A structural risk-neutral model for pricing and hedging power - - PowerPoint PPT Presentation

Position of the problem Spot model Pricing & hedging Risk premium vs error model Conclusion

A structural risk-neutral model for pricing and hedging power derivatives

FiME Research Centre Monthly Seminar - Paris Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 1 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model Conclusion

Agenda

1

Position of the problem Electricity prices modeling Related works

2

Spot model Design Estimation

3

Pricing & hedging Futures Options

4

Risk premium vs error model

5

Conclusion

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 2 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 3 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 3 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 3 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 3 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 3 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 3 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 3 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 3 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 3 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 4 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 4 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 4 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 4 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 4 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 4 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 4 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 4 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 4 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 4 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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 Options Pirrong & Jermakyan (00) × × Barlow (02) × Kanamura & Ohashi (07) × Cartea & Villaplana (08) × × Coulon & Howison (09) × × Lyle & Elliot (09) × × ×

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 5 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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 Options Pirrong & Jermakyan (00) × × Barlow (02) × Kanamura & Ohashi (07) × Cartea & Villaplana (08) × × Coulon & Howison (09) × × Lyle & Elliot (09) × × ×

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 5 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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 Options Pirrong & Jermakyan (00) × × Barlow (02) × Kanamura & Ohashi (07) × Cartea & Villaplana (08) × × Coulon & Howison (09) × × Lyle & Elliot (09) × × ×

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 5 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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 Options Pirrong & Jermakyan (00) × × Barlow (02) × Kanamura & Ohashi (07) × Cartea & Villaplana (08) × × Coulon & Howison (09) × × Lyle & Elliot (09) × × ×

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 5 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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 Options Pirrong & Jermakyan (00) × × Barlow (02) × Kanamura & Ohashi (07) × Cartea & Villaplana (08) × × Coulon & Howison (09) × × Lyle & Elliot (09) × × ×

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 5 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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 Options Pirrong & Jermakyan (00) × × Barlow (02) × Kanamura & Ohashi (07) × Cartea & Villaplana (08) × × Coulon & Howison (09) × × Lyle & Elliot (09) × × ×

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 5 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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 Options Pirrong & Jermakyan (00) × × Barlow (02) × Kanamura & Ohashi (07) × Cartea & Villaplana (08) × × Coulon & Howison (09) × × Lyle & Elliot (09) × × ×

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 5 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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 Options Pirrong & Jermakyan (00) × × Barlow (02) × Kanamura & Ohashi (07) × Cartea & Villaplana (08) × × Coulon & Howison (09) × × Lyle & Elliot (09) × × ×

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 5 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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 Options Pirrong & Jermakyan (00) × × Barlow (02) × Kanamura & Ohashi (07) × Cartea & Villaplana (08) × × Coulon & Howison (09) × × Lyle & Elliot (09) × × ×

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 5 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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 Options A., Campi, Nguyen & Touzi (09) × × improved SRN model × × ×

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 6 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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 Options A., Campi, Nguyen & Touzi (09) × × improved SRN model × × ×

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 6 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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 Options A., Campi, Nguyen & Touzi (09) × × improved SRN model × × ×

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 6 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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 Options A., Campi, Nguyen & Touzi (09) × × improved SRN model × × ×

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 6 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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 Options A., Campi, Nguyen & Touzi (09) × × improved SRN model × × ×

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 6 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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 Options A., Campi, Nguyen & Touzi (09) × × improved SRN model × × ×

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 6 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model Conclusion Design Estimation

Initial SRN Model

Variables n fuels, 1 ≤ i ≤ n Dt demand (MW) C i

t

capacities (en MW) Si

t

fuel prices hi heat rates (hiSi

t en e/MWh, ր en i)

Electricity price (e/MWh)

• Pt =

n

• i=1

hiSi

t1{ i−1

k=1 C k t ≤ Dt ≤ i k=1 C k t }

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 7 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model Conclusion Design Estimation

Initial SRN Model

Variables n fuels, 1 ≤ i ≤ n Dt demand (MW) C i

t

capacities (en MW) Si

t

fuel prices hi heat rates (hiSi

t en e/MWh, ր en i)

Electricity price (e/MWh)

• Pt =

n

• i=1

hiSi

t1{ i−1

k=1 C k t ≤ Dt ≤ i k=1 C k t }

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 7 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model Conclusion Design Estimation

Initial SRN Model

Variables n fuels, 1 ≤ i ≤ n Dt demand (MW) C i

t

capacities (en MW) Si

t

fuel prices hi heat rates (hiSi

t en e/MWh, ր en i)

