commodity price modeling in edf. parameter estimation and ... · commodity price modeling in edf....
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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
Workshop on Electricity, Energy and Commodities Risk Managemen t - Fields Institute, Toronto, 2013 2 / 41
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
Workshop on Electricity, Energy and Commodities Risk Managemen t - Fields Institute, Toronto, 2013 3 / 41
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
Workshop on Electricity, Energy and Commodities Risk Managemen t - Fields Institute, Toronto, 2013 4 / 41
Spot market
Daily spot for fuels
(semi)Hourly spot price for power
0 4 8 12 16 20 24
40
60
Hours
Pric
e(e
/MW
h)
Spot price Peak price Base price
Figure : French power spot prices on January 12th 2012
Workshop on Electricity, Energy and Commodities Risk Managemen t - Fields Institute, Toronto, 2013 5 / 41
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
Time
Workshop on Electricity, Energy and Commodities Risk Managemen t - Fields Institute, Toronto, 2013 6 / 41
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
Time2011
Q1 Q2 Q3
1 2 3 4 5
t0
15/12/2010
Workshop on Electricity, Energy and Commodities Risk Managemen t - Fields Institute, Toronto, 2013 7 / 41
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
Time2011
Q1 Q2 Q3
1 2 3 4 5
t0
15/12/2010
2012
Q1 Q2 Q3
1 2 311 12
t1
25/10/2011
Workshop on Electricity, Energy and Commodities Risk Managemen t - Fields Institute, Toronto, 2013 8 / 41
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
Time2011
Q1 Q2 Q3
1 2 3 4 5
t0
15/12/2010
2012
Q1 Q2 Q3
1 2 311 12
t1
25/10/2011 Zoom
October 2011 November 2011
W1 W2 W3 W4 W1 W2 W3 W4W1 W2 W3
25/10/2011
Time
Workshop on Electricity, Energy and Commodities Risk Managemen t - Fields Institute, Toronto, 2013 9 / 41
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
Workshop on Electricity, Energy and Commodities Risk Managemen t - Fields Institute, Toronto, 2013 10 / 41
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 ]
Workshop on Electricity, Energy and Commodities Risk Managemen t - Fields Institute, Toronto, 2013 11 / 41
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
Workshop on Electricity, Energy and Commodities Risk Managemen t - Fields Institute, Toronto, 2013 12 / 41
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
Workshop on Electricity, Energy and Commodities Risk Managemen t - Fields Institute, Toronto, 2013 13 / 41
Two-factor model description
FormulationdFt(T )
Ft(T )= σs(t)e
−α(T−t)dW (s)t + σl(t)dW (l)
t
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∑
i=0
[
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)
t
Workshop on Electricity, Energy and Commodities Risk Managemen t - Fields Institute, Toronto, 2013 14 / 41
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
Workshop on Electricity, Energy and Commodities Risk Managemen t - Fields Institute, Toronto, 2013 15 / 41
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
n
with ν(δt) = e2αδt−1
2α
Marginal volatility: average of instantaneous return variance
MV 2(T , θ, α, σs, σl) =1N
N∑
n=1
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)
)2
Workshop on Electricity, Energy and Commodities Risk Managemen t - Fields Institute, Toronto, 2013 16 / 41
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
[
σ2s
1 − e−2αt
2α+ σ
2l t]
+
∫ t
0σse−α(t−u)dW (s)
u
︸ ︷︷ ︸
X (s)t
+
∫ t
0σldW (l)
u
︸ ︷︷ ︸
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
Workshop on Electricity, Energy and Commodities Risk Managemen t - Fields Institute, Toronto, 2013 17 / 41
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)
Workshop on Electricity, Energy and Commodities Risk Managemen t - Fields Institute, Toronto, 2013 18 / 41
Results: estimation on UK Power
1WAH 1MAH 1QAH 1SAH 6SAH0.08
0.1
0.12
0.14
0.16
0.18
0.2
Products
Vol
atili
ty
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%
Workshop on Electricity, Energy and Commodities Risk Managemen t - Fields Institute, Toronto, 2013 19 / 41
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
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
Month
euro
s/M
Wh
