Risk Neutralmodeltohedgingandpricinpowerderivatives Seminar
Transcript of Risk Neutralmodeltohedgingandpricinpowerderivatives Seminar
8/13/2019 Risk Neutralmodeltohedgingandpricinpowerderivatives Seminar
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Position of the problemSpot model
Pricing & hedgingConclusion
A structural risk-neutral modelfor pricing and hedging power derivatives
Rene Aıd, Luciano Campi, Nicolas LangreneUniversities Paris 13 & Paris 7
EDF R&D - FiME Research Centre
Rene A ıd, Luciano Campi, Nicolas Langrene Universities Paris 13 & Paris 7 EDF R&D - FiME Research C
A structural risk-neutral model for pricing and hedging power derivatives 1 / 45
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Position of the problemSpot model
Pricing & hedgingConclusion
Contents
1 Position of the problemElectricity prices modelingRelated works
2 Spot modelDesignEstimation
3 Pricing & hedgingDynamics of fuels etc
Local risk minimizationFuturesOptionsHedging with futures on electricity
4 Conclusion
Rene A ıd, Luciano Campi, Nicolas Langrene Universities Paris 13 & Paris 7 EDF R&D - FiME Research C
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Position of the problemSpot model
Pricing & hedgingConclusion
Electricity prices modelingRelated works
Looking for a power spot price model
Applicationspricing of derivatives on the spotasset valuation (strip of hourly fuel spread options)
hedgingenergy market risk management
Model requirements
realisticrobusttractableconsistent
Rene A ıd, Luciano Campi, Nicolas Langrene Universities Paris 13 & Paris 7 EDF R&D - FiME Research C
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Position of the problemSpot model
Pricing & hedgingConclusion
Electricity prices modelingRelated works
Two types of modeling
Modeling futures pricespros modeling the real available instrumentscons introduction of many parameters to reconstruct
hourly futures prices
Modeling spot prices1 Exogeneous
pros tractabilitycons correlation
2 Equilibriumpros correlationcons complexity
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Position of the problemSpot model
Pricing & hedgingConclusion
Electricity prices modelingRelated works
This talk
Objectivespricing and hedging power derivatives...... using an improved version of Aıd, C., Nguyen & Touzi
(09) Structural Risk-Neutral modelSpot Futures Options
Aıd, C., Nguyen & Touzi (09) × ×improved SRN model × × ×
Rene A ıd, Luciano Campi, Nicolas Langrene Universities Paris 13 & Paris 7 EDF R&D - FiME Research CA structural risk-neutral model for pricing and hedging power derivatives 6 / 45
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Position of the problemSpot model
Pricing & hedgingConclusion
DesignEstimation
Initial SRN Model
Variablesn fuels, 1 ≤ i ≤ n
D t demand (MW)C i
t capacities (en MW)
S i t fuel priceshi heat rates (hi S i t en e /MWh, en i )
Electricity price (e /MWh)
P t =n
i =1
hi S i t 1{ i − 1k =1 C k
t ≤ D t ≤i k =1 C k
t }
Rene A ıd, Luciano Campi, Nicolas Langrene Universities Paris 13 & Paris 7 EDF R&D - FiME Research CA structural risk-neutral model for pricing and hedging power derivatives 7 / 45
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Position of the problemSpot model
Pricing & hedgingConclusion
DesignEstimation
Initial SRN Model
Variablesn fuels, 1 ≤ i ≤ n
D t demand (MW)C i
t capacities (en MW)
S i t fuel priceshi heat rates (hi S i t en e /MWh, en i )
Electricity price (e /MWh)
P t =n
i =1
hi S i t 1{ i − 1k =1 C k
t ≤ D t ≤i k =1 C k
t }
Rene A ıd, Luciano Campi, Nicolas Langrene Universities Paris 13 & Paris 7 EDF R&D - FiME Research CA structural risk-neutral model for pricing and hedging power derivatives 7 / 45
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Position of the problemSpot model
Pricing & hedgingConclusion
DesignEstimation
Initial SRN Model
Variablesn fuels, 1 ≤ i ≤ n
D t demand (MW)C i
t capacities (en MW)
S i t fuel priceshi heat rates (hi S i t en e /MWh, en i )
Electricity price (e /MWh)
P t =n
i =1
hi S i t 1{ i − 1k =1 C k
t ≤ D t ≤i k =1 C k
t }
Rene A ıd, Luciano Campi, Nicolas Langrene Universities Paris 13 & Paris 7 EDF R&D - FiME Research CA structural risk-neutral model for pricing and hedging power derivatives 7 / 45
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Position of the problemSpot model
Pricing & hedgingConclusion
DesignEstimation
Initial SRN Model
Variablesn fuels, 1 ≤ i ≤ n
D t demand (MW)C i
t capacities (en MW)
S i t fuel priceshi heat rates (hi S i t en e /MWh, en i )
Electricity price (e /MWh)
P t =n
i =1
hi S i t 1{ i − 1k =1 C k
t ≤ D t ≤i k =1 C k
t }
Rene A ıd, Luciano Campi, Nicolas Langrene Universities Paris 13 & Paris 7 EDF R&D - FiME Research CA structural risk-neutral model for pricing and hedging power derivatives 7 / 45
P i i f h bl
