Model Risk and Power Plant Valuation - uni-due.de€¦ · Seite 3/30Model Risk and Power Plant Valuation j Motivation and Introduction Motivation and Introduction: Questions I How

Model Risk and Power Plant Valuation

Energy & Finance Conference 2013, Essen,9-11 October 2013

Anna NazarovaBased on a joint work with Karl Bannor, Rudiger Kiesel and Matthias Scherer | Chair for Energy Trading and Finance |University of Duisburg-Essen

Seite 2/30 Model Risk and Power Plant Valuation |

OutlookMotivation and Introduction

Theoretical Aspects

Spread Options and Power Plant Valuation

Empirical Investigation of the Model Risk

Questions and discussion

References

Appendix

Anna Nazarova | Chair for Energy Trading and Finance, University of Duisburg-Essen |

Seite 3/30 Model Risk and Power Plant Valuation | Motivation and Introduction

Motivation and Introduction: Questions

I How significant is the impact of the model’s choice on the valueof a given instrument?

I How to assess the value of the parameters’ uncertainty?

I What is the main driver of the model risk in the energy markets?


Seite 4/30 Model Risk and Power Plant Valuation | Motivation and Introduction

General Approach

I We consider the model risk inherent in the valuation procedure offossil power plants.

I We focus on a gas-fired power plant as flexible and low-carbonsource of electricity which is an important building block in termsof the switch to a low-carbon energy generation.

I We model the generated financial streams as the spread optionand investigate the reinvestment problem.

I To capture model risk we use a methodology recently establishedin a series of papers: [Cont, 2006, Bannor and Scherer, 2013].


Seite 5/30 Model Risk and Power Plant Valuation | Theoretical Aspects

Risk-Captured Price

Having

I a contingent claim X ,I a parameter space Θ,I a distribution R on the parameters,I a parameterised family of valuation measures (Qθ)θ∈Θ,I a law-invariant, normalised convex risk measure ρ

results in a risk-captured price of a contingent claim X by

Γ(X ) := ρ(θ 7→ Eθ[X ]).



Visualisation of the Steps of Parameter Risk-Capturing Valuation

Quantifies parameter risk of derivative price

Model: complex financial market

Discounted derivative payout X

Parameter space Θ Derivative price Eθ[X]

Probability measure R on Θ

Ask price: Г(X)= ρ(θ → Eθ[X])

Bid price: -Г(-X)

Risk measure ρ

Derivative price distribution

induced by R and θ → Eθ[X]

Pricing function θ → Eθ[X]



Risk-Capturing Functional: AVaR Example

I Define AVaR w.r.t. the significance level α ∈ (0,1) of somerandom variable X as

AVaRα(X ) =1α

∫ α

0qX (1− β)dβ,

I where qX (γ) is γ quantile of the random variable X .

I The AVaR measures the risk which may occur according to thepreviously specified model Qθ.

I When calculating the parameter risk-captured price of X beinginduced by the AVaR, risk-neutral prices (Eθ[X ])θ∈Θ w.r.t.different models (Qθ)θ∈Θ are compared and subsumed by theAVaR risk measure. Hence, the AVaR is used to quantify theparameter risk we are exposed to when pricing X .


Seite 8/30 Model Risk and Power Plant Valuation | Spread Options and Power Plant Valuation

Clean Spark Spread Option and Virtual Power Plant

I We model the daily profit (or loss) of the virtual power plantposition as

Vt = maxPt − hGt − ηEt ,0,

I Pt - is the power price;I Gt - is the gas price;I Et - is the carbon certificate price;I h - is the heat rate of the power plant;I η - CO2 emission rate of the power plant.


Seite 9/30 Model Risk and Power Plant Valuation | Spread Options and Power Plant Valuation

Energy Price Models

LetI (Ω,P,F ,Ft,t∈[0,T ]) be a complete filtered probability space;I carbon price

dEt = αE Et dt + σE Et dW Et ;

I gas priceGt = eg(t)+Zt ,

dZt = −αG Zt dt + σG dW Gt ;

I power pricePt = ef (t)+Xt +Yt ,

base signal: dXt = −αP Xt dt + σP dW Pt ,

spike signal: dYt = −β Yt dt + Jt dNt .I dependence structure

W E , W G and N are mutually independent processes,dW P

t dW Gt = ρdt .


Seite 10/30 Model Risk and Power Plant Valuation | Empirical Investigation of the Model Risk

Data

I Phelix day base: It is the average price of the hours 1 to 24 forelectricity traded on the spot market. It is calculated for allcalendar days of the year as the simple average of the auctionprices for the hours 1 to 24 in the market area Germany/Austria,EUR/MWh.

I NCG daily price: Delivery is possible at the virtual trading hub inthe market areas of NetConnect Germany GmbH & Co KG,EUR/MWh.

