Position of the problem
Spot model
Pricing & hedging
Conclusion
A structural risk-neutral model
for pricing and hedging power derivatives
Energy Finance 2010 Conference - Duisburg University - Essen
René Aı̈d, Luciano Campi, Nicolas Langrené
Paris-Dauphine University - Paris Diderot University
EDF R&D - FiME Research Centre
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research Centre
1 / 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Agenda
1
Position of the problem
Electricity prices modeling
Related works
2
Spot model
Design
Estimation
3
Pricing & hedging
Futures
Options
4
Conclusion
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research Centre
2 / 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Electricity prices modeling
Related works
Looking for a power spot price model
Applications
pricing of derivatives on the spot
asset valuation (strip of hourly fuel spread options)
hedging
energy market risk management
Model requirements
realistic
robust
tractable
consistent
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research Centre
3 / 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Electricity prices modeling
Related works
Looking for a power spot price model
Applications
pricing of derivatives on the spot
asset valuation (strip of hourly fuel spread options)
hedging
energy market risk management
Model requirements
realistic
robust
tractable
consistent
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research Centre
3 / 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Electricity prices modeling
Related works
Looking for a power spot price model
Applications
pricing of derivatives on the spot
asset valuation (strip of hourly fuel spread options)
hedging
energy market risk management
Model requirements
realistic
robust
tractable
consistent
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research Centre
3 / 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Electricity prices modeling
Related works
Looking for a power spot price model
Applications
pricing of derivatives on the spot
asset valuation (strip of hourly fuel spread options)
hedging
energy market risk management
Model requirements
realistic
robust
tractable
consistent
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research Centre
3 / 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Electricity prices modeling
Related works
Looking for a power spot price model
Applications
pricing of derivatives on the spot
asset valuation (strip of hourly fuel spread options)
hedging
energy market risk management
Model requirements
realistic
robust
tractable
consistent
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research Centre
3 / 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Electricity prices modeling
Related works
Looking for a power spot price model
Applications
pricing of derivatives on the spot
asset valuation (strip of hourly fuel spread options)
hedging
energy market risk management
Model requirements
realistic
robust
tractable
consistent
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research Centre
3 / 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Electricity prices modeling
Related works
Looking for a power spot price model
Applications
pricing of derivatives on the spot
asset valuation (strip of hourly fuel spread options)
hedging
energy market risk management
Model requirements
realistic
robust
tractable
consistent
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research Centre
3 / 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Electricity prices modeling
Related works
Looking for a power spot price model
Applications
pricing of derivatives on the spot
asset valuation (strip of hourly fuel spread options)
hedging
energy market risk management
Model requirements
realistic
robust
tractable
consistent
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research Centre
3 / 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Electricity prices modeling
Related works
Looking for a power spot price model
Applications
pricing of derivatives on the spot
asset valuation (strip of hourly fuel spread options)
hedging
energy market risk management
Model requirements
realistic
robust
tractable
consistent
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research Centre
3 / 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Electricity prices modeling
Related works
Modeling strategies
Modeling futures prices
pros modeling the real available instruments
cons introduction of many parameters to reconstruct
hourly futures prices
Modeling spot prices
1
Exogeneous
pros tractability
cons dependancies
2
Equilibrium
pros dependancies
cons complexity
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research Centre
4 / 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Electricity prices modeling
Related works
Modeling strategies
Modeling futures prices
pros modeling the real available instruments
cons introduction of many parameters to reconstruct
hourly futures prices
Modeling spot prices
1
Exogeneous
pros tractability
cons dependancies
2
Equilibrium
pros dependancies
cons complexity
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research Centre
4 / 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Electricity prices modeling
Related works
Modeling strategies
Modeling futures prices
pros modeling the real available instruments
cons introduction of many parameters to reconstruct
hourly futures prices
Modeling spot prices
1
Exogeneous
pros tractability
cons dependancies
2
Equilibrium
pros dependancies
cons complexity
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research Centre
4 / 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Electricity prices modeling
Related works
Modeling strategies
Modeling futures prices
pros modeling the real available instruments
cons introduction of many parameters to reconstruct
hourly futures prices
Modeling spot prices
1
Exogeneous
pros tractability
cons dependancies
2
Equilibrium
pros dependancies
cons complexity
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research Centre
4 / 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Electricity prices modeling
Related works
Modeling strategies
Modeling futures prices
pros modeling the real available instruments
cons introduction of many parameters to reconstruct
hourly futures prices
Modeling spot prices
1
Exogeneous
pros tractability
cons dependancies
2
Equilibrium
pros dependancies
cons complexity
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research Centre
4 / 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Electricity prices modeling
Related works
Modeling strategies
Modeling futures prices
pros modeling the real available instruments
cons introduction of many parameters to reconstruct
hourly futures prices
Modeling spot prices
1
Exogeneous
pros tractability