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Position of the problemSpot model
Pricing & hedgingConclusion
DesignEstimation
Initial SRN Model
Variablesn fuels, 1 ≤ i ≤ n
D t demand (MW)C i
t capacities (en MW)
S i t fuel priceshi heat rates (hi S i t en e /MWh, en i )
Electricity price (e /MWh)
P t =n
i =1
hi S i t 1{ i − 1k =1 C k
t ≤ D t ≤i k =1 C k
t }
Rene A ıd, Luciano Campi, Nicolas Langrene Universities Paris 13 & Paris 7 EDF R&D - FiME Research CA structural risk-neutral model for pricing and hedging power derivatives 7 / 45
P iti f th bl
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Position of the problemSpot model
Pricing & hedgingConclusion
DesignEstimation
Initial SRN Model
Variablesn fuels, 1 ≤ i ≤ n
D t demand (MW)C i
t capacities (en MW)
S i t fuel priceshi heat rates (hi S i t en e /MWh, en i )
Electricity price (e /MWh)
P t =n
i =1
hi S i t 1{ i − 1k =1 C k
t ≤ D t ≤i k =1 C k
t }
Rene A ıd, Luciano Campi, Nicolas Langrene Universities Paris 13 & Paris 7 EDF R&D - FiME Research CA structural risk-neutral model for pricing and hedging power derivatives 7 / 45
Position of the problem
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Position of the problemSpot model
Pricing & hedgingConclusion
DesignEstimation
Initial SRN Model
Variablesn fuels, 1 ≤ i ≤ n
D t demand (MW)C i
t capacities (en MW)
S i t fuel priceshi heat rates (hi S i t en e /MWh, en i )
Electricity price (e /MWh)
P t =n
i =1
hi S i t 1{ i − 1k =1 C k
t ≤ D t ≤i k =1 C k
t }
Rene A ıd, Luciano Campi, Nicolas Langrene Universities Paris 13 & Paris 7 EDF R&D - FiME Research CA structural risk-neutral model for pricing and hedging power derivatives 7 / 45
Position of the problem
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Position of the problemSpot model
Pricing & hedgingConclusion
DesignEstimation
Initial SRN model
ProsConsistency between electricity prices and fuel prices
Consistency between electricity prices and demand
Rene A ıd, Luciano Campi, Nicolas Langrene Universities Paris 13 & Paris 7 EDF R&D - FiME Research CA structural risk-neutral model for pricing and hedging power derivatives 8 / 45
Position of the problem
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Position of the problemSpot model
Pricing & hedgingConclusion
DesignEstimation
Initial SRN model
ProsConsistency between electricity prices and fuel prices
Consistency between electricity prices and demand
ConsMarginal fuel cost is not the spot price
1 Non-convex technical constraints2 Strategic behaviour (Horta csu & Puller, RAND J. of
Economics 2008)3 Fixed cost recovery problem for peak-load generation plants
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Position of the problem
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Position of the problemSpot model
Pricing & hedgingConclusion
DesignEstimation
Initial SRN Model - illustration
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Position of the problem
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pSpot model
Pricing & hedgingConclusion
DesignEstimation
Initial SRN Model - illustration
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Position of the problem
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Spot modelPricing & hedging
Conclusion
DesignEstimation
Improved SRN model
Marginal fuel cost
P t := n
i =1 hi S i t 1{ i − 1k =1 C k
t ≤ D t ≤ i k =1 C k
t }
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Position of the problemS d l D i
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Spot modelPricing & hedging
Conclusion
DesignEstimation
Improved SRN model
Marginal fuel cost
P t := n
i =1 hi S i t 1{ i − 1k =1 C k
t ≤ D t ≤ i k =1 C k
t }
Available capacity C t := nk =1 C k t
Rene A ıd, Luciano Campi, Nicolas Langrene Universities Paris 13 & Paris 7 EDF R&D - FiME Research CA structural risk-neutral model for pricing and hedging power derivatives 12 / 45
Position of the problemS t d l D i
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Spot modelPricing & hedging
Conclusion
DesignEstimation
Improved SRN model
Marginal fuel cost
P t := n
i =1 hi S i t 1{ i − 1k =1 C k
t ≤ D t ≤ i k =1 C k
t }
Available capacity C t := nk =1 C k t
Price spikes occur when the electric system is under stress, i.e.C t − D t is small
Rene A ıd, Luciano Campi, Nicolas Langrene Universities Paris 13 & Paris 7 EDF R&D - FiME Research CA structural risk-neutral model for pricing and hedging power derivatives 12 / 45
Position of the problemSpot model Design
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Spot modelPricing & hedging
Conclusion
DesignEstimation
Improved SRN model
Marginal fuel cost
P t := n
i =1 hi S i t 1{ i − 1k =1 C k
t ≤ D t ≤ i k =1 C k
t }
Available capacity C t := nk =1 C k t
Price spikes occur when the electric system is under stress, i.e.C t − D t is small
y t := P t
P t as a (nonlinear) function of x t := C t − D t
