2F calibrated on spot

Figure : Calibration results on UK power

parameter Estimation on MV Estimation on spotσs 19.1% 84.5%α 1.37 162.65σl 9.8% 9.8%

Results: European options pricing (UK Power)

Mar−13 Jun−13 Sep−13 Jan−14 Apr−14 Jul−14 Oct−14 Feb−15 May−150.5

Figure : European option princing on UK Power

Empirical volatility Estimation on MV Estimation on spotMtM (e) 64 252 60 787 52 295

Relative difference – -5% -19%

Conclusion

2 factor model objectivesRepresent spot prices and forward productsCan be calibrated on historical or implied volatility, and on spot pricesCan fit perfectly the forward products at a given date

Caibration issues: need approximationsConstant and equally weighted shaping factorsConstant volatility function σs(t) and σl(t)No statistical resultsCalibration methods have a high impact on indicators

A "benchmark" value can give e reference for forward derivativesMore difficult for options on spot prices

Some other issuesNo specific event (spikes, negative prices...)Weak relationship between commodities

Two different works

Fundamental approach

I1t Nuclear

Production Cost(eur/GWh)

Capacity(GW)

Figure : Structural model [Aïd et al. (2012)]

I1t Nuclear

Capacity(GW)

DemandOff Peak

Dt ∈ I2t

I1t Nuclear

Capacity(GW)

DemandOff Peak

Dt ∈ I2t

Mt = h2S2t

hi Sit 1Dt∈Iit

C it − Dt

I1t Nuclear

Capacity(GW)

DemandOff Peak

Dt ∈ I2t

DemandPeak

Dt ∈ I5t

Mt = h2S2t

Mt = h5S5t

hi Sit 1Dt∈Iit

C it − Dt

I1t Nuclear

Capacity(GW)

DemandOff Peak

Dt ∈ I2t

DemandPeak

Dt ∈ I5t

Mt = h2S2t

Mt = h5S5t

hi Sit 1Dt∈Iit

C it − Dt

St =γ

Power spot price modeling

St = g (Cmaxt − Dt)

hiSit1Dt∈I i

withDt demand at t ,C i

t production capacity of fuel i ,Cmax

t total capacity,Si

t spot price of fuel i ,hi heat rate of production with fuel i .I it capacity interval where fuel i is marginal.

g(x) = min(M, γ

)1x>0 + M1x<0

0 2 40

Residual capacity (GWh)

γ=15 γ=10 γ=5

Results on spot prices

Figure : Spot price reconstruction from the structural model

Consequence on forward prices

No-arbitrage condition

Ft(T ) = EQ[ST |Ft ]

Induced relation between forward prices

Ft(T ) =

g (CmaxT − DT ) 1DT∈I i

︸︷︷︸

GiT (t,Ct ,Dt )

hiF it (T )

Final reconstruction (assuming F it (T

′) = F it (T , θ) ∀T ′ ∈ [T ; T + θ]:

Ft(T , θ) =n∑

T+θ∑

T ′=T

GiT ′(t ,Ct ,Dt)

︸︷︷︸

stochastic weights

hiF it (T , θ)

Modeling demand and capacities

Deterministic part + stochastic part (Ornstein-Uhlenbeck)

Dt = fD(t) + ZD(t)

C it = fi(t) + Zi(t)

Deterministic part:

fD(t) = d (D)1 + d (D)

2 cos(

2π(t − d (D)3 )

+ weekD(t)

fi(t) = d (i)1 + d (i)

2 cos(

2π(t − d (i)3 )

+ week i(t)

Stochastic part: Ornstein-Uhlenbeck process

dZD(t) = −αDZD(t)dt + βDdW Dt

dZi(t) = −αiZi(t)dt + βidW it

Study on forward reconstruction

F (t ,T , θ) =

T+θ∑

T ′=T

GiT ′(t ,Ct ,Dt)

hiF i(t ,T , θ)

GiT ′(t ,Ct ,Dt) = E

g (CmaxT − DT ) 1DT∈I i

Model specifications3 types of capacities: nuclear, (gas/coal), oilCarbon taken into accountEstimation of Demand and Capacities model parameters d (∗)

1 , d (∗)2 , d (∗)