cons dependancies
2
Equilibrium
pros dependancies
cons complexity
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research Centre
4 / 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Electricity prices modeling
Related works
Modeling strategies
Modeling futures prices
pros modeling the real available instruments
cons introduction of many parameters to reconstruct
hourly futures prices
Modeling spot prices
1
Exogeneous
pros tractability
cons dependancies
2
Equilibrium
pros dependancies
cons complexity
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research Centre
4 / 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Electricity prices modeling
Related works
Modeling strategies
Modeling futures prices
pros modeling the real available instruments
cons introduction of many parameters to reconstruct
hourly futures prices
Modeling spot prices
1
Exogeneous
pros tractability
cons dependancies
2
Equilibrium
pros dependancies
cons complexity
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research Centre
4 / 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Electricity prices modeling
Related works
Modeling strategies
Modeling futures prices
pros modeling the real available instruments
cons introduction of many parameters to reconstruct
hourly futures prices
Modeling spot prices
1
Exogeneous
pros tractability
cons dependancies
2
Equilibrium
pros dependancies
cons complexity
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research Centre
4 / 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Electricity prices modeling
Related works
Modeling strategies
Modeling futures prices
pros modeling the real available instruments
cons introduction of many parameters to reconstruct
hourly futures prices
Modeling spot prices
1
Exogeneous
pros tractability
cons dependancies
2
Equilibrium
pros dependancies
cons complexity
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research Centre
4 / 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Electricity prices modeling
Related works
Related works
Electricity prices exogeneous dynamics
Deng (00), Benth et al. (03, 07, 09), Burger et al. (04), Kolodnyi
(04), Cartea & Figueroa (05), Geman & Roncoroni (06)
Equilibrium model
Spot
Futures
Pirrong & Jermakyan (00)
×
Barlow (02)
×
Kanamura & Ohashi (07)
×
Cartea & Villaplana (08)
×
×
Coulon & Howison (09)
×
×
Lyle & Elliot (09)
×
×
Options
×
×
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research Centre
5 / 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Electricity prices modeling
Related works
Related works
Electricity prices exogeneous dynamics
Deng (00), Benth et al. (03, 07, 09), Burger et al. (04), Kolodnyi
(04), Cartea & Figueroa (05), Geman & Roncoroni (06)
Equilibrium model
Spot
Futures
Pirrong & Jermakyan (00)
×
Barlow (02)
×
Kanamura & Ohashi (07)
×
Cartea & Villaplana (08)
×
×
Coulon & Howison (09)
×
×
Lyle & Elliot (09)
×
×
Options
×
×
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research Centre
5 / 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Electricity prices modeling
Related works
Related works
Electricity prices exogeneous dynamics
Deng (00), Benth et al. (03, 07, 09), Burger et al. (04), Kolodnyi
(04), Cartea & Figueroa (05), Geman & Roncoroni (06)
Equilibrium model
Spot
Futures
Pirrong & Jermakyan (00)
×
Barlow (02)
×
Kanamura & Ohashi (07)
×
Cartea & Villaplana (08)
×
×
Coulon & Howison (09)
×
×
Lyle & Elliot (09)
×
×
Options
×
×
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research Centre
5 / 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Electricity prices modeling
Related works
Related works
Electricity prices exogeneous dynamics
Deng (00), Benth et al. (03, 07, 09), Burger et al. (04), Kolodnyi
(04), Cartea & Figueroa (05), Geman & Roncoroni (06)
Equilibrium model
Spot
Futures
Pirrong & Jermakyan (00)
×
Barlow (02)
×
Kanamura & Ohashi (07)
×
Cartea & Villaplana (08)
×
×
Coulon & Howison (09)
×
×
Lyle & Elliot (09)
×
×
Options
×
×
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research Centre
5 / 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Electricity prices modeling
Related works
Related works
Electricity prices exogeneous dynamics
Deng (00), Benth et al. (03, 07, 09), Burger et al. (04), Kolodnyi
(04), Cartea & Figueroa (05), Geman & Roncoroni (06)
Equilibrium model
Spot
Futures
Pirrong & Jermakyan (00)
×
Barlow (02)
×
Kanamura & Ohashi (07)
×
Cartea & Villaplana (08)
×
×
Coulon & Howison (09)
×
×
Lyle & Elliot (09)
×
×
Options
×
×
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research Centre
5 / 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Electricity prices modeling
Related works
Related works
Electricity prices exogeneous dynamics
Deng (00), Benth et al. (03, 07, 09), Burger et al. (04), Kolodnyi
(04), Cartea & Figueroa (05), Geman & Roncoroni (06)
Equilibrium model
Spot
Futures
Pirrong & Jermakyan (00)
×
Barlow (02)
×
Kanamura & Ohashi (07)
×
Cartea & Villaplana (08)
×
×
Coulon & Howison (09)
×
×
Lyle & Elliot (09)
×
×
Options
×
×
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research Centre
5 / 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Electricity prices modeling
Related works
Related works
Electricity prices exogeneous dynamics
Deng (00), Benth et al. (03, 07, 09), Burger et al. (04), Kolodnyi
(04), Cartea & Figueroa (05), Geman & Roncoroni (06)
Equilibrium model
Spot
Futures
Pirrong & Jermakyan (00)
×
Barlow (02)
×
Kanamura & Ohashi (07)
×
Cartea & Villaplana (08)
×
×
Coulon & Howison (09)
×
×
Lyle & Elliot (09)
×
×
Options
×
×
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research Centre
5 / 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Electricity prices modeling
Related works
Related works
Electricity prices exogeneous dynamics
Deng (00), Benth et al. (03, 07, 09), Burger et al. (04), Kolodnyi
(04), Cartea & Figueroa (05), Geman & Roncoroni (06)
Equilibrium model
Spot
Futures
Pirrong & Jermakyan (00)
×
Barlow (02)
×
Kanamura & Ohashi (07)
×
Cartea & Villaplana (08)
×
×
Coulon & Howison (09)
×
×
Lyle & Elliot (09)
×
×
Options
×
×
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research Centre
5 / 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Electricity prices modeling
Related works
Related works
Electricity prices exogeneous dynamics
Deng (00), Benth et al. (03, 07, 09), Burger et al. (04), Kolodnyi
(04), Cartea & Figueroa (05), Geman & Roncoroni (06)
Equilibrium model
Spot
Futures
Pirrong & Jermakyan (00)
×
Barlow (02)
×
Kanamura & Ohashi (07)
×
Cartea & Villaplana (08)
×
×
Coulon & Howison (09)
×
×
Lyle & Elliot (09)
×
×
Options
×
×
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research Centre
5 / 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Electricity prices modeling
Related works
This talk
Objectives
pricing and hedging power derivatives...