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Position of the problemSpot model Design
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Spot modelPricing & hedging
Conclusion
DesignEstimation
Improved SRN model - Estimation
Figure: PowerNext - 19th hours
Nov, 13th 06 to April 30th 10
ObservationDecreasing relationDifficult estimation
IdeaQuantiles
Rene A ıd, Luciano Campi, Nicolas Langrene Universities Paris 13 & Paris 7 EDF R&D - FiME Research CA structural risk-neutral model for pricing and hedging power derivatives 13 / 45
Position of the problemSpot model Design
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Spot modelPricing & hedging
Conclusion
DesignEstimation
Improved SRN model - Estimation
Figure: PowerNext - 19th hours
Nov, 13th 06 to April 30th 10
ObservationDecreasing relationDifficult estimation
IdeaQuantiles
Rene A ıd, Luciano Campi, Nicolas Langrene Universities Paris 13 & Paris 7 EDF R&D - FiME Research CA structural risk-neutral model for pricing and hedging power derivatives 13 / 45
Position of the problemSpot model Design
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pPricing & hedging
Conclusion
gEstimation
Improved SRN model - Estimation
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Position of the problemSpot model Design
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pPricing & hedging
Conclusion
gEstimation
Improved SRN model - Estimation
Rene A ıd, Luciano Campi, Nicolas Langrene Universities Paris 13 & Paris 7 EDF R&D - FiME Research CA structural risk-neutral model for pricing and hedging power derivatives 15 / 45
Position of the problemSpot model Design
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Pricing & hedgingConclusion
Estimation
Improved SRN model - Estimation
Rene A ıd, Luciano Campi, Nicolas Langrene Universities Paris 13 & Paris 7 EDF R&D - FiME Research CA structural risk-neutral model for pricing and hedging power derivatives 16 / 45
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Position of the problemSpot model
P i i & h d iDesignE i i
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Pricing & hedgingConclusion
Estimation
Improved SRN model - Estimation
Rene A ıd, Luciano Campi, Nicolas Langrene Universities Paris 13 & Paris 7 EDF R&D - FiME Research CA structural risk-neutral model for pricing and hedging power derivatives 18 / 45
Position of the problemSpot model
Pricing & hedgingDesignEstimation
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Pricing & hedgingConclusion
Estimation
Improved SRN model - Estimation
Rene A ıd, Luciano Campi, Nicolas Langrene Universities Paris 13 & Paris 7 EDF R&D - FiME Research CA structural risk-neutral model for pricing and hedging power derivatives 19 / 45
Position of the problemSpot model
Pricing & hedgingDesignEstimation
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Pricing & hedgingConclusion
Estimation
Improved SRN model - Estimation
Estimated relation : y t = γ x νt
Improved SRN model
P t = g n
k =1
C k t − D t ×
n
i =1
hi S i t 1{ i − 1k =1 C k
t ≤ D t ≤ i k =1 C k
t }
with scarcity function
g (x ) := min γ
x ν , M 1{x > 0} + M 1{x 0}
Rene A ıd, Luciano Campi, Nicolas Langrene Universities Paris 13 & Paris 7 EDF R&D - FiME Research CA structural risk-neutral model for pricing and hedging power derivatives 20 / 45
Position of the problemSpot model
Pricing & hedgingDesignEstimation
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Pricing & hedgingConclusion
Estimation
Improved SRN model - Back-testing
Rene A ıd, Luciano Campi, Nicolas Langrene Universities Paris 13 & Paris 7 EDF R&D - FiME Research CA structural risk-neutral model for pricing and hedging power derivatives 21 / 45
Position of the problemSpot model
Pricing & hedgingDesignEstimation
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Pricing & hedgingConclusion
Estimation
Improved SRN model - Back-testing
Rene A ıd, Luciano Campi, Nicolas Langrene Universities Paris 13 & Paris 7 EDF R&D - FiME Research CA structural risk-neutral model for pricing and hedging power derivatives 22 / 45
Position of the problemSpot model
Pricing & hedgingDesignEstimation
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Pricing & hedgingConclusion
Estimation
Improved SRN model - Backtesting
Rene A ıd, Luciano Campi, Nicolas Langrene Universities Paris 13 & Paris 7 EDF R&D - FiME Research CA structural risk-neutral model for pricing and hedging power derivatives 23 / 45
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Position of the problemSpot model
Pricing & hedging
Dynamics of fuels etcLocal risk minimizationFuturesOptions
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Conclusion OptionsHedging with futures on electricity
Dynamics of fuels
Assume r constant, convenience yields and storage costs zero forsimplicity.