Empirical Vol2F calibrated on MV2F calibrated on spot
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%
Workshop on Electricity, Energy and Commodities Risk Managemen t - Fields Institute, Toronto, 2013 20 / 41
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
Workshop on Electricity, Energy and Commodities Risk Managemen t - Fields Institute, Toronto, 2013 21 / 41
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
Workshop on Electricity, Energy and Commodities Risk Managemen t - Fields Institute, Toronto, 2013 22 / 41
Fundamental approach
Wind
I1t Nuclear
I2t
Coal
I3t
Gas
I4t
Oil
I5t
Production Cost(eur/GWh)
Capacity(GW)
0
Cmaxt
Figure : Structural model [Aïd et al. (2012)]
Workshop on Electricity, Energy and Commodities Risk Managemen t - Fields Institute, Toronto, 2013 23 / 41
Fundamental approach
Wind
I1t Nuclear
I2t
Coal
I3t
Gas
I4t
Oil
I5t
Production Cost(eur/GWh)
Capacity(GW)
0
DemandOff Peak
Dt ∈ I2t
Cmaxt
Figure : Structural model [Aïd et al. (2012)]
Workshop on Electricity, Energy and Commodities Risk Managemen t - Fields Institute, Toronto, 2013 24 / 41
Fundamental approach
Wind
I1t Nuclear
I2t
Coal
I3t
Gas
I4t
Oil
I5t
Production Cost(eur/GWh)
Capacity(GW)
0
DemandOff Peak
Dt ∈ I2t
Cmaxt
Mt = h2S2t
Rt
Mt =n
∑
i=1
hi Sit 1Dt∈Iit
Rt =n
∑
i=1
C it − Dt
Figure : Structural model [Aïd et al. (2012)]
Workshop on Electricity, Energy and Commodities Risk Managemen t - Fields Institute, Toronto, 2013 25 / 41
Fundamental approach
Wind
I1t Nuclear
I2t
Coal
I3t
Gas
I4t
Oil
I5t
Production Cost(eur/GWh)
Capacity(GW)
0
DemandOff Peak
Dt ∈ I2t
DemandPeak
Dt ∈ I5t
Cmaxt
Mt = h2S2t
Mt = h5S5t
Rt
Rt
Mt =n
∑
i=1
hi Sit 1Dt∈Iit
Rt =n
∑
i=1
C it − Dt
Figure : Structural model [Aïd et al. (2012)]
Workshop on Electricity, Energy and Commodities Risk Managemen t - Fields Institute, Toronto, 2013 26 / 41
Fundamental approach
Wind
I1t Nuclear
I2t
Coal
I3t
Gas
I4t
Oil
I5t
Production Cost(eur/GWh)
Capacity(GW)
0
DemandOff Peak
Dt ∈ I2t
DemandPeak
Dt ∈ I5t
Cmaxt
Mt = h2S2t
Mt = h5S5t
Rt
Rt
Mt =n
∑
i=1
hi Sit 1Dt∈Iit
Rt =n
∑
i=1
C it − Dt
St =γ
Rν
t
Mt
Figure : Structural model [Aïd et al. (2012)]
Workshop on Electricity, Energy and Commodities Risk Managemen t - Fields Institute, Toronto, 2013 27 / 41
Power spot price modeling
St = g (Cmaxt − Dt)
n∑
i=1
hiSit1Dt∈I i
t
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
xν
)1x>0 + M1x<0
0 2 40
10
20
30
Residual capacity (GWh)
γ=15 γ=10 γ=5
Workshop on Electricity, Energy and Commodities Risk Managemen t - Fields Institute, Toronto, 2013 28 / 41
Results on spot prices
Figure : Spot price reconstruction from the structural model
Workshop on Electricity, Energy and Commodities Risk Managemen t - Fields Institute, Toronto, 2013 29 / 41
Consequence on forward prices
No-arbitrage condition
Ft(T ) = EQ[ST |Ft ]
Induced relation between forward prices
Ft(T ) =
n∑
i=1
E∗[
g (CmaxT − DT ) 1DT∈I i
T|Ft
]
︸ ︷︷ ︸
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∑
i=1
(1θ
T+θ∑
T ′=T
GiT ′(t ,Ct ,Dt)
)
︸ ︷︷ ︸
stochastic weights
hiF it (T , θ)
Workshop on Electricity, Energy and Commodities Risk Managemen t - Fields Institute, Toronto, 2013 30 / 41
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
Workshop on Electricity, Energy and Commodities Risk Managemen t - Fields Institute, Toronto, 2013 31 / 41
Study on forward reconstruction
F (t ,T , θ) =
n∑
i=1
(1θ
T+θ∑
T ′=T
GiT ′(t ,Ct ,Dt)
)
hiF i(t ,T , θ)
GiT ′(t ,Ct ,Dt) = E
∗[
g (CmaxT − DT ) 1DT∈I i
T|Ft
]
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
Workshop on Electricity, Energy and Commodities Risk Managemen t - Fields Institute, Toronto, 2013 32 / 41
Electricity forward reconstruction: Month-ahead
01/09 10/09 08/10 05/11 03/12 12/12
30
40
50
60
70
Product : 1MAH
Pric
e(e
ur/M
Wh)
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
Workshop on Electricity, Energy and Commodities Risk Managemen t - Fields Institute, Toronto, 2013 33 / 41
Electricity forward reconstruction: Quarter-ahead
01/09 11/09 08/10 06/11 03/12 12/12
30
40
50
60
70
Product : 1QAH
Pric
e(e
ur/M
Wh)
Market Price Model Price
Figure : 1-Quarter-ahead reconstruction: observed prices (blue) and reconstructed forward prices fromcomplete structural model with (red) and without (pink) the scarcity function