... using an improved version of A., Campi Nguyen & Touzi
(09) Structural Risk-Neutral model
Spot
Futures
A., Campi, Nguyen & Touzi (09) ×
×
×
×
improved SRN model
Options
×
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research Centre
6 / 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Electricity prices modeling
Related works
This talk
Objectives
pricing and hedging power derivatives...
... using an improved version of A., Campi Nguyen & Touzi
(09) Structural Risk-Neutral model
Spot
Futures
A., Campi, Nguyen & Touzi (09) ×
×
×
×
improved SRN model
Options
×
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research Centre
6 / 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Electricity prices modeling
Related works
This talk
Objectives
pricing and hedging power derivatives...
... using an improved version of A., Campi Nguyen & Touzi
(09) Structural Risk-Neutral model
Spot
Futures
A., Campi, Nguyen & Touzi (09) ×
×
×
×
improved SRN model
Options
×
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research Centre
6 / 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Electricity prices modeling
Related works
This talk
Objectives
pricing and hedging power derivatives...
... using an improved version of A., Campi Nguyen & Touzi
(09) Structural Risk-Neutral model
Spot
Futures
A., Campi, Nguyen & Touzi (09) ×
×
×
×
improved SRN model
Options
×
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research Centre
6 / 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Electricity prices modeling
Related works
This talk
Objectives
pricing and hedging power derivatives...
... using an improved version of A., Campi Nguyen & Touzi
(09) Structural Risk-Neutral model
Spot
Futures
A., Campi, Nguyen & Touzi (09) ×
×
×
×
improved SRN model
Options
×
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research Centre
6 / 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Electricity prices modeling
Related works
This talk
Objectives
pricing and hedging power derivatives...
... using an improved version of A., Campi Nguyen & Touzi
(09) Structural Risk-Neutral model
Spot
Futures
A., Campi, Nguyen & Touzi (09) ×
×
×
×
improved SRN model
Options
×
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research Centre
6 / 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Initial SRN Model
Variables
n
fuels, 1 ≤ i ≤ n
Dt
demand (MW)
Cti
Sti
capacities (en MW)
hi
fuel prices
heat rates (hi Sti en e/MWh, % en i)
Electricity price (e/MWh)
bt =
P
n
X
hi Sti 1{Pi−1 Ctk ≤ Dt ≤ Pi
k=1
k=1
Ctk }
i=1
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research Centre
7 / 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Initial SRN Model
Variables
n
fuels, 1 ≤ i ≤ n
Dt
demand (MW)
Cti
Sti
capacities (en MW)
hi
fuel prices
heat rates (hi Sti en e/MWh, % en i)
Electricity price (e/MWh)
bt =
P
n
X
hi Sti 1{Pi−1 Ctk ≤ Dt ≤ Pi
k=1
k=1
Ctk }
i=1
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research Centre
7 / 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Initial SRN Model
Variables
n
fuels, 1 ≤ i ≤ n
Dt
demand (MW)
Cti
Sti
capacities (en MW)
hi
fuel prices
heat rates (hi Sti en e/MWh, % en i)
Electricity price (e/MWh)
bt =
P
n
X
hi Sti 1{Pi−1 Ctk ≤ Dt ≤ Pi
k=1
k=1
Ctk }
i=1
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research Centre
7 / 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Initial SRN Model
Variables
n
fuels, 1 ≤ i ≤ n
Dt
demand (MW)
Cti
Sti
capacities (en MW)
hi
fuel prices
heat rates (hi Sti en e/MWh, % en i)
Electricity price (e/MWh)
bt =
P
n
X
hi Sti 1{Pi−1 Ctk ≤ Dt ≤ Pi
k=1
k=1
Ctk }
i=1
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research Centre
7 / 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Initial SRN Model
Variables
n
fuels, 1 ≤ i ≤ n
Dt
demand (MW)
Cti
Sti
capacities (en MW)
hi
fuel prices
heat rates (hi Sti en e/MWh, % en i)
Electricity price (e/MWh)
bt =
P
n
X
hi Sti 1{Pi−1 Ctk ≤ Dt ≤ Pi
k=1
k=1
Ctk }
i=1
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research Centre
7 / 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Initial SRN Model
Variables
n
fuels, 1 ≤ i ≤ n
Dt
demand (MW)
Cti
Sti
capacities (en MW)
hi
fuel prices
heat rates (hi Sti en e/MWh, % en i)
Electricity price (e/MWh)
bt =
P
n
X
hi Sti 1{Pi−1 Ctk ≤ Dt ≤ Pi
k=1
k=1
Ctk }
i=1
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research Centre
7 / 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Initial SRN Model
Variables
n
fuels, 1 ≤ i ≤ n
Dt
demand (MW)
Cti
Sti
capacities (en MW)
hi
fuel prices
heat rates (hi Sti en e/MWh, % en i)
Electricity price (e/MWh)
bt =
P
n
X
hi Sti 1{Pi−1 Ctk ≤ Dt ≤ Pi
k=1
k=1
Ctk }
i=1
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research Centre
7 / 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Initial SRN model
Pros
Consistency between electricity prices and fuel prices