Fuelsn ≥ 1 fuels (as coal, gas, oil ...) whose cost hi S i to produce 1
MWh of electricity follows
dS i t = S i t (µ i t dt + σ i
t dW S , i t )
where W S , i are correlated BMs and coeff’s are chosen so that
h1S 1 < . . . < hn S n (model spreads Y i = hi +1 S i +1 − hi S i asindependent geometric BMs).
NA and completeness assumptionThere exists a unique risk-neutral probability Q ∼ P for S .
Rene A ıd, Luciano Campi, Nicolas Langrene Universities Paris 13 & Paris 7 EDF R&D - FiME Research CA structural risk-neutral model for pricing and hedging power derivatives 25 / 45
Position of the problemSpot model
Pricing & hedgingC l i
Dynamics of fuels etcLocal risk minimizationFuturesOptions
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Conclusion OptionsHedging with futures on electricity
Local risk minimization I
Roughly speaking, let X be a multidimensional (discounted) priceprocess
Introduced by Follmer-Schweizer (1991)Under regularity condition, any payoff H with maturity T
H = H 0 + T
0ξ H
t dX t + LH T
where H 0 ∈R and LH martingale orthogonal to X .
T 0 ξ dX is the hedgeable part, LH T the residual risk, H 0 + LH T
the cost of the strategyHow to compute H 0, ξ H , LH ? Easy when X is a martingale :use Kunita-Watanabe decomposition !
Rene A ıd, Luciano Campi, Nicolas Langrene Universities Paris 13 & Paris 7 EDF R&D - FiME Research CA structural risk-neutral model for pricing and hedging power derivatives 26 / 45
Position of the problemSpot model
Pricing & hedgingC l i
Dynamics of fuels etcLocal risk minimizationFuturesOptions
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Conclusion pHedging with futures on electricity
Local risk minimization I
Roughly speaking, let X be a multidimensional (discounted) priceprocess
Introduced by Follmer-Schweizer (1991)Under regularity condition, any payoff H with maturity T
H = H 0 + T
0ξ H
t dX t + LH T
where H 0 ∈R and LH martingale orthogonal to X .
T 0 ξ dX is the hedgeable part, LH T the residual risk, H 0 + LH T
the cost of the strategyHow to compute H 0, ξ H , LH ? Easy when X is a martingale :use Kunita-Watanabe decomposition !
Rene A ıd, Luciano Campi, Nicolas Langrene Universities Paris 13 & Paris 7 EDF R&D - FiME Research CA structural risk-neutral model for pricing and hedging power derivatives 26 / 45
Position of the problemSpot model
Pricing & hedgingConclusion
Dynamics of fuels etcLocal risk minimizationFuturesOptions
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Conclusion pHedging with futures on electricity
Local risk minimization I
Roughly speaking, let X be a multidimensional (discounted) priceprocess
Introduced by Follmer-Schweizer (1991)Under regularity condition, any payoff H with maturity T
H = H 0 + T
0ξ H
t dX t + LH T
where H 0 ∈R and LH martingale orthogonal to X .
T 0 ξ dX is the hedgeable part, LH T the residual risk, H 0 + LH T
the cost of the strategyHow to compute H 0, ξ H , LH ? Easy when X is a martingale :use Kunita-Watanabe decomposition !
Rene A ıd, Luciano Campi, Nicolas Langrene Universities Paris 13 & Paris 7 EDF R&D - FiME Research CA structural risk-neutral model for pricing and hedging power derivatives 26 / 45
Position of the problemSpot model
Pricing & hedgingConclusion
Dynamics of fuels etcLocal risk minimizationFuturesOptions
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Conclusion Hedging with futures on electricity
Local risk minimization I
Roughly speaking, let X be a multidimensional (discounted) priceprocess
Introduced by Follmer-Schweizer (1991)Under regularity condition, any payoff H with maturity T
H = H 0 + T
0ξ H
t dX t + LH T
where H 0 ∈R and LH martingale orthogonal to X .
T 0 ξ dX is the hedgeable part, LH T the residual risk, H 0 + LH T
the cost of the strategyHow to compute H 0, ξ H , LH ? Easy when X is a martingale :use Kunita-Watanabe decomposition !
Rene A ıd, Luciano Campi, Nicolas Langrene Universities Paris 13 & Paris 7 EDF R&D - FiME Research CA structural risk-neutral model for pricing and hedging power derivatives 26 / 45
Position of the problemSpot model
Pricing & hedgingConclusion
Dynamics of fuels etcLocal risk minimizationFuturesOptions
d h f l
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Conclusion Hedging with futures on electricity
Local risk minimization II
When X is not a martingale but not far from being so ...Follmer-Schweizer (1991) : there exists a risk-neutral
Q for X
s.t.
H = E [H ] + T
0 ξ H t dX t + LH
T
H 0 =
E [H ], ξ H =
ξ H , LH =
LH
Q is called minimal equivalent martingale measureHobson (2005) : in diffusion models with non-tradable assets,
E [H ] is an upper bound for U -indifference bid price.
