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

Position of the problem Spot model Electricity prices modeling Pricing & hedging Related works Risk premium vs error model Conclusion 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 Electricity prices modeling Pricing & hedging Related works Risk premium vs error model Conclusion 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 Electricity prices modeling Pricing & hedging Related works Risk premium vs error model Conclusion 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 Design Pricing & hedging Estimation Risk premium vs error model Conclusion Initial SRN Model Variables fuels, 1 ≤ i ≤ n n D t demand (MW) C i capacities (en MW) t S i fuel prices t heat rates ( h i S i h i t en e /MWh, ր en i ) Electricity price ( e /MWh) n � � h i S i P t = t 1 { � i − 1 t ≤ D t ≤ � i t } k =1 C k k =1 C k i =1 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 Design Pricing & hedging Estimation Risk premium vs error model Conclusion 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 Design Pricing & hedging Estimation Risk premium vs error model Conclusion 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 Non-convex technical constraints (may lead to negative prices) 1 Strategic behaviour (Horta¸ csu & Puller, RAND J. of 2 Economics 2008) Fixed cost recovery problem for peak-load generation plants 3 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 Design Pricing & hedging Estimation Risk premium vs error model Conclusion 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 Design Pricing & hedging Estimation Risk premium vs error model Conclusion Improved SRN model P t := � n Marginal fuel cost � i =1 h i S i t 1 { � i − 1 t ≤ D t ≤ � i k =1 C k 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 Design Pricing & hedging Estimation Risk premium vs error model Conclusion Improved SRN model P t := � n Marginal fuel cost � i =1 h i S i t 1 { � i − 1 t ≤ D t ≤ � i k =1 C k 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 Design Pricing & hedging Estimation Risk premium vs error model Conclusion Improved SRN model P t := � n Marginal fuel cost � i =1 h i S i t 1 { � i − 1 t ≤ D t ≤ � i k =1 C k 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 − D t 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 Design Pricing & hedging Estimation Risk premium vs error model Conclusion Improved SRN model P t := � n Marginal fuel cost � i =1 h i S i t 1 { � i − 1 t ≤ D t ≤ � i k =1 C k 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 − D t 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 Design Pricing & hedging Estimation Risk premium vs error model Conclusion Improved SRN model P t := � n Marginal fuel cost � i =1 h i S i t 1 { � i − 1 t ≤ D t ≤ � i k =1 C k 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 − D t is small Corresponds to peak-load fixed cost problem recovery... y t := P t P t as a (nonlinear) function of x t := C t − D 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 Design Pricing & hedging Estimation Risk premium vs error model Conclusion Improved SRN model - Estimation Observation Decreasing relation Difficult estimation Idea Quantile Figure: PowerNext - 19th hours Nov, 13th 06 to April 30th 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 13 / 43

Position of the problem Spot model Design Pricing & hedging Estimation Risk premium vs error model Conclusion 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 Design Pricing & hedging Estimation Risk premium vs error model Conclusion Improved SRN model - Estimation γ Estimated relation : y t = x ν t Improved SRN model � � � � n n � � C k h i S i P t = g t − D t × t 1 { � i − 1 t ≤ D t ≤ � i t } k =1 C k k =1 C k k =1 i =1 with scarcity function � γ � g ( x ) := min x ν , M 1 { x > 0 } + M 1 { x � 0 } 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 Design Pricing & hedging Estimation Risk premium vs error model Conclusion 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 Design Pricing & hedging Estimation Risk premium vs error model Conclusion 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 Design Pricing & hedging Estimation Risk premium vs error model Conclusion 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 Futures Pricing & hedging Options Risk premium vs error model Conclusion 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)) 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 Futures Pricing & hedging Options Risk premium vs error model Conclusion Futures � � � Q Futures prices F e e − r ( T − t ) P T t ( T ) = E t n � F e h i G T i ( t , C t , D t ) F i t ( T ) = t ( T ) i =1 with : � � � � n � G T C k i ( t , C t , D t ) = E t g T − D T 1 { � i − 1 T ≤ D T ≤ � i T } k =1 C k k =1 C k k =1 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 Futures Pricing & hedging Options Risk premium vs error model Conclusion Futures prices - hedging Demand & capacities dD t = a ( t , D t ) dt + b ( t , D t ) dW D t � � � � dW C , i dC i t , C i t , C i t = α i dt + β i t t t Futures price dynamics t ( T ) = � n � � dF e G T i ( t , C t , D t ) dF i t ( T ) + F i t ( T ) dG T i ( t , C t , D t ) i =1 h i n � ∂ G T t ) dW C , k dG T i ( t , C t , D t ) β k ( t , C k i ( t , C t , D t ) = t ∂ c k k =1 + ∂ G T ∂ z ( t , C t , D t ) b ( t , D t ) dW D i 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 26 / 43

Position of the problem Spot model Futures Pricing & hedging Options Risk premium vs error model Conclusion 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 Futures Pricing & hedging Options Risk premium vs error model Conclusion 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 Futures Pricing & hedging Options Risk premium vs error model Conclusion Futures prices - hedging To go further, need to choose dynamics for demand and capacities deterministic part for seasonality + Ornstein-Uhlenbeck G T explicite as function of extended incomplete i Goodwin-Staton integral : � ∞ 1 ( y + z ) ν e − z 2 dz � G ( x , y ; ν ) = x 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 Futures Pricing & hedging Options Risk premium vs error model Conclusion Futures prices - hedging To go further, need to choose dynamics for demand and capacities deterministic part for seasonality + Ornstein-Uhlenbeck G T explicite as function of extended incomplete i Goodwin-Staton integral : � ∞ 1 ( y + z ) ν e − z 2 dz � G ( x , y ; ν ) = x ... 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 Futures Pricing & hedging Options Risk premium vs error model Conclusion 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 Futures Pricing & hedging Options Risk premium vs error model Conclusion 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 Futures Pricing & hedging Options Risk premium vs error model Conclusion 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 Futures Pricing & hedging Options Risk premium vs error model Conclusion 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 Futures Pricing & hedging Options Risk premium vs error model Conclusion 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 Futures Pricing & hedging Options Risk premium vs error model Conclusion 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