Consistency between electricity prices and demand
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research Centre
8 / 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Initial SRN model
Pros
Consistency between electricity prices and fuel prices
Consistency between electricity prices and demand
Cons
Marginal fuel cost is not the spot price
1
2
3
Non-convex technical constraints (may lead to negative prices)
Strategic behaviour (Hortaçsu & Puller, RAND J. of
Economics 2008)
Fixed cost recovery problem for peak-load generation plants
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research Centre
8 / 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Initial SRN model
Pros
Consistency between electricity prices and fuel prices
Consistency between electricity prices and demand
Cons
Marginal fuel cost is not the spot price
1
2
3
Non-convex technical constraints (may lead to negative prices)
Strategic behaviour (Hortaçsu & Puller, RAND J. of
Economics 2008)
Fixed cost recovery problem for peak-load generation plants
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research Centre
8 / 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Initial SRN model
Pros
Consistency between electricity prices and fuel prices
Consistency between electricity prices and demand
Cons
Marginal fuel cost is not the spot price
1
2
3
Non-convex technical constraints (may lead to negative prices)
Strategic behaviour (Hortaçsu & Puller, RAND J. of
Economics 2008)
Fixed cost recovery problem for peak-load generation plants
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research Centre
8 / 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Initial SRN model
Pros
Consistency between electricity prices and fuel prices
Consistency between electricity prices and demand
Cons
Marginal fuel cost is not the spot price
1
2
3
Non-convex technical constraints (may lead to negative prices)
Strategic behaviour (Hortaçsu & Puller, RAND J. of
Economics 2008)
Fixed cost recovery problem for peak-load generation plants
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research Centre
8 / 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Initial SRN model
Pros
Consistency between electricity prices and fuel prices
Consistency between electricity prices and demand
Cons
Marginal fuel cost is not the spot price
1
2
3
Non-convex technical constraints (may lead to negative prices)
Strategic behaviour (Hortaçsu & Puller, RAND J. of
Economics 2008)
Fixed cost recovery problem for peak-load generation plants
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research Centre
8 / 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Initial SRN model
Pros
Consistency between electricity prices and fuel prices
Consistency between electricity prices and demand
Cons
Marginal fuel cost is not the spot price
1
2
3
Non-convex technical constraints (may lead to negative prices)
Strategic behaviour (Hortaçsu & Puller, RAND J. of
Economics 2008)
Fixed cost recovery problem for peak-load generation plants
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research Centre
8 / 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Initial SRN model
Pros
Consistency between electricity prices and fuel prices
Consistency between electricity prices and demand
Cons
Marginal fuel cost is not the spot price
1
2
3
Non-convex technical constraints (may lead to negative prices)
Strategic behaviour (Hortaçsu & Puller, RAND J. of
Economics 2008)
Fixed cost recovery problem for peak-load generation plants
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research Centre
8 / 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Initial SRN Model - illustration
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research Centre
9 / 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Initial SRN Model - illustration
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research 10
Centre
/ 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Initial SRN Model - illustration
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research 11
Centre
/ 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Improved SRN model
bt := Pn hi S i 1 Pi−1 k
P
Marginal fuel cost P
t {
i=1
C ≤ Dt ≤ i
k=1
t
k=1
Ctk }
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research 12
Centre
/ 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Improved SRN model
bt := Pn hi S i 1 Pi−1 k
P
Marginal fuel cost P
t {
i=1
C ≤ Dt ≤ i
k=1
Available capacity C t :=
t
k=1
Ctk }
Pn
k
k=1 Ct
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research 12
Centre
/ 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Improved SRN model
bt := Pn hi S i 1 Pi−1 k
P
Marginal fuel cost P
t {
i=1
C ≤ Dt ≤ i
k=1
Available capacity C t :=
t
k=1
Ctk }
Pn
k
k=1 Ct
Price spikes occur when the electric system is under stress, i.e.