Rene A ıd, Luciano Campi, Nicolas Langrene Universities Paris 13 & Paris 7 EDF R&D - FiME Research CA structural risk-neutral model for pricing and hedging power derivatives 27 / 45
Position of the problemSpot model
Pricing & hedgingConclusion
Dynamics of fuels etcLocal risk minimizationFuturesOptionsH d i i h f l i i
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Conclusion Hedging with futures on electricity
Local risk minimization II
When X is not a martingale but not far from being so ...Follmer-Schweizer (1991) : there exists a risk-neutral
Q for X
s.t.
H = E [H ] + T
0 ξ H t dX t + LH
T
H 0 =
E [H ], ξ H =
ξ H , LH =
LH
Q is called minimal equivalent martingale measureHobson (2005) : in diffusion models with non-tradable assets,
E [H ] is an upper bound for U -indifference bid price.
Rene A ıd, Luciano Campi, Nicolas Langrene Universities Paris 13 & Paris 7 EDF R&D - FiME Research CA structural risk-neutral model for pricing and hedging power derivatives 27 / 45
Position of the problemSpot model
Pricing & hedgingConclusion
Dynamics of fuels etcLocal risk minimizationFuturesOptionsHedging ith f t res on electricit
8/13/2019 Risk Neutralmodeltohedgingandpricinpowerderivatives Seminar
http://slidepdf.com/reader/full/risk-neutralmodeltohedgingandpricinpowerderivatives-seminar 45/68
Hedging with futures on electricity
Local risk minimization II
When X is not a martingale but not far from being so ...Follmer-Schweizer (1991) : there exists a risk-neutral
Q for X
s.t.
H = E [H ] + T
0 ξ H t dX t + LH
T
H 0 =
E [H ], ξ H =
ξ H , LH =
LH
Q is called minimal equivalent martingale measureHobson (2005) : in diffusion models with non-tradable assets,
E [H ] is an upper bound for U -indifference bid price.
Rene A ıd, Luciano Campi, Nicolas Langrene Universities Paris 13 & Paris 7 EDF R&D - FiME Research CA structural risk-neutral model for pricing and hedging power derivatives 27 / 45
Position of the problemSpot model
Pricing & hedgingConclusion
Dynamics of fuels etcLocal risk minimizationFuturesOptionsHedging with futures on electricity
8/13/2019 Risk Neutralmodeltohedgingandpricinpowerderivatives Seminar
http://slidepdf.com/reader/full/risk-neutralmodeltohedgingandpricinpowerderivatives-seminar 46/68
Hedging with futures on electricity
Local risk minimization II
When X is not a martingale but not far from being so ...Follmer-Schweizer (1991) : there exists a risk-neutral
Q for X
s.t.
H = E [H ] + T
0 ξ H t dX t + LH
T
H 0 =
E [H ], ξ H =
ξ H , LH =
LH
Q is called minimal equivalent martingale measureHobson (2005) : in diffusion models with non-tradable assets,
E [H ] is an upper bound for U -indifference bid price.
Rene A ıd, Luciano Campi, Nicolas Langrene Universities Paris 13 & Paris 7 EDF R&D - FiME Research CA structural risk-neutral model for pricing and hedging power derivatives 27 / 45
Position of the problemSpot modelPricing & hedging
Conclusion
Dynamics of fuels etcLocal risk minimizationFuturesOptionsHedging with futures on electricity
8/13/2019 Risk Neutralmodeltohedgingandpricinpowerderivatives Seminar
http://slidepdf.com/reader/full/risk-neutralmodeltohedgingandpricinpowerderivatives-seminar 47/68
Hedging with futures on electricity
Local risk minimization II
When X is not a martingale but not far from being so ...Follmer-Schweizer (1991) : there exists a risk-neutral
Q for X
s.t.
H = E [H ] + T
0 ξ H t dX t + LH
T
H 0 =
E [H ], ξ H =
ξ H , LH =
LH
Q is called minimal equivalent martingale measureHobson (2005) : in diffusion models with non-tradable assets,
E [H ] is an upper bound for U -indifference bid price.
