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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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



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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



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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

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 4 / 45

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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 × × ×


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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 }


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DesignEstimation

Initial SRN Model


D t demand (MW)C i




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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DesignEstimation

Initial SRN Model


D t demand (MW)C i




P t =n

i =1

hi S i t 1{ i − 1k =1 C k

t ≤ D t ≤i k =1 C k

t }


P i i f h bl

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DesignEstimation

Initial SRN Model


D t demand (MW)C i




P t =n

i =1

hi S i t 1{ i − 1k =1 C k

t ≤ D t ≤i k =1 C k

t }


P iti f th bl

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DesignEstimation

Initial SRN Model


D t demand (MW)C i




P t =n

i =1

hi S i t 1{ i − 1k =1 C k

t ≤ D t ≤i k =1 C k

t }


Position of the problem

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DesignEstimation

Initial SRN Model


D t demand (MW)C i




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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DesignEstimation

Initial SRN model

ProsConsistency between electricity prices and fuel prices

Consistency between electricity prices and demand



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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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DesignEstimation

Initial SRN Model - illustration


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pSpot model


DesignEstimation

Initial SRN Model - illustration



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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 }


Position of the problemS d l D i

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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


Position of the problemS t d l D i

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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 }


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


Position of the problemSpot model Design

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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 }


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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Conclusion

DesignEstimation

Improved SRN model - Estimation

Figure: PowerNext - 19th hours

Nov, 13th 06 to April 30th 10

ObservationDecreasing relationDifficult estimation

IdeaQuantiles



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Conclusion

DesignEstimation


Figure: PowerNext - 19th hours

Nov, 13th 06 to April 30th 10

ObservationDecreasing relationDifficult estimation

IdeaQuantiles



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pPricing & hedging

Conclusion

gEstimation




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Estimation



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P i i & h d iDesignE i i




Estimation




Pricing & hedgingDesignEstimation

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Estimation





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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}




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Estimation

Improved SRN model - Back-testing




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Estimation

Improved SRN model - Back-testing




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Estimation

Improved SRN model - Backtesting


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Pricing & hedging

Dynamics of fuels etcLocal risk minimizationFuturesOptions



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 .



Pricing & hedgingC l i


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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 !



Pricing & hedgingC l i


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Conclusion pHedging with futures on electricity




H = H 0 + T

0ξ H

t dX t + LH T








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Conclusion Hedging with futures on electricity




H = H 0 + T

0ξ H

t dX t + LH T








d h f l

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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.




Dynamics of fuels etcLocal risk minimizationFuturesOptionsH d i i h f l i i

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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






Dynamics of fuels etcLocal risk minimizationFuturesOptionsHedging ith f t res on electricit

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Hedging with futures on electricity



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






Dynamics of fuels etcLocal risk minimizationFuturesOptionsHedging with futures on electricity

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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




Position of the problemSpot modelPricing & hedging

Conclusion


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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




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Conclusion





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 }



Conclusion


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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.



Conclusion


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g g y


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).



Conclusion


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Futures prices - hedging : spot simulations



Conclusion


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Conclusion


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Numerical test

Hedging a 3-monthselectricity futures with adelivery period of 1 hour

with a daily rebalancedbasket of futures

contracts on fuels



Conclusion


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Remarks

Positive values arelosses

Far from maturity :perfect hedge ;electricity futures isequivalent to a

basket of fuelsClose to maturity :inefficient hedge



Conclusion


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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 ).



Conclusion


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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



Conclusion


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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

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Conclusion

C l i



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



Conclusion

C l i

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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


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Conclusion

R f



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



Conclusion

References

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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

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risk neutralmodeltohedgingandpricinpowerderivatives seminar

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