C t − Dt is small
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research 12
Centre
/ 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Improved SRN model
bt := Pn hi S i 1 Pi−1 k
P
Marginal fuel cost P
t {
i=1
C ≤ Dt ≤ i
k=1
Available capacity C t :=
t
k=1
Ctk }
Pn
k
k=1 Ct
Price spikes occur when the electric system is under stress, i.e.
C t − Dt is small
Corresponds to peak-load fixed cost problem recovery...
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research 12
Centre
/ 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Improved SRN model
bt := Pn hi S i 1 Pi−1 k
P
Marginal fuel cost P
t {
i=1
C ≤ Dt ≤ i
k=1
Available capacity C t :=
t
k=1
Ctk }
Pn
k
k=1 Ct
Price spikes occur when the electric system is under stress, i.e.
C t − Dt is small
Corresponds to peak-load fixed cost problem recovery...
yt :=
Pt
bt
P
as a (nonlinear) function of xt := C t − Dt
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research 12
Centre
/ 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Improved SRN model - Estimation
Observation
Decreasing relation
Difficult estimation
Idea
Quantile
Figure: PowerNext - 19th hours
Nov, 13th 06 to April 30th 10
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research 13
Centre
/ 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Improved SRN model - Estimation
Observation
Decreasing relation
Difficult estimation
Idea
Quantile
Figure: PowerNext - 19th hours
Nov, 13th 06 to April 30th 10
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research 13
Centre
/ 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Improved SRN model - Estimation
Observation
Decreasing relation
Difficult estimation
Idea
Quantile
Figure: PowerNext - 19th hours
Nov, 13th 06 to April 30th 10
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research 13
Centre
/ 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Improved SRN model - Estimation
Observation
Decreasing relation
Difficult estimation
Idea
Quantile
Figure: PowerNext - 19th hours
Nov, 13th 06 to April 30th 10
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research 13
Centre
/ 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Improved SRN model - Estimation
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research 14
Centre
/ 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Improved SRN model - Estimation
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research 15
Centre
/ 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Improved SRN model - Estimation
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research 16
Centre
/ 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Improved SRN model - Estimation
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research 17
Centre
/ 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Improved SRN model - Estimation
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research 18
Centre
/ 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Improved SRN model - Estimation
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research 19
Centre
/ 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Improved SRN model - Estimation
γ
xtν
Estimated relation : yt =
Improved SRN model
Pt = g
n
X
!
Ctk − Dt
k=1
×
n
X
!
hi Sti 1{Pi−1 Ctk ≤ Dt ≤ Pi
k=1
k=1
Ctk }
i=1
with scarcity function
g (x) := min
γ
,
M
1{x>0} + M1{x60}
xν
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research 20
Centre
/ 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Improved SRN model - Back-testing
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research 21
Centre
/ 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Improved SRN model - Back-testing
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research 22
Centre
/ 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Design
Estimation
Improved SRN model - Backtesting
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research 23
Centre
/ 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Futures
Options
Pricing & hedging
Pricing
incomplete market
need for a hedging criterion
Super-replication, utility indifference or mean-variance
our choice : Local Risk Minimization
Local Risk Minimization (Pham (00), Schweizer (01))
b
valuation : expected discounted payoff under Q
allows to decompose contingent claim between hedgeable part
(fuels) and non-hedgeable part (demand, capacities)
allows explicit formulas
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research 24
Centre
/ 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Futures
Options
Pricing & hedging
Pricing
incomplete market
need for a hedging criterion
Super-replication, utility indifference or mean-variance
our choice : Local Risk Minimization
Local Risk Minimization (Pham (00), Schweizer (01))
b
valuation : expected discounted payoff under Q
allows to decompose contingent claim between hedgeable part
(fuels) and non-hedgeable part (demand, capacities)
allows explicit formulas
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research 24
Centre
/ 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Futures
Options
Pricing & hedging
Pricing
incomplete market
need for a hedging criterion
Super-replication, utility indifference or mean-variance
our choice : Local Risk Minimization
Local Risk Minimization (Pham (00), Schweizer (01))
b
valuation : expected discounted payoff under Q
allows to decompose contingent claim between hedgeable part