Rene A ıd, Luciano Campi, Nicolas Langrene Universities Paris 13 & Paris 7 EDF R&D - FiME Research CA structural risk-neutral model for pricing and hedging power derivatives 27 / 45
8/13/2019 Risk Neutralmodeltohedgingandpricinpowerderivatives Seminar
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Position of the problemSpot modelPricing & hedging
Conclusion
Dynamics of fuels etcLocal risk minimizationFuturesOptionsHedging with futures on electricity
8/13/2019 Risk Neutralmodeltohedgingandpricinpowerderivatives Seminar
http://slidepdf.com/reader/full/risk-neutralmodeltohedgingandpricinpowerderivatives-seminar 49/68
Hedging with futures on electricity
Futures
Under our assumptions, we can prove the following
Futures prices F e t (T ) =
E t [P T ]
F e t (T ) =
n
i =1
hi G T i (t , C t , D t ) F i t (T )
with :
G T i (t , C t , D t ) = E t g
n
k =1
C k T − D T 1{ i − 1
k =1 C k T ≤ D T ≤ i
k =1 C k T }
Rene A ıd, Luciano Campi, Nicolas Langrene Universities Paris 13 & Paris 7 EDF R&D - FiME Research CA structural risk-neutral model for pricing and hedging power derivatives 29 / 45
Position of the problemSpot modelPricing & hedging
Conclusion
Dynamics of fuels etcLocal risk minimizationFuturesOptionsHedging with futures on electricity
8/13/2019 Risk Neutralmodeltohedgingandpricinpowerderivatives Seminar
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g g y
Futures prices - hedging
Futures price dynamicsdF e
t (T ) = ni =1 hi G T
i (t , C t , D t )dF i t (T ) + F i t (T )dG T i (t , C t , D t )
dG T i (t , C t , D t ) =n
k =1∂ G
T
i ∂ c k
(t , C t , D t )β k (t , C k t )dW C ,k t
+ ∂ G T
i
∂ z (t , C t , D t )b (t , D t )dW D
t
so thatdF e
t (T ) = θS t dW t + θC
t dW C t + θD
t dW D t
for adapted suitable processes θS , θC , θD , which are explicitlycomputable.
Rene A ıd, Luciano Campi, Nicolas Langrene Universities Paris 13 & Paris 7 EDF R&D - FiME Research CA structural risk-neutral model for pricing and hedging power derivatives 30 / 45
Position of the problemSpot modelPricing & hedging
Conclusion
Dynamics of fuels etcLocal risk minimizationFuturesOptionsHedging with futures on electricity
8/13/2019 Risk Neutralmodeltohedgingandpricinpowerderivatives Seminar
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g g y
Futures prices - hedging
To go further, need to choose more specic dynamics fordemand and capacitiesdeterministic part for seasonality + Ornstein-UhlenbeckG T
i explicite as function of extended incomplete Goodwin-Staton integral :
G (x , y ; ν ) =
∞
x
1(y + z )ν e − z 2 dz
... for which efficient numerical algorithms are provided in Aıd,C. & Langrene (10).
Rene A ıd, Luciano Campi, Nicolas Langrene Universities Paris 13 & Paris 7 EDF R&D - FiME Research CA structural risk-neutral model for pricing and hedging power derivatives 31 / 45
Position of the problemSpot modelPricing & hedging
Conclusion
Dynamics of fuels etcLocal risk minimizationFuturesOptionsHedging with futures on electricity
8/13/2019 Risk Neutralmodeltohedgingandpricinpowerderivatives Seminar
http://slidepdf.com/reader/full/risk-neutralmodeltohedgingandpricinpowerderivatives-seminar 52/68
Futures prices - hedging : spot simulations
Rene A ıd, Luciano Campi, Nicolas Langrene Universities Paris 13 & Paris 7 EDF R&D - FiME Research CA structural risk-neutral model for pricing and hedging power derivatives 32 / 45
Position of the problemSpot modelPricing & hedging
Conclusion
Dynamics of fuels etcLocal risk minimizationFuturesOptionsHedging with futures on electricity
8/13/2019 Risk Neutralmodeltohedgingandpricinpowerderivatives Seminar
http://slidepdf.com/reader/full/risk-neutralmodeltohedgingandpricinpowerderivatives-seminar 53/68
Futures prices - hedging : spot simulations
Rene A ıd, Luciano Campi, Nicolas Langrene Universities Paris 13 & Paris 7 EDF R&D - FiME Research CA structural risk-neutral model for pricing and hedging power derivatives 33 / 45
Position of the problemSpot modelPricing & hedging
Conclusion
Dynamics of fuels etcLocal risk minimizationFuturesOptionsHedging with futures on electricity
8/13/2019 Risk Neutralmodeltohedgingandpricinpowerderivatives Seminar
http://slidepdf.com/reader/full/risk-neutralmodeltohedgingandpricinpowerderivatives-seminar 54/68