Workshop on Electricity, Energy and Commodities Risk Managemen t - Fields Institute, Toronto, 2013 34 / 41
Electricity forward reconstruction: Year-ahead
01/09 10/09 08/10 05/11 03/12 12/12
35
40
45
50
55
60
Product : 1YAH
Pric
e(e
ur/M
Wh)
Market Price Model Price
Figure : 1-Year-ahead reconstruction: observed prices (blue) and reconstructed forward prices fromcomplete structural model with (red) and without (pink) the scarcity function
Workshop on Electricity, Energy and Commodities Risk Managemen t - Fields Institute, Toronto, 2013 35 / 41
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
Workshop on Electricity, Energy and Commodities Risk Managemen t - Fields Institute, Toronto, 2013 36 / 41
Forward reconstruction with model uncertainty
04-Jan-2011 15-Mar-2011 31-May-2011 09-Aug-2011 19-Oct-2011 30-Dec-2011
50
55
60
Days
Pric
e(e
ur/M
Wh)
Market Price Model Price Confidence Bounds
Workshop on Electricity, Energy and Commodities Risk Managemen t - Fields Institute, Toronto, 2013 37 / 41
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)
Workshop on Electricity, Energy and Commodities Risk Managemen t - Fields Institute, Toronto, 2013 38 / 41
Calibration results
07/11 10/11 02/12 05/12 08/12 11/12−1.5
−1
−0.5
0
0.5
1
Dates
GW
Figure : Calibrated ε(T ) on observed forward prices: 1MAH→ 6MAH, 1QAH→ 3QAH and 1YAH observedon 28-Jun-2011.
Workshop on Electricity, Energy and Commodities Risk Managemen t - Fields Institute, Toronto, 2013 39 / 41
Calibration results
01/11 01/12 01/13−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
Dates
GW
1MAH1QAH1YAH
Figure : Evolution of ε(T ) over time
Workshop on Electricity, Energy and Commodities Risk Managemen t - Fields Institute, Toronto, 2013 40 / 41
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
Workshop on Electricity, Energy and Commodities Risk Managemen t - Fields Institute, Toronto, 2013 41 / 41
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.
Workshop on Electricity, Energy and Commodities Risk Managemen t - Fields Institute, Toronto, 2013 42 / 41
Results: Earnings at Risk (UK Power)
Q2_2013 Q3_2013 Q4_2013 Q1_2014 Q2_2014 Q3_2014 Q4_2014 Q1_20152
4
6
8
10
12
14
16
18
20
Month
euro
s/M
Wh
Empirical Vol2F calibrated on MV2F calibrated on spot
Figure : Earnig-at-Risk estimation
Workshop on Electricity, Energy and Commodities Risk Managemen t - Fields Institute, Toronto, 2013 43 / 41
Results: Estimation on Fench Power
1WAH 1MAH 2MAH 3MAH 1QAH 2QAH 3QAH 1YAH 2YAH0
0.2
0.4
0.6
0.8
1
1.2
1.4
Products
Vol
atili
ty
Empirical Vol2F calibrated on MV2F calibrated on spot
Figure : Estimation results on French power: volatility reconstruction
parameter Estimation on MV Estimation on spotσs 45% 302%α 8.73 88.15σl 11% 11%
Workshop on Electricity, Energy and Commodities Risk Managemen t - Fields Institute, Toronto, 2013 44 / 41
Results: European options pricing (French Power)
Mar−13 Jun−13 Sep−13 Jan−14 Apr−14 Jul−14 Oct−14 Feb−15 May−150
0.5
1
1.5
2
2.5
3
3.5
4
Period of EaR calculus
euro
s/M
Wh
Empirical Vol2F calibrated on MV2F calibrated on spot
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%
Workshop on Electricity, Energy and Commodities Risk Managemen t - Fields Institute, Toronto, 2013 45 / 41
Results: Earnings at Risk (French Power)
Q2_2013 Q3_2013 Q4_2013 Q1_2014 Q2_2014 Q3_2014 Q4_2014 Q1_20152
3
4
5
6
7
8
9
10
Period of EaR calculus
euro
s/M
Wh
Empirical Vol2F calibrated on MV2F calibrated on spot
Figure : Earning-at-Risk estimation
Workshop on Electricity, Energy and Commodities Risk Managemen t - Fields Institute, Toronto, 2013 46 / 41
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)
Workshop on Electricity, Energy and Commodities Risk Managemen t - Fields Institute, Toronto, 2013 47 / 41
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)
)
Workshop on Electricity, Energy and Commodities Risk Managemen t - Fields Institute, Toronto, 2013 48 / 41
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
Time.
By noting T2 = T1 + τ , we have V∞(α) −−−−→T1→∞
2α(eατ−1)2
τ2σ2
lσ2
sT1
When T1 → ∞, the behavior of V∞(α) is linear in T1
V∞(α) increases when α increases
Workshop on Electricity, Energy and Commodities Risk Managemen t - Fields Institute, Toronto, 2013 49 / 41