(fuels) and non-hedgeable part (demand, capacities)
allows explicit formulas
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research 24
Centre
/ 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Futures
Options
Pricing & hedging
Pricing
incomplete market
need for a hedging criterion
Super-replication, utility indifference or mean-variance
our choice : Local Risk Minimization
Local Risk Minimization (Pham (00), Schweizer (01))
b
valuation : expected discounted payoff under Q
allows to decompose contingent claim between hedgeable part
(fuels) and non-hedgeable part (demand, capacities)
allows explicit formulas
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research 24
Centre
/ 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Futures
Options
Pricing & hedging
Pricing
incomplete market
need for a hedging criterion
Super-replication, utility indifference or mean-variance
our choice : Local Risk Minimization
Local Risk Minimization (Pham (00), Schweizer (01))
b
valuation : expected discounted payoff under Q
allows to decompose contingent claim between hedgeable part
(fuels) and non-hedgeable part (demand, capacities)
allows explicit formulas
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research 24
Centre
/ 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Futures
Options
Pricing & hedging
Pricing
incomplete market
need for a hedging criterion
Super-replication, utility indifference or mean-variance
our choice : Local Risk Minimization
Local Risk Minimization (Pham (00), Schweizer (01))
b
valuation : expected discounted payoff under Q
allows to decompose contingent claim between hedgeable part
(fuels) and non-hedgeable part (demand, capacities)
allows explicit formulas
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research 24
Centre
/ 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Futures
Options
Pricing & hedging
Pricing
incomplete market
need for a hedging criterion
Super-replication, utility indifference or mean-variance
our choice : Local Risk Minimization
Local Risk Minimization (Pham (00), Schweizer (01))
b
valuation : expected discounted payoff under Q
allows to decompose contingent claim between hedgeable part
(fuels) and non-hedgeable part (demand, capacities)
allows explicit formulas
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research 24
Centre
/ 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Futures
Options
Pricing & hedging
Pricing
incomplete market
need for a hedging criterion
Super-replication, utility indifference or mean-variance
our choice : Local Risk Minimization
Local Risk Minimization (Pham (00), Schweizer (01))
b
valuation : expected discounted payoff under Q
allows to decompose contingent claim between hedgeable part
(fuels) and non-hedgeable part (demand, capacities)
allows explicit formulas
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research 24
Centre
/ 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Futures
Options
Futures
b −r (T −t)
Futures prices Fte (T ) = EQ
PT
t e
Fte
(T ) =
n
X
hi GiT(t, Ct , Dt ) Fti (T )
i=1
with :
"
GiT(t,Ct ,Dt ) = Et g
n
X
!
CTk − DT
#
1{Pi−1 C k ≤ DT ≤ Pi
k=1 T
k
k=1 CT
}
k=1
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research 25
Centre
/ 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Futures
Options
Futures prices - hedging
Demand & capacities
dDt = a (t, Dt ) dt + b (t, Dt ) dWtD
dCti = αi t, Cti dt + βi t, Cti dWtC ,i
Futures price dynamics
P
dFte (T ) = ni=1 hi GiT (t, Ct , Dt )dFti (T ) + Fti (T )dGiT (t, Ct , Dt )
dGiT (t, Ct , Dt )
=
+
n
X
∂G T
i
k=1
∂GiT
∂z
∂ck
(t, Ct , Dt )βk (t, Ctk )dWtC ,k
(t, Ct , Dt )b(t, Dt )dWtD
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
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University
and hedging
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Position of the problem
Spot model
Pricing & hedging
Conclusion
Futures
Options
Futures prices - hedging
To go further, need to choose dynamics for demand and
capacities
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
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Position of the problem
Spot model
Pricing & hedging
Conclusion
Futures
Options
Futures prices - hedging
To go further, need to choose dynamics for demand and
capacities
deterministic part for seasonality + Ornstein-Uhlenbeck
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
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University
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Position of the problem
Spot model
Pricing & hedging
Conclusion
Futures
Options
Futures prices - hedging
To go further, need to choose dynamics for demand and
capacities
deterministic part for seasonality + Ornstein-Uhlenbeck
GiT explicite as function of extended incomplete
Goodwin-Staton integral :
Z ∞
1
−z 2
e
dz
G (x, y ; ν) =
νe
(y + z)
x
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
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Position of the problem
Spot model
Pricing & hedging
Conclusion
Futures
Options
Futures prices - hedging
To go further, need to choose dynamics for demand and
capacities
deterministic part for seasonality + Ornstein-Uhlenbeck
GiT explicite as function of extended incomplete
Goodwin-Staton integral :
Z ∞
1
−z 2
e
dz
G (x, y ; ν) =
νe
(y + z)
x
... for which efficient numerical algorithms are provided in A.,
Campi & Langrené (10).