3and week∗(t) on two years of historical data from RTE1

Observation of commodity forward products from Platts 2

1www.rte-france.fr2www.platts.fr

Electricity forward reconstruction: Month-ahead

01/09 10/09 08/10 05/11 03/12 12/12

Product : 1MAH

Market Price Model Price

Figure : 1-Month-ahead reconstruction: observed prices (blue) and reconstructed forward prices fromcomplete structural model with (red) and without (pink) the scarcity function

Electricity forward reconstruction: Quarter-ahead

01/09 11/09 08/10 06/11 03/12 12/12

Product : 1QAH

Figure : 1-Quarter-ahead reconstruction: observed prices (blue) and reconstructed forward prices fromcomplete structural model with (red) and without (pink) the scarcity function

Electricity forward reconstruction: Year-ahead

01/09 10/09 08/10 05/11 03/12 12/12

Product : 1YAH

Figure : 1-Year-ahead reconstruction: observed prices (blue) and reconstructed forward prices fromcomplete structural model with (red) and without (pink) the scarcity function

Forward reconstruction with model uncertainty

fD(t) = d (D)1 + d (D)

2 cos(

2π(t − d (D)3 )

+ weekD(t)

fi(t) = d (i)1 + d (i)

2 cos(

2π(t − d (i)3 )

+ week i(t)

dZD(t) = −αDZD(t)dt + βDdW Dt

dZi(t) = −αiZi(t)dt + βidW it

Parameters of demand and capacities models are estimated onhistorical data

Confidence intervals can also be estimatedA confidence interval can be dedeuced on forward prices

from 20 model parameterswith 400 Monte Carlo simulation on each parameter’s confidence interval

Forward reconstruction with model uncertainty

04-Jan-2011 15-Mar-2011 31-May-2011 09-Aug-2011 19-Oct-2011 30-Dec-2011

Market Price Model Price Confidence Bounds

Calibration

Calibration problemAt a given date t , we observe several forward products (Ft(T , θ))(T , θ)Objective of calibration: determine model parameters to exactly retrieve allthe forward products

Impossible for the current model

Introduction of a "risk premium" ε(T ) in the demand process

fD(t) = ε(T ) + d (D)1 + d (D)

2 cos(

2π(t − d (D)3 )

+ weekD(t)

fi(t) = d (i)1 + d (i)

2 cos(

2π(t − d (i)3 )

+ week i(t)

Calibration results

07/11 10/11 02/12 05/12 08/12 11/12−1.5

−0.5

Figure : Calibrated ε(T ) on observed forward prices: 1MAH→ 6MAH, 1QAH→ 3QAH and 1YAH observedon 28-Jun-2011.

Calibration results

01/11 01/12 01/13−3.5

−2.5

−1.5

−0.5

1MAH1QAH1YAH

Figure : Evolution of ε(T ) over time

Conclusion

Satisfactory reconstruction of forward productsEnough random factors to produce non trivial long term pricesLong term products seem to be strongly linked with fundamentalsA risk aversion is observable

The calibration on observed forward products is possibleThe "implied" function ε(T ) is reasonableShort-term forward products present more risk aversion

Perspectives2-factor model

Keep volatility functions σs(t) and σl (t)Semi-parametric estimationSome first statistical resultsimprove the link between commodities

Structural modelDeepen the parameter uncertainty propagationCalibration on options

Bibliography

R. Aïd, L. Campi and N. Langrené, " A structural risk-neutral model for pricing and hedging powerderivatives ", hal-00525800, to appear in Mathematical Finance

F. E. Benth and S. Koekebakker, "Stochastic modeling of financial electricity contracts", EnergyEconomics, vol.30, pp.1116–1157, 2008

S. Goutte, N. Oudjane et F. Russo, " Variance Optimal Hedging for discrete time processes withindependent increments. Application to Electricity Markets ", to appear in Journal of ComputationalFinance, 2011.