Electricity price (e/MWh)

• Pt =

n

• i=1

hiSi

t1{ i−1

k=1 C k t ≤ Dt ≤ i k=1 C k t }

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 7 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model Conclusion Design Estimation

Initial SRN Model

Variables n fuels, 1 ≤ i ≤ n Dt demand (MW) C i

t

capacities (en MW) Si

t

fuel prices hi heat rates (hiSi

t en e/MWh, ր en i)

Electricity price (e/MWh)

• Pt =

n

• i=1

hiSi

t1{ i−1

k=1 C k t ≤ Dt ≤ i k=1 C k t }

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 7 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model Conclusion Design Estimation

Initial SRN Model

Variables n fuels, 1 ≤ i ≤ n Dt demand (MW) C i

t

capacities (en MW) Si

t

fuel prices hi heat rates (hiSi

t en e/MWh, ր en i)

Electricity price (e/MWh)

• Pt =

n

• i=1

hiSi

t1{ i−1

k=1 C k t ≤ Dt ≤ i k=1 C k t }

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 7 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model Conclusion Design Estimation

Initial SRN Model

Variables n fuels, 1 ≤ i ≤ n Dt demand (MW) C i

t

capacities (en MW) Si

t

fuel prices hi heat rates (hiSi

t en e/MWh, ր en i)

Electricity price (e/MWh)

• Pt =

n

• i=1

hiSi

t1{ i−1

k=1 C k t ≤ Dt ≤ i k=1 C k t }

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 7 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model Conclusion Design Estimation

Initial SRN Model

Variables n fuels, 1 ≤ i ≤ n Dt demand (MW) C i

t

capacities (en MW) Si

t

fuel prices hi heat rates (hiSi

t en e/MWh, ր en i)

Electricity price (e/MWh)

• Pt =

n

• i=1

hiSi

t1{ i−1

k=1 C k t ≤ Dt ≤ i k=1 C k t }

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 7 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model Conclusion Design Estimation

Initial SRN model

Pros Consistency between electricity prices and fuel prices Consistency between electricity prices and demand

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 8 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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

Non-convex technical constraints (may lead to negative prices)

2

Strategic behaviour (Horta¸ csu & Puller, RAND J. of Economics 2008)

3

Fixed cost recovery problem for peak-load generation plants

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 8 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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

Non-convex technical constraints (may lead to negative prices)

2

Strategic behaviour (Horta¸ csu & Puller, RAND J. of Economics 2008)

3

Fixed cost recovery problem for peak-load generation plants

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 8 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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

Non-convex technical constraints (may lead to negative prices)

2

Strategic behaviour (Horta¸ csu & Puller, RAND J. of Economics 2008)

3

Fixed cost recovery problem for peak-load generation plants

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 8 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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

Non-convex technical constraints (may lead to negative prices)

2

Strategic behaviour (Horta¸ csu & Puller, RAND J. of Economics 2008)

3

Fixed cost recovery problem for peak-load generation plants

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 8 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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

Non-convex technical constraints (may lead to negative prices)

2

Strategic behaviour (Horta¸ csu & Puller, RAND J. of Economics 2008)

3

Fixed cost recovery problem for peak-load generation plants

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 8 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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

Non-convex technical constraints (may lead to negative prices)

2

Strategic behaviour (Horta¸ csu & Puller, RAND J. of Economics 2008)

3

Fixed cost recovery problem for peak-load generation plants

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 8 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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

Non-convex technical constraints (may lead to negative prices)

2

Strategic behaviour (Horta¸ csu & Puller, RAND J. of Economics 2008)

3

Fixed cost recovery problem for peak-load generation plants

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 8 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model Conclusion Design Estimation

Initial SRN Model - illustration

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 9 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model Conclusion Design Estimation

Initial SRN Model - illustration

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 10 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model Conclusion Design Estimation

Initial SRN Model - illustration

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 11 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model Conclusion Design Estimation

Improved SRN model

Marginal fuel cost Pt := n

i=1 hiSi t1{ i−1

k=1 C k t ≤ Dt ≤ i k=1 C k t } Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 12 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model Conclusion Design Estimation

Improved SRN model

Marginal fuel cost Pt := n

i=1 hiSi t1{ i−1

k=1 C k t ≤ Dt ≤ i k=1 C k t }

Available capacity C t := n

k=1 C k t

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 12 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model Conclusion Design Estimation

Improved SRN model

Marginal fuel cost Pt := n

i=1 hiSi t1{ i−1

k=1 C k t ≤ Dt ≤ i k=1 C k t }

Available capacity C t := n

k=1 C k t

Price spikes occur when the electric system is under stress, i.e. C t − Dt is small

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 12 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model Conclusion Design Estimation

Improved SRN model

Marginal fuel cost Pt := n

i=1 hiSi t1{ i−1

k=1 C k t ≤ Dt ≤ i k=1 C k t }

Available capacity C t := n

k=1 C k t

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...