Futures prices - hedging : spot simulations
Rene A ıd, Luciano Campi, Nicolas Langrene Universities Paris 13 & Paris 7 EDF R&D - FiME Research CA structural risk-neutral model for pricing and hedging power derivatives 34 / 45
Position of the problemSpot modelPricing & hedging
Conclusion
Dynamics of fuels etcLocal risk minimizationFuturesOptionsHedging with futures on electricity
8/13/2019 Risk Neutralmodeltohedgingandpricinpowerderivatives Seminar
http://slidepdf.com/reader/full/risk-neutralmodeltohedgingandpricinpowerderivatives-seminar 55/68
Futures prices - hedging : spot simulations
Rene A ıd, Luciano Campi, Nicolas Langrene Universities Paris 13 & Paris 7 EDF R&D - FiME Research CA structural risk-neutral model for pricing and hedging power derivatives 35 / 45
Position of the problemSpot modelPricing & hedging
Conclusion
Dynamics of fuels etcLocal risk minimizationFuturesOptionsHedging with futures on electricity
8/13/2019 Risk Neutralmodeltohedgingandpricinpowerderivatives Seminar
http://slidepdf.com/reader/full/risk-neutralmodeltohedgingandpricinpowerderivatives-seminar 56/68
Futures prices - hedging : spot simulations
Rene A ıd, Luciano Campi, Nicolas Langrene Universities Paris 13 & Paris 7 EDF R&D - FiME Research CA structural risk-neutral model for pricing and hedging power derivatives 36 / 45
Position of the problemSpot modelPricing & hedging
Conclusion
Dynamics of fuels etcLocal risk minimizationFuturesOptionsHedging with futures on electricity
8/13/2019 Risk Neutralmodeltohedgingandpricinpowerderivatives Seminar
http://slidepdf.com/reader/full/risk-neutralmodeltohedgingandpricinpowerderivatives-seminar 57/68
Futures prices - hedging
Numerical test
Hedging a 3-monthselectricity futures with adelivery period of 1 hour
with a daily rebalancedbasket of futures
contracts on fuels
Rene A ıd, Luciano Campi, Nicolas Langrene Universities Paris 13 & Paris 7 EDF R&D - FiME Research CA structural risk-neutral model for pricing and hedging power derivatives 37 / 45
Position of the problemSpot modelPricing & hedging
Conclusion
Dynamics of fuels etcLocal risk minimizationFuturesOptionsHedging with futures on electricity
8/13/2019 Risk Neutralmodeltohedgingandpricinpowerderivatives Seminar
http://slidepdf.com/reader/full/risk-neutralmodeltohedgingandpricinpowerderivatives-seminar 58/68
Futures prices - hedging
Numerical test
Hedging a 3-monthselectricity futures with adelivery period of 1 hour
with a daily rebalancedbasket of futures
contracts on fuels
Rene A ıd, Luciano Campi, Nicolas Langrene Universities Paris 13 & Paris 7 EDF R&D - FiME Research CA structural risk-neutral model for pricing and hedging power derivatives 37 / 45
Position of the problemSpot modelPricing & hedging
Conclusion
Dynamics of fuels etcLocal risk minimizationFuturesOptionsHedging with futures on electricity
8/13/2019 Risk Neutralmodeltohedgingandpricinpowerderivatives Seminar
http://slidepdf.com/reader/full/risk-neutralmodeltohedgingandpricinpowerderivatives-seminar 59/68
Futures prices - hedging
Remarks
Positive values arelosses
Far from maturity :perfect hedge ;electricity futures isequivalent to a
basket of fuelsClose to maturity :inefficient hedge
Rene A ıd, Luciano Campi, Nicolas Langrene Universities Paris 13 & Paris 7 EDF R&D - FiME Research CA structural risk-neutral model for pricing and hedging power derivatives 38 / 45
Position of the problemSpot modelPricing & hedging
Conclusion
Dynamics of fuels etcLocal risk minimizationFuturesOptionsHedging with futures on electricity
8/13/2019 Risk Neutralmodeltohedgingandpricinpowerderivatives Seminar
http://slidepdf.com/reader/full/risk-neutralmodeltohedgingandpricinpowerderivatives-seminar 60/68
Spread options
Spread option with a 2 fuel modelThe price π0 at time t = 0 of a call spread option with pay-off H = ( P T −h1S 1T −K )+ is given by :
π 0 = R 2f C 1T − D T
(z )f C 2T (c ) φ1(c , z )1{ z > 0} + φ2(c , z )1{ z ≤ 0} dcdz ,
φ1 = ( g −1)BS 0(σ1 , K )1{ g > 1}
φ2 = g ∞
0f Y 1T
(y )BS 0 σ 2, K + (1 −g )y
g 1{ g ≤ 1} + 1{ g > 1} 1{ y < K
g − 1 } dy
+ gY 20 N
r − σ 2
12 T −ln K
(g − 1)Y 10
σ 1√ T + ( g −1) BS 0 σ 1,
K g −1
1{ g > 1}
with g := g (c + z ).