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
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Position of the problem
Spot model
Pricing & hedging
Conclusion
Futures
Options
Futures prices - hedging : spot simulations
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
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Position of the problem
Spot model
Pricing & hedging
Conclusion
Futures
Options
Futures prices - hedging : spot simulations
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
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University
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Position of the problem
Spot model
Pricing & hedging
Conclusion
Futures
Options
Futures prices - hedging : spot simulations
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
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University
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Position of the problem
Spot model
Pricing & hedging
Conclusion
Futures
Options
Futures prices - hedging
Numerical test
Hedging an electricity
futures with a delivery
period of 1 hour
with a daily rebalanced
basket of futures
contracts on fuels
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
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University
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Position of the problem
Spot model
Pricing & hedging
Conclusion
Futures
Options
Futures prices - hedging
Numerical test
Hedging an electricity
futures with a delivery
period of 1 hour
with a daily rebalanced
basket of futures
contracts on fuels
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
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Position of the problem
Spot model
Pricing & hedging
Conclusion
Futures
Options
Futures prices - hedging
Numerical test
Hedging an electricity
futures with a delivery
period of 1 hour
with a daily rebalanced
basket of futures
contracts on fuels
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
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Position of the problem
Spot model
Pricing & hedging
Conclusion
Futures
Options
Futures prices - hedging
Numerical test
Hedging an electricity
futures with a delivery
period of 1 hour
with a daily rebalanced
basket of futures
contracts on fuels
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
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Centre
/ 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Futures
Options
Futures prices - hedging
Remarks
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
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power- derivatives
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Position of the problem
Spot model
Pricing & hedging
Conclusion
Futures
Options
Futures prices - hedging
Remarks
Positive values are
losses
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
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Position of the problem
Spot model
Pricing & hedging
Conclusion
Futures
Options
Futures prices - hedging
Remarks
Positive values are
losses
Far from maturity :
perfect hedge ;
electricity futures is
equivalent to a
basket of fuels
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
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Position of the problem
Spot model
Pricing & hedging
Conclusion
Futures
Options
Futures prices - hedging
Remarks
Positive values are
losses
Far from maturity :
perfect hedge ;
electricity futures is
equivalent to a
basket of fuels
Close to maturity :
inefficient hedge
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
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University
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Position of the problem
Spot model
Pricing & hedging
Conclusion
Futures
Options
Spread options (do not panic)
Spread option with a 2 fuel model
The price π0 at time t = 0 of a call spread option with pay-off
H = (PT − h1 ST1 − K )+ is given by :
Z
π0 =
fC 1 −DT (z)fC 2 (c) φ1 (c, z)1{z>0} + φ2 (c, z)1{z≤0} dcdz,
R2
T
T
φ1 = (g − 1)BS0 (σ1 , K )1{g >1}
Z ∞
K + (1 − g )y ˆ
fY 1 (y )BS0 σ2 ,
1{g ≤1} + 1{g >1} 1{y < K } dy
φ2 = g
T
g −1
g
0




+ gY02 N 
r−
σ12
2
T − ln
√
σ1 T
K
(g −1)Y01


 + (g − 1) BS0

K

σ1 ,
 1{g >1}
g −1
with g := g (c + z).
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
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University
and hedging
EDF R&D
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Position of the problem
Spot model
Pricing & hedging
Conclusion
Futures
Options
Spread options (do not panic)
Spread option with a 2 fuel model
The price π0 at time t = 0 of a call spread option with pay-off
H = (PT − h1 ST1 − K )+ is given by :
Z
π0 =
fC 1 −DT (z)fC 2 (c) φ1 (c, z)1{z>0} + φ2 (c, z)1{z≤0} dcdz,
R2
T
T
φ1 = (g − 1)BS0 (σ1 , K )1{g >1}
Z ∞
K + (1 − g )y ˆ
fY 1 (y )BS0 σ2 ,
1{g ≤1} + 1{g >1} 1{y < K } dy
φ2 = g
T
g −1
g
0




+ gY02 N 
r−
σ12
2
T − ln
√
σ1 T
K
(g −1)Y01


 + (g − 1) BS0

K

σ1 ,
 1{g >1}
g −1
with g := g (c + z).
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
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Centre
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Position of the problem
Spot model
Pricing & hedging
Conclusion
Futures
Options
Spread options (do not panic)
Spread option with a 2 fuel model
The price π0 at time t = 0 of a call spread option with pay-off
H = (PT − h1 ST1 − K )+ is given by :
Z
π0 =
fC 1 −DT (z)fC 2 (c) φ1 (c, z)1{z>0} + φ2 (c, z)1{z≤0} dcdz,
R2
T
T
φ1 = (g − 1)BS0 (σ1 , K )1{g >1}
Z ∞
K + (1 − g )y ˆ
fY 1 (y )BS0 σ2 ,
1{g ≤1} + 1{g >1} 1{y < K } dy
φ2 = g
T
g −1
g
0




+ gY02 N 
r−
σ12
2
T − ln
√
σ1 T
K
(g −1)Y01


 + (g − 1) BS0

K

σ1 ,
 1{g >1}
g −1
with g := g (c + z).
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research 33
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Position of the problem
Spot model
Pricing & hedging
Conclusion
Futures
Options
Spread options
semi-explicit formula : numerical integration
partial hedging with futures on fuels and electricity
applied on European dark spread call option with a period of
delivery of 1 hour
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research 34
Centre
/ 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Futures
Options
Spread options
semi-explicit formula : numerical integration
partial hedging with futures on fuels and electricity
applied on European dark spread call option with a period of
delivery of 1 hour
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research 34
Centre
/ 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Futures
Options
Spread options
semi-explicit formula : numerical integration
partial hedging with futures on fuels and electricity
applied on European dark spread call option with a period of
delivery of 1 hour
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research 34
Centre
/ 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Futures
Options
Spread options
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research 35
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Position of the problem
Spot model
Pricing & hedging
Conclusion
Futures
Options
Spread options
seasonality pattern
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
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Position of the problem
Spot model
Pricing & hedging
Conclusion
Futures
Options
Spread options
seasonality pattern
information on planned outages
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research 35
Centre
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Position of the problem
Spot model
Pricing & hedging
Conclusion
Conclusion
Conclusions
SRN electricity spot price model with a scarcity function
allows futures and derivatives pricing and hedging
nevertheless, only fuels dependancies can he hedged...