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 12 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model Conclusion Design Estimation

Improved SRN model

Marginal fuel cost Pt := n

i=1 hiSi t1{ i−1

k=1 C k t ≤ Dt ≤ i k=1 C k t }

Available capacity C t := n

k=1 C k t

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

• Pt as a (nonlinear) function of xt := C t − Dt

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 12 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model Conclusion Design Estimation

Improved SRN model - Estimation

Figure: PowerNext - 19th hours Nov, 13th 06 to April 30th 10

Observation Decreasing relation Difficult estimation Idea Quantile

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 13 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model Conclusion Design Estimation

Improved SRN model - Estimation

Figure: PowerNext - 19th hours Nov, 13th 06 to April 30th 10

Observation Decreasing relation Difficult estimation Idea Quantile

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 13 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model Conclusion Design Estimation

Improved SRN model - Estimation

Figure: PowerNext - 19th hours Nov, 13th 06 to April 30th 10

Observation Decreasing relation Difficult estimation Idea Quantile

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 13 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model Conclusion Design Estimation

Improved SRN model - Estimation

Figure: PowerNext - 19th hours Nov, 13th 06 to April 30th 10

Observation Decreasing relation Difficult estimation Idea Quantile

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 13 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model Conclusion Design Estimation

Improved SRN model - Estimation

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 14 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model Conclusion Design Estimation

Improved SRN model - Estimation

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 15 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model Conclusion Design Estimation

Improved SRN model - Estimation

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 16 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model Conclusion Design Estimation

Improved SRN model - Estimation

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 17 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model Conclusion Design Estimation

Improved SRN model - Estimation

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 18 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model Conclusion Design Estimation

Improved SRN model - Estimation

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 19 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model Conclusion Design Estimation

Improved SRN model - Estimation

Estimated relation : yt =

γ xν

t

Improved SRN model

Pt = g

• n
• k=1

C k

t − Dt

• ×
• n
• i=1

hiSi

t1{ i−1

k=1 C k t ≤ Dt ≤ i k=1 C k t }

• with scarcity function

g (x) := min γ xν , M

• 1{x>0} + M1{x0}

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 20 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model Conclusion Design Estimation

Improved SRN model - Back-testing

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 21 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model Conclusion Design Estimation

Improved SRN model - Back-testing

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 22 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model Conclusion Design Estimation

Improved SRN model - Backtesting

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 23 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model Conclusion Futures Options

Pricing & hedging

Pricing incomplete market need for a hedging criterion Super-replication, utility indifference or mean-variance

• ur choice : Local Risk Minimization

Local Risk Minimization (Pham (00), Schweizer (01)) valuation : expected discounted payoff under Q allows to decompose contingent claim between hedgeable part (fuels) and non-hedgeable part (demand, capacities) allows explicit formulas

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 24 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model Conclusion Futures Options

Pricing & hedging

Pricing incomplete market need for a hedging criterion Super-replication, utility indifference or mean-variance

• ur choice : Local Risk Minimization

Local Risk Minimization (Pham (00), Schweizer (01)) valuation : expected discounted payoff under Q allows to decompose contingent claim between hedgeable part (fuels) and non-hedgeable part (demand, capacities) allows explicit formulas

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 24 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model Conclusion Futures Options

Pricing & hedging

Pricing incomplete market need for a hedging criterion Super-replication, utility indifference or mean-variance

• ur choice : Local Risk Minimization

Local Risk Minimization (Pham (00), Schweizer (01)) valuation : expected discounted payoff under Q allows to decompose contingent claim between hedgeable part (fuels) and non-hedgeable part (demand, capacities) allows explicit formulas

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 24 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model Conclusion Futures Options

Pricing & hedging

Pricing incomplete market need for a hedging criterion Super-replication, utility indifference or mean-variance

• ur choice : Local Risk Minimization

Local Risk Minimization (Pham (00), Schweizer (01)) valuation : expected discounted payoff under Q allows to decompose contingent claim between hedgeable part (fuels) and non-hedgeable part (demand, capacities) allows explicit formulas

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 24 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model Conclusion Futures Options