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 d, Luciano

Statistical Emulators for Pricing and Hedging Longevity Risk Products Jimmy Risk August 6, 2015

Arbitrage-Free Pricing of XVA Motivation Model Hedging Agostino Capponi Arbitrage Columbia

Semi-parametric Pricing and Hedging of Volatility and Hybrid Derivatives Peter Carr Based on

Basis risk in static versus dynamic longevity-risk hedging Luca Regis 1 joint work with: Clemente

Robust pricing and hedging via neural SDEs Lukasz Szpruch University of Edinburgh, The Alan

On robust pricing and hedging and the resulting notions of weak arbitrage Jan Ob l oj

Risks & Hedging - Summary Treasury division Risk Mngt Process ID & Measure the

Pricing and hedging of mortality-linked securities Helena Aro Instructor: Teemu Pennanen

Hedging credit index tranches Investigating versions of the standard model Christopher C. Finger

Lecture 3.1: Option Pricing The one and two period binomial option pricing models Models:

Asset Pricing Chapter VII. The Capital Asset Pricing Model: Another View About Risk June 20,

Pricing and hedging of derivatives in illiquid markets Pierre Patie RiskLab ETH Z urich

On Market-Making and Delta-Hedging 1 Market Makers 2 Market-Making and Bond-Pricing On

FX risk hedging at EADS 1 2 Reasons for EADS FX risk management policy Reasons for EADS FX risk

Asset Pricing Chapter IV. Measuring Risk and Risk Aversion June 20, 2006 Asset Pricing 4.1

Delta and Gamma hedging of mortality and interest-rate risk Elisa Luciano 1 , Luca Regis 2 , Elena

REGULARITY FOR SINGULAR RISK-NEUTRAL VALUATION EQUATIONS Kolmogorov Equations in Physics and

HEPATITIS AWARENESS MONTH WEBINAR 2020 04/27/2020 JANE PAN, EXECUTIVE DIRECTOR

Overview Four supportive steps to help young children identify their feelings, learn healthy

Experimental choices and field behavior on impatience, saving and smoking Matthias Sutter

Option Pricing with Semi-Markov Switching Lvy Process Financial Mathematics Lunch Talk Yi

Soft Actor-Critic Zikun Chen, Minghan Li Jan. 28, 2020 Soft Actor-Critic: Ofg-Policy Maximum

Kernel-based Reinforcement Learning in Robust Markov Decision Processes Shiau Hong Lim, Arnaud

Specification of Concretization and Symbolization Policies in Symbolic Execution S ebastien

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 d, Luciano

Statistical Emulators for Pricing and Hedging Longevity Risk Products Jimmy Risk August 6, 2015

Arbitrage-Free Pricing of XVA Motivation Model Hedging Agostino Capponi Arbitrage Columbia

Semi-parametric Pricing and Hedging of Volatility and Hybrid Derivatives Peter Carr Based on

Basis risk in static versus dynamic longevity-risk hedging Luca Regis 1 joint work with: Clemente

Robust pricing and hedging via neural SDEs Lukasz Szpruch University of Edinburgh, The Alan

On robust pricing and hedging and the resulting notions of weak arbitrage Jan Ob l oj

Risks &amp; Hedging - Summary Treasury division Risk Mngt Process ID &amp; Measure the

Pricing and hedging of mortality-linked securities Helena Aro Instructor: Teemu Pennanen

Hedging credit index tranches Investigating versions of the standard model Christopher C. Finger

Lecture 3.1: Option Pricing The one and two period binomial option pricing models Models:

Asset Pricing Chapter VII. The Capital Asset Pricing Model: Another View About Risk June 20,

Pricing and hedging of derivatives in illiquid markets Pierre Patie RiskLab ETH Z urich

On Market-Making and Delta-Hedging 1 Market Makers 2 Market-Making and Bond-Pricing On

FX risk hedging at EADS 1 2 Reasons for EADS FX risk management policy Reasons for EADS FX risk

Asset Pricing Chapter IV. Measuring Risk and Risk Aversion June 20, 2006 Asset Pricing 4.1

Delta and Gamma hedging of mortality and interest-rate risk Elisa Luciano 1 , Luca Regis 2 , Elena

REGULARITY FOR SINGULAR RISK-NEUTRAL VALUATION EQUATIONS Kolmogorov Equations in Physics and

HEPATITIS AWARENESS MONTH WEBINAR 2020 04/27/2020 JANE PAN, EXECUTIVE DIRECTOR

Overview Four supportive steps to help young children identify their feelings, learn healthy

Experimental choices and field behavior on impatience, saving and smoking Matthias Sutter

Option Pricing with Semi-Markov Switching Lvy Process Financial Mathematics Lunch Talk Yi

Soft Actor-Critic Zikun Chen, Minghan Li Jan. 28, 2020 Soft Actor-Critic: Ofg-Policy Maximum

Kernel-based Reinforcement Learning in Robust Markov Decision Processes Shiau Hong Lim, Arnaud

Specification of Concretization and Symbolization Policies in Symbolic Execution S ebastien

Risks & Hedging - Summary Treasury division Risk Mngt Process ID & Measure the