S. Goutte, N. Oudjane et F. Russo, " Variance Optimal Hedging for continuous time processes withindependent increments and applications ", Stochastics,http://dx.doi.org/10.1080/17442508.2013.77440, 2013

R. Kiesel, G. Schidlmayr and R. H. Börger, "A two-factor model for electricity forward market",Quantitative Finance, vol. 9, n◦ 3, 2009. Available at SSRN: http://ssrn.com/abstract=982784

A. Nguyen Huu, Valorisation financière sur les marchés d’électricité, PhD. Thesis, Université ParisDauphine, 2012.

A. Nguyen Huu, " A note on super-hedging for investors-producers " , FiME lab. repoprt, rr-fime-12-01,Université Paris Dauphine, 2012.

A. Monfort et O. Féron, " Joint econometric modeling of spot elecricity prices, forwards and options ",Review of Derivatives Research, vol.15, n◦3, pp. 217–256, 2012

X. Warin, " Gas storage hedging ", FiME lab. report, rr-fime-11-04, Université Paris Dauphine, 2011.

Results: Earnings at Risk (UK Power)

Q2_2013 Q3_2013 Q4_2013 Q1_2014 Q2_2014 Q3_2014 Q4_2014 Q1_20152

Figure : Earnig-at-Risk estimation

Results: Estimation on Fench Power

1WAH 1MAH 2MAH 3MAH 1QAH 2QAH 3QAH 1YAH 2YAH0

Products

Figure : Estimation results on French power: volatility reconstruction

parameter Estimation on MV Estimation on spotσs 45% 302%α 8.73 88.15σl 11% 11%

Results: European options pricing (French Power)

Mar−13 Jun−13 Sep−13 Jan−14 Apr−14 Jul−14 Oct−14 Feb−15 May−150

Period of EaR calculus

Figure : European option pricing on French Power

Empirical volatility Estimation on MV Estimation on spotMtM (e) 41 068 40 326 27 649

Relative difference – -2% -33%

Results: Earnings at Risk (French Power)

Q2_2013 Q3_2013 Q4_2013 Q1_2014 Q2_2014 Q3_2014 Q4_2014 Q1_20152

Period of EaR calculus

Figure : Earning-at-Risk estimation

Recent work on parameters estimation

Model formulation: keep σs(t) and σl(t) as function.

dFt(T )

Ft(T )= σs(t)e−α(T−t)dW (s)

t + σl(t)dW (l)t

Approximation Ft(T , θ) = Ft(T ) ⇒ statistical (asymptotic) results onestimates

Semi-parametric estimationσs(t) and σl(t) are not parametrizedFocus on estimation of α (considering σs(t) and σl(t) as nuisanceparameters)Non parametric estimation of functions σs(t) and σl(t)

Identification and estimation with two maturities

Observations (Ftn(T1))tn∈[0 ; T1]and (Ftn(T2))tn∈[0 ; T2]

of two forwardproducts of respective delivery T1 and T2.We will suppose T1 < T2 and tN = T1 (i.e. the number of jointlyobservale forward products is N).

Let us note ∆F ni (T ) = ln

(Fti (T )

Fti−1 (T )

Estimation of αDefine the function g : (−1, 1) ∋ x 7→ − 1

T2−T1ln(

1+x1−x

∈ R.

Let q̂ =∑n

i=1(∆Fni (T2)−∆Fn

i (T1))2

∑ni=1

(∆Fni (T2))

2−(∆Fn

i (T1))2) .

Estimator of α α̂ = g(q̂1q̂∈(−1,1)

Identification and estimation with two maturities

Theorem

Suppose

σs and σl are continuous on [0 ; T1],

∃ (κ,K ) ∈ (R+∗ )

2, ∀i = 1, ...,n, κT1

n ≤ |ti − ti−1| ≤ K T1n

Then√

n (α̂− α) −−−−→n→+∞

N (0,V∞(α)) with

V∞(α) =T1

(T2 − T1)2

eα(T2−T1) − 1)2

∫ T1

0 e−2α(T1−t)σ2s (t)σ

2l (t)dHt

(∫ T1

0 e−2α(T1−t)σ2s (t)dt

)2 (1)

with Ht = limn→+∞nT1

ti≤t(ti − ti−1)2 the Asymptotic Quadratic Variation of

By noting T2 = T1 + τ , we have V∞(α) −−−−→T1→∞

2α(eατ−1)2

τ2σ2

When T1 → ∞, the behavior of V∞(α) is linear in T1

V∞(α) increases when α increases

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