Pricing & hedging

Pricing incomplete market need for a hedging criterion Super-replication, utility indifference or mean-variance

• ur choice : Local Risk Minimization

Local Risk Minimization (Pham (00), Schweizer (01)) valuation : expected discounted payoff under Q allows to decompose contingent claim between hedgeable part (fuels) and non-hedgeable part (demand, capacities) allows explicit formulas

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 24 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model Conclusion Futures Options

Pricing & hedging

Pricing incomplete market need for a hedging criterion Super-replication, utility indifference or mean-variance

• ur choice : Local Risk Minimization

Local Risk Minimization (Pham (00), Schweizer (01)) valuation : expected discounted payoff under Q allows to decompose contingent claim between hedgeable part (fuels) and non-hedgeable part (demand, capacities) allows explicit formulas

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 24 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model Conclusion Futures Options

Pricing & hedging

Pricing incomplete market need for a hedging criterion Super-replication, utility indifference or mean-variance

• ur choice : Local Risk Minimization

Local Risk Minimization (Pham (00), Schweizer (01)) valuation : expected discounted payoff under Q allows to decompose contingent claim between hedgeable part (fuels) and non-hedgeable part (demand, capacities) allows explicit formulas

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 24 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model Conclusion Futures Options

Pricing & hedging

Pricing incomplete market need for a hedging criterion Super-replication, utility indifference or mean-variance

• ur choice : Local Risk Minimization

Local Risk Minimization (Pham (00), Schweizer (01)) valuation : expected discounted payoff under Q allows to decompose contingent claim between hedgeable part (fuels) and non-hedgeable part (demand, capacities) allows explicit formulas

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 24 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model Conclusion Futures Options

Futures

Futures prices F e

t (T) = E

• Q

t

• e−r(T−t)PT
• F e

t (T) = n

• i=1

hiG T

i (t, Ct, Dt) F i t (T)

with :

G T

i (t,Ct,Dt) = Et

• g
• n
• k=1

C k

T − DT

• 1{

i−1

k=1C k T ≤ DT ≤ i k=1C k T}

• Ren´

e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 25 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model Conclusion Futures Options

Futures prices - hedging

Demand & capacities dDt = a (t, Dt) dt + b (t, Dt) dW D

t

dC i

t = αi

• t, C i

t

• dt + βi
• t, C i

t

• dW C,i

t

Futures price dynamics dF e

t (T) = n i=1 hi

• G T

i (t, Ct, Dt)dF i t (T) + F i t (T)dG T i (t, Ct, Dt)

• dG T

i (t, Ct, Dt) = n

• k=1

∂G T

i

∂ck (t, Ct, Dt)βk(t, C k

t )dW C,k t

+ ∂G T

i

∂z (t, Ct, Dt)b(t, Dt)dW D

t

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Position of the problem Spot model Pricing & hedging Risk premium vs error model Conclusion Futures Options

Futures prices - hedging

To go further, need to choose dynamics for demand and capacities

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 27 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model Conclusion Futures Options

Futures prices - hedging

To go further, need to choose dynamics for demand and capacities deterministic part for seasonality + Ornstein-Uhlenbeck

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 27 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model Conclusion Futures Options

Futures prices - hedging

To go further, need to choose dynamics for demand and capacities deterministic part for seasonality + Ornstein-Uhlenbeck G T

i

explicite as function of extended incomplete Goodwin-Staton integral :

• G (x, y; ν) =

∞

x

1 (y + z)ν e−z2dz

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 27 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model Conclusion Futures Options

Futures prices - hedging

To go further, need to choose dynamics for demand and capacities deterministic part for seasonality + Ornstein-Uhlenbeck G T

i

explicite as function of extended incomplete Goodwin-Staton integral :

• G (x, y; ν) =

∞

x

1 (y + z)ν e−z2dz ... for which efficient numerical algorithms are provided in A., Campi & Langren´ e (10).

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 27 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model Conclusion Futures Options

Futures prices - hedging : spot simulations

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 28 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model Conclusion Futures Options

Futures prices - hedging : spot simulations

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 29 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model Conclusion Futures Options

Futures prices - hedging : spot simulations

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 30 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model Conclusion Futures Options

Futures prices - hedging : spot simulations

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 31 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model Conclusion Futures Options

Futures prices - hedging : spot simulations

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 32 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 33 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 33 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 33 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 33 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model Conclusion Futures Options

Futures prices - hedging

Remarks

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 34 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model Conclusion Futures Options

Futures prices - hedging

Remarks Positive values are losses

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 34 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 34 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 34 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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 − h1S1

T − K)+ is given by :

π0 =

• R2 fC1

T −DT (z)fC2 T (c)

• φ1(c, z)1{z>0} + φ2(c, z)1{z≤0}
• dcdz,

φ1 = (g − 1)BS0(σ1, K)1{g>1} φ2 = g ∞ ˆ fY 1

T (y)BS0

• σ2, K + (1 − g)y

g 1{g≤1} + 1{g>1}1{y<

K g−1 }

• dy

+   gY 2

0 N

  

• r −

σ2

1

2

• T − ln
• K

(g−1)Y 1

• σ1

√ T    + (g − 1) BS0

• σ1,

K g − 1

• 

  1{g>1} with g := g(c + z).