I Emission certificate daily price: One EU emission allowanceconfers the right to emit 1 tonne of carbon dioxide or 1 tonne ofcarbon dioxide equivalent, EUR/EUA.

I Observation period: 25.09.2009 - 08.06.2012.



Power, Gas, and Carbon Prices, 25.09.2009 - 08.06.2012

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Clean Spark Spread, 25.09.2009 - 08.06.2012

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29.05.20

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29.06.20

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29.07.20

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29.08.20

09

29.09.20

09

29.10.20

09

29.11.20

09

29.12.20

09

29.01.20

10

28.02.20

10

31.03.20

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30.04.20

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31.05.20

10

30.06.20

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31.07.20

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31.08.20

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30.09.20

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31.10.20

10

30.11.20

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31.12.20

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31.01.20

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28.02.20

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31.03.20

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30.04.20

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31.05.20

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30.06.20

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31.07.20

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31.08.20

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31.10.20

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31.01.20

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29.02.20

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30.04.20

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12

Spark Spread

Value

Date

Spark Spread



Estimating the Model Parameters

I Estimation of the seasonal trends and deseasonalisation ofpower and gas.

I Separation of the power base and spike signals.

I Estimation of the mean-reverting rates.

I Estimation of the power base signal Xt .

I Estimation of the spike signal Yt .

I Estimation of the correlation.

Following the above steps, we estimate the set of parameters withmainly an MLE approach

αE , σE ,g(t), αG, σG, f (t), αP , β, σP , λ, µs, σs, ρ.



General Procedure: Step 1

I After estimating all the parameters of our prices, we simulatethem for the future time period and compute for every day t thespark spread value Vt given as

Vt = maxPt − hGt − ηEt ,0.

I Then, by fixing all the parameters except for the chosen one andsetting the shift value ξ (e.g. 1%), we compute shifted up anddown spark spread values as

V upt (θ + ξ),

V downt (θ − ξ).




I We compute the value of the power plant (VPP) by means ofMonte Carlo simulations. For fixed large N and T = 3 years wehave

VPP(t ,T ) =1N

N∑i=1

VPPi (t ,T ),

VPPi (t ,T ) =

∫ T

te−r(s−t)Vi (s) ds.

I For the chosen shift ξ we also compute

VPPup(t ,T ; θ) = VPP(t ,T ; θ + ξ),

VPPdown(t ,T ; θ) = VPP(t ,T ; θ − ξ).




I We continue with sensitivity measuring of the VPP w.r.t. theparameter θ with the central finite difference [Glasserman, 2004]

∇θVPP(t ,T ) :=∂VPP∂θ

=VPPup(t ,T ; θ)− VPPdown(t ,T ; θ)

2ξ

I Finally, we compute the bid and ask prices by using aclosed-form approximation formula for the AVaR to get therisk-captured prices by subtracting and adding risk-adjustmentvalue to VPP(t ,T ) respectively. This risk-adjustment value iscomputed as

ϕ(Φ−1(1− α))

α

√(∇θVPP)′ · Σθ · ∇θVPP

N,

denoting by Σθ the asymptotic covariance matrix of the estimatorfor the parameter θ [McNeil et al., 2005].



Risk Values Results

I parameter risk in spike size: Laplace and Gaussian distributions;I parameter risk in correlation;I parameter risk in gas signal;I joint parameter risk in gas and base power signal;I joint parameter risk in gas, power and emissions (all processes

except of jump size parameter).



Resulting values for the relative width of the bid-ask spread forvarious model risk sources. α1 = 0.01 (the highest risk-aversion),α2 = 0.1, α3= 0.5

Jumps size distributionGaussian Laplace

α1 α2 α3 α1 α2 α3

Mod

elR

isk Jumps 111.9% 73.71% 33.51% 163.5% 107.7% 48.96%

Correlation 6.95% 4.58% 2.08% 3.29% 2.17% 0.99%Gas and power base 6.48% 4.27% 1.94% 3.07% 2.02% 0.92%

Gas 6.11% 4.03% 1.83% 2.89% 1.91% 0.87%Gas, power and carbon 8.21% 5.41% 2.46% 3.83% 2.52% 1.15%



Parameter-risk implied bid-ask spread w.r.t. jump sizedistribution: Gaussian

500 1000 1500 2000 2500 3000 3500 4000 4500 50001

1.5

2

2.5

3

3.5

4

4.5

5

5.5

Simulations

Pric

e V

alue

Bid and ask prices accounting for the parameter risk in jump distribution with normal jumps

AVaR

0.01AskPrice

AVaR0.01

BidPrice

AVaR0.1

AskPrice

AVaR0.1

BidPrice

AVaR0.5

AskPrice

AVaR0.5

BidPrice

500 1000 1500 2000 2500 3000 3500 4000 4500 5000

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

Simulations

Bid

−A

sk D

elta

Val

ue

Relative bid−ask spread width accounting for the parameter risk in jump distribution with normal jumps