... and present work only treated hourly futures
Perspectives
comparison with ”real” quoted futures
comparison with calibration procedure
American options for investment problem
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research 36
Centre
/ 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Conclusion
Conclusions
SRN electricity spot price model with a scarcity function
allows futures and derivatives pricing and hedging
nevertheless, only fuels dependancies can he hedged...
... and present work only treated hourly futures
Perspectives
comparison with ”real” quoted futures
comparison with calibration procedure
American options for investment problem
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research 36
Centre
/ 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Conclusion
Conclusions
SRN electricity spot price model with a scarcity function
allows futures and derivatives pricing and hedging
nevertheless, only fuels dependancies can he hedged...
... and present work only treated hourly futures
Perspectives
comparison with ”real” quoted futures
comparison with calibration procedure
American options for investment problem
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research 36
Centre
/ 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Conclusion
Conclusions
SRN electricity spot price model with a scarcity function
allows futures and derivatives pricing and hedging
nevertheless, only fuels dependancies can he hedged...
... and present work only treated hourly futures
Perspectives
comparison with ”real” quoted futures
comparison with calibration procedure
American options for investment problem
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research 36
Centre
/ 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Conclusion
Conclusions
SRN electricity spot price model with a scarcity function
allows futures and derivatives pricing and hedging
nevertheless, only fuels dependancies can he hedged...
... and present work only treated hourly futures
Perspectives
comparison with ”real” quoted futures
comparison with calibration procedure
American options for investment problem
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research 36
Centre
/ 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Conclusion
Conclusions
SRN electricity spot price model with a scarcity function
allows futures and derivatives pricing and hedging
nevertheless, only fuels dependancies can he hedged...
... and present work only treated hourly futures
Perspectives
comparison with ”real” quoted futures
comparison with calibration procedure
American options for investment problem
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research 36
Centre
/ 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Conclusion
Conclusions
SRN electricity spot price model with a scarcity function
allows futures and derivatives pricing and hedging
nevertheless, only fuels dependancies can he hedged...
... and present work only treated hourly futures
Perspectives
comparison with ”real” quoted futures
comparison with calibration procedure
American options for investment problem
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research 36
Centre
/ 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Conclusion
Conclusions
SRN electricity spot price model with a scarcity function
allows futures and derivatives pricing and hedging
nevertheless, only fuels dependancies can he hedged...
... and present work only treated hourly futures
Perspectives
comparison with ”real” quoted futures
comparison with calibration procedure
American options for investment problem
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research 36
Centre
/ 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Conclusion
Conclusions
SRN electricity spot price model with a scarcity function
allows futures and derivatives pricing and hedging
nevertheless, only fuels dependancies can he hedged...
... and present work only treated hourly futures
Perspectives
comparison with ”real” quoted futures
comparison with calibration procedure
American options for investment problem
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research 36
Centre
/ 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
Conclusion
Conclusions
SRN electricity spot price model with a scarcity function
allows futures and derivatives pricing and hedging
nevertheless, only fuels dependancies can he hedged...
... and present work only treated hourly futures
Perspectives
comparison with ”real” quoted futures
comparison with calibration procedure
American options for investment problem
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
and hedging
EDF R&D
power- derivatives
FiME Research 36
Centre
/ 39
Position of the problem
Spot model
Pricing & hedging
Conclusion
References
A., Campi, Nguyen Huu & Touzi, Int. J. Theoretical &
Applied Finance, 2010
Barlow, Math. Finance, 2002
Benth & Koekebakker, J. of Energy Economics, 2007
Benth & Vos, Tech. Rep., Math. Dept. Oslo, 2009
Benth, Ekeland, Hauge & Nielsen, Applied Math. Finance,
2003
Burger, Klar, Müller & Schlindlmayr, Quantitative Finance,
2004
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
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Position of the problem
Spot model
Pricing & hedging
Conclusion
References
Cartea & Figueroa, Applied Math. Finance, 2005
Cartea & Villaplana, J. of Banking & Finance, 2008
Coulon & Howison, J. of Energy Markets, 2009
Deng, Tech. Rept.,California Energy Institute, 2000
Geman & Roncoroni, J. of Business, 2006
Kanamura & Ohashi, Energy Economics, 2007
Kolodnyi, J. of Engineering Mathematics, 2004
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
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Position of the problem
Spot model
Pricing & hedging
Conclusion
References
Lyle & Elliott, Energy Economics, 2009
Pham, Math. Meth. of Operations Research, 2000
Pirrong & Jermakyan, J. of Banking & Finance, 2008 1
Schweizer, Handbook Math. Finance, Cambridge Univ. Press,
2001
1. Olin Business School Tech. Rep. 2000
René Aı̈d, Luciano Campi, Nicolas Langrené Paris-Dauphine
A structural University
risk-neutral- Paris
modelDiderot
for pricing
University
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