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 35 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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 − h1S1

T − K)+ is given by :

π0 =

• R2 fC1

T −DT (z)fC2 T (c)

• φ1(c, z)1{z>0} + φ2(c, z)1{z≤0}
• dcdz,

φ1 = (g − 1)BS0(σ1, K)1{g>1} φ2 = g ∞ ˆ fY 1

T (y)BS0

• σ2, K + (1 − g)y

g 1{g≤1} + 1{g>1}1{y<

K g−1 }

• dy

+   gY 2

0 N

  

• r −

σ2

1

2

• T − ln
• K

(g−1)Y 1

• σ1

√ T    + (g − 1) BS0

• σ1,

K g − 1

• 

  1{g>1} with g := g(c + z).

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 35 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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 − h1S1

T − K)+ is given by :

π0 =

• R2 fC1

T −DT (z)fC2 T (c)

• φ1(c, z)1{z>0} + φ2(c, z)1{z≤0}
• dcdz,

φ1 = (g − 1)BS0(σ1, K)1{g>1} φ2 = g ∞ ˆ fY 1

T (y)BS0

• σ2, K + (1 − g)y

g 1{g≤1} + 1{g>1}1{y<

K g−1 }

• dy

+   gY 2

0 N

  

• r −

σ2

1

2

• T − ln
• K

(g−1)Y 1

• σ1

√ T    + (g − 1) BS0

• σ1,

K g − 1

• 

  1{g>1} with g := g(c + z).

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 35 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 36 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 36 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 36 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model Conclusion Futures Options

Spread options

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 37 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model Conclusion Futures Options

Spread options

seasonality pattern

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 37 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model Conclusion Futures Options

Spread options

seasonality pattern information on planned outages

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 37 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model Conclusion

Risk premium vs error model

Comparison between real quoted electricity futures and estimated price using the spot model Risk premium

F e

t (T, θ) −

F e

t (T, θ)

with estimated electricity futures price

• F e

t (T, θ) =

θ F e

t (T + θ) dθ

with :

F e

t (T) = n

• i=1

hiG T

i (t, Ct, Dt) F i t (T)

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 38 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model Conclusion

Risk premium vs error model

Estimation done on August, 28th, 2010 for baseload month electricity futures on PowerNext.

SEPT10 OCT10 NOV10 DEC10 JAN11 Quoted 49.5 55.69 62. 60.45 61.36 Estimation 52.2 53.1 55.2 55.5 53.4 Premium

• 2.7

2.59 6.8 4.95 7.96 Relative error (%)

• 5.5

4.7 11 8.2 13 Implied excess demand (GW)

• 0.4

0.3 0.63 0.46 0.79

Is there a way to make a distinction between risk premium and error model ?

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 39 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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 dynamics comparison with calibration procedure American options for investment problem

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 40 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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 dynamics comparison with calibration procedure American options for investment problem

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 40 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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 dynamics comparison with calibration procedure American options for investment problem

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 40 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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 dynamics comparison with calibration procedure American options for investment problem

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 40 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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 dynamics comparison with calibration procedure American options for investment problem

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 40 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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 dynamics comparison with calibration procedure American options for investment problem

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 40 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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 dynamics comparison with calibration procedure American options for investment problem

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 40 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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 dynamics comparison with calibration procedure American options for investment problem

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 40 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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 dynamics comparison with calibration procedure American options for investment problem

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 40 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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 dynamics comparison with calibration procedure American options for investment problem

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 40 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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, M¨ uller & Schlindlmayr, Quantitative Finance, 2004

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 41 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 42 / 43

Position of the problem Spot model Pricing & hedging Risk premium vs error model 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

Ren´ e A¨ ıd, Luciano Campi, Nicolas Langren´ e Paris-Dauphine University - Paris Diderot University EDF R&D - FiME Research Centre A structural risk-neutral model for pricing and hedging power derivatives 43 / 43