AVaR

0.01 Bid−Ask Delta

AVaR0.1

Bid−Ask Delta

AVaR0.5

Bid−Ask Delta



Parameter-risk implied bid-ask spread w.r.t. jump sizedistribution: Laplace

500 1000 1500 2000 2500 3000 3500 4000 4500 5000−2

0

2

4

6

8

10

12

14

16

Simulations

Pric

e V

alue

Bid and ask prices accounting for the parameter risk in jump distribution with Laplace jumps

AVaR

0.01AskPrice

AVaR0.01

BidPrice

AVaR0.1

AskPrice

AVaR0.1

BidPrice

AVaR0.5

AskPrice

AVaR0.5

BidPrice

500 1000 1500 2000 2500 3000 3500 4000 4500 50000.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

Simulations

Bid

−A

sk D

elta

Val

ue

Relative bid−ask spread width accounting for the parameter risk in jump distribution with Laplace jumps

AVaR


AVaR0.1

Bid−Ask Delta

AVaR0.5

Bid−Ask Delta



Parameter-risk implied bid-ask spread w.r.t. correlationparameter, Gaussian jumps

500 1000 1500 2000 2500 3000 3500 4000 4500 50003.05

3.1

3.15

3.2

3.25

3.3

3.35

3.4

3.45

Simulations

Pric

e V

alue

Bid and ask prices accounting for the parameter risk in correlation with normal jumps

AVaR

0.01AskPrice

AVaR0.01

BidPrice

AVaR0.1

AskPrice

AVaR0.1

BidPrice

AVaR0.5

AskPrice

AVaR0.5

BidPrice

500 1000 1500 2000 2500 3000 3500 4000 4500 50000.02

0.03

0.04

0.05

0.06

0.07

0.08

Simulations

Bid

−A

sk D

elta

Val

ue

Relative bid−ask spread width accounting for the parameter risk in correlation with normal jumps

AVaR


AVaR0.1

Bid−Ask Delta

AVaR0.5

Bid−Ask Delta



Parameter-risk implied bid-ask spread w.r.t. correlationparameter, Laplace jumps

500 1000 1500 2000 2500 3000 3500 4000 4500 50006.7

6.8

6.9

7

7.1

7.2

7.3

7.4

7.5

7.6

7.7

Simulations

Pric

e V

alue

Bid and ask prices accounting for the parameter risk in correlation with Laplace jumps

AVaR

0.01AskPrice

AVaR0.01

BidPrice

AVaR0.1

AskPrice

AVaR0.1

BidPrice

AVaR0.5

AskPrice

AVaR0.5

BidPrice

500 1000 1500 2000 2500 3000 3500 4000 4500 50000.005

0.01

0.015

0.02

0.025

0.03

0.035

Simulations

Bid

−A

sk D

elta

Val

ue

Relative bid−ask spread width accounting for the parameter risk in correlation with Laplace jumps

AVaR


AVaR0.1

Bid−Ask Delta

AVaR0.5

Bid−Ask Delta


Seite 23/30 Model Risk and Power Plant Valuation | Questions and discussion

Conclusive Remarks

I We suggested a methodology to quantify model risk in powerplant valuation approaches (spread options).

I We studied various sources of risks and found out that thecorrelation and spike risks are dominating in the energy sector.

I We managed to estimate the lower boundary for the total modelrisk in terms of the chosen model.

I Future possible application in the energy markets could be ageneration of an hourly power forward curve and valuationprocedures for storages.


Seite 24/30 Model Risk and Power Plant Valuation | References

References

Bannor, K. and Scherer, M. (2013).Capturing parameter risk with convex risk measures.To appear in European Actuarial Journal.

Cont, R. (2006).Model uncertainty and its impact on the pricing of derivative instruments.Mathematical Finance, 16(3):519–547.

Glasserman, P. (2004).Monte Carlo methods in financial engineering, volume 53.Springer.

McNeil, A. J., Frey, R., and Embrechts, P. (2005).Quantitative risk management: concepts, techniques, and tools.Princeton university press.

Thank you for your attention!


Seite 25/30 Model Risk and Power Plant Valuation | Appendix

Parameter-risk implied bid-ask spread w.r.t. the gas and powerbase processes, Gaussian jumps

500 1000 1500 2000 2500 3000 3500 4000 4500 50003.1

3.15

3.2

3.25

3.3

3.35

3.4

3.45

Simulations

Pric

e V

alue

Bid and ask prices accounting for the parameter risk in base power and gas signals with normal jumps

AVaR

0.01AskPrice

AVaR0.01

BidPrice

AVaR0.1

AskPrice

AVaR0.1

BidPrice

AVaR0.5

AskPrice

AVaR0.5

BidPrice

500 1000 1500 2000 2500 3000 3500 4000 4500 50000.01

0.02

0.03

0.04

0.05

0.06

0.07

Simulations

Bid

−A

sk D

elta

Val

ue

Relative bid−ask spread width accounting for the parameter risk in base power and gas signals with normal jumps

AVaR


AVaR0.1

Bid−Ask Delta

AVaR0.5

Bid−Ask Delta



Parameter-risk implied bid-ask spread w.r.t. the gas and powerbase processes, Laplace jumps

500 1000 1500 2000 2500 3000 3500 4000 4500 50006.7

6.8

6.9

7

7.1

7.2

7.3

7.4

7.5

7.6

7.7

Simulations

Pric

e V

alue

Bid and ask prices accounting for the parameter risk in base power and gas signals with Laplace jumps

AVaR

0.01AskPrice

AVaR0.01

BidPrice

AVaR0.1

AskPrice

AVaR0.1

BidPrice

AVaR0.5

AskPrice

AVaR0.5

BidPrice

500 1000 1500 2000 2500 3000 3500 4000 4500 50000.005

0.01

0.015

0.02

0.025

0.03

0.035

Simulations

Bid

−A

sk D

elta

Val

ue

Relative bid−ask spread width accounting for the parameter risk in base power and gas signals with Laplace jumps

AVaR


AVaR0.1

Bid−Ask Delta

AVaR0.5

Bid−Ask Delta



Parameter-risk implied bid-ask spread w.r.t. the gas priceprocess, Gaussian jumps

500 1000 1500 2000 2500 3000 3500 4000 4500 50003.1

3.15

3.2

3.25

3.3

3.35

3.4

3.45

Simulations

Pric

e V

alue

Bid and ask prices accounting for the parameter risk in gas signals with normal jumps

AVaR

0.01AskPrice

AVaR0.01

BidPrice

AVaR0.1

AskPrice

AVaR0.1

BidPrice

AVaR0.5

AskPrice

AVaR0.5

BidPrice

500 1000 1500 2000 2500 3000 3500 4000 4500 50000.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0.055

0.06

0.065

Simulations

Bid

−A

sk D

elta

Val

ue

Relative bid−ask spread width accounting for the parameter risk in gas signals with normal jumps

AVaR


AVaR0.1

Bid−Ask Delta

AVaR0.5

Bid−Ask Delta



Parameter-risk implied bid-ask spread w.r.t. the gas priceprocess, Laplace jumps

500 1000 1500 2000 2500 3000 3500 4000 4500 50006.7

6.8

6.9

7

7.1

7.2

7.3

7.4

7.5

7.6

7.7

Simulations

Pric

e V

alue

Bid and ask prices accounting for the parameter risk in gas signals with Laplace jumps

AVaR

0.01AskPrice

AVaR0.01

BidPrice

AVaR0.1

AskPrice

AVaR0.1

BidPrice

AVaR0.5

AskPrice

AVaR0.5

BidPrice

500 1000 1500 2000 2500 3000 3500 4000 4500 50000.005

0.01

0.015

0.02

0.025

0.03

Simulations

Bid

−A

sk D

elta

Val

ue

Relative bid−ask spread width accounting for the parameter risk in gas signals with Laplace jumps

AVaR


AVaR0.1

Bid−Ask Delta

AVaR0.5

Bid−Ask Delta



Parameter-risk implied bid-ask spread w.r.t. all the parameters,except of the Gaussian jump size

500 1000 1500 2000 2500 3000 3500 4000 4500 50003.05

3.1

3.15

3.2

3.25

3.3

3.35

3.4

3.45

Simulations

Pric

e V

alue

Bid and ask prices accounting for the parameter risk in diffusion components with normal jumps

AVaR

0.01AskPrice

AVaR0.01

BidPrice

AVaR0.1

AskPrice

AVaR0.1

BidPrice

AVaR0.5

AskPrice

AVaR0.5

BidPrice

500 1000 1500 2000 2500 3000 3500 4000 4500 50000.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Simulations

Bid

−A

sk D

elta

Val

ue

Relative bid−ask spread width accounting for the parameter risk in diffusion components with normal jumps

AVaR


AVaR0.1

Bid−Ask Delta

AVaR0.5

Bid−Ask Delta



Parameter-risk implied bid-ask spread w.r.t. all the parameters,except of the Laplace jump size

500 1000 1500 2000 2500 3000 3500 4000 4500 50006.7

6.8

6.9

7

7.1

7.2

7.3

7.4

7.5

7.6

7.7

Simulations

Pric

e V

alue

Bid and ask prices accounting for the parameter risk in diffusion components with Laplace jumps