Rene A ıd, Luciano Campi, Nicolas Langrene Universities Paris 13 & Paris 7 EDF R&D - FiME Research CA structural risk-neutral model for pricing and hedging power derivatives 39 / 45
Position of the problemSpot modelPricing & hedging
Conclusion
Dynamics of fuels etcLocal risk minimizationFuturesOptionsHedging with futures on electricity
8/13/2019 Risk Neutralmodeltohedgingandpricinpowerderivatives Seminar
http://slidepdf.com/reader/full/risk-neutralmodeltohedgingandpricinpowerderivatives-seminar 61/68
Spread options
semi-explicit formula : numerical integration
partial hedging with futures on fuels and electricity,semi-explicit formulae for partial hedging strategy (not onlyfor spread options)applied on European dark spread (i.e. energy - gas) call optionwith a period of delivery of 1 hour
Rene A ıd, Luciano Campi, Nicolas Langrene Universities Paris 13 & Paris 7 EDF R&D - FiME Research CA structural risk-neutral model for pricing and hedging power derivatives 40 / 45
Position of the problemSpot modelPricing & hedging
Conclusion
Dynamics of fuels etcLocal risk minimizationFuturesOptionsHedging with futures on electricity
8/13/2019 Risk Neutralmodeltohedgingandpricinpowerderivatives Seminar
http://slidepdf.com/reader/full/risk-neutralmodeltohedgingandpricinpowerderivatives-seminar 62/68
Hedging with futures on electricity
Consider H = ϕ(F e T (T
∗
), F T (T ∗
), C T , D T ) with T ∗
> T .By Markov, its Q -price in t is φ(t , F t , C t , D t ) with φ(t , x , c , z )regularH ’s decomposition into hedgeable part/residual risk is
H = E [H ] + T
0 ξ t dF t + T
0 ξ e t dF e t + LT
where
ξ e
t = 1
|| (θC t , θ
D t )||
2i
θC , i t β i ∂ c i φ + θD
t b ∂ z φ
ξ i t = ∂ y i φ + hi G T ∗
i
|| (θC t , θD
t )|| 2i
θC , i t β i ∂ c i φ + θD
t b ∂ z φ
LT can be computed explicitly as wellRene A ıd, Luciano Campi, Nicolas Langrene Universities Paris 13 & Paris 7 EDF R&D - FiME Research CA structural risk-neutral model for pricing and hedging power derivatives 41 / 45
8/13/2019 Risk Neutralmodeltohedgingandpricinpowerderivatives Seminar
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Position of the problemSpot modelPricing & hedging
Conclusion
C l i
8/13/2019 Risk Neutralmodeltohedgingandpricinpowerderivatives Seminar
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Conclusion
ConclusionsSRN electricity spot price model with a scarcity functionallows futures and derivatives pricing and hedgingnevertheless, only fuels dependent part can be hedged ...... unless we use energy future for partially hedging demandand capacities risk (see the paper Aıd-C.-Langrene)
Perspectives
comparison with ”real”quoted futures, calibration dynamicsutility-based pricing of futures, options ...optimal investment/production problem, optimal switching
Rene A ıd, Luciano Campi, Nicolas Langrene Universities Paris 13 & Paris 7 EDF R&D - FiME Research CA structural risk-neutral model for pricing and hedging power derivatives 42 / 45
Position of the problemSpot modelPricing & hedging
Conclusion
C l i
8/13/2019 Risk Neutralmodeltohedgingandpricinpowerderivatives Seminar
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Conclusion
ConclusionsSRN electricity spot price model with a scarcity functionallows futures and derivatives pricing and hedgingnevertheless, only fuels dependent part can be hedged ...... unless we use energy future for partially hedging demandand capacities risk (see the paper Aıd-C.-Langrene)
Perspectives
comparison with ”real”quoted futures, calibration dynamicsutility-based pricing of futures, options ...optimal investment/production problem, optimal switching
Rene A ıd, Luciano Campi, Nicolas Langrene Universities Paris 13 & Paris 7 EDF R&D - FiME Research CA structural risk-neutral model for pricing and hedging power derivatives 42 / 45
8/13/2019 Risk Neutralmodeltohedgingandpricinpowerderivatives Seminar
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Position of the problemSpot modelPricing & hedging
Conclusion
R f
8/13/2019 Risk Neutralmodeltohedgingandpricinpowerderivatives Seminar
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References
Cartea & Figueroa, Applied Math. Finance , 2005Cartea & Villaplana, J. of Banking & Finance , 2008
Coulon & Howison, J. of Energy Markets , 2009Deng, Tech. Rept.,California Energy Institute , 2000Geman & Roncoroni, J. of Business , 2006Kanamura & Ohashi, Energy Economics , 2007
Kolodnyi, J. of Engineering Mathematics , 2004
Rene A ıd, Luciano Campi, Nicolas Langrene Universities Paris 13 & Paris 7 EDF R&D - FiME Research CA structural risk-neutral model for pricing and hedging power derivatives 44 / 45
Position of the problemSpot modelPricing & hedging
Conclusion
References
8/13/2019 Risk Neutralmodeltohedgingandpricinpowerderivatives Seminar
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References
Lyle & Elliott, Energy Economics , 2009
Pham, Math. Meth. of Operations Research , 2000Pirrong & Jermakyan, J. of Banking & Finance , 2008 1
Schweizer, Handbook Math. Finance, Cambridge Univ. Press ,2001
1. Olin Business School Tech. Rep. 2000Rene A ıd, Luciano Campi, Nicolas Langrene Universities Paris 13 & Paris 7 EDF R&D - FiME Research CA structural risk-neutral model for pricing and hedging power derivatives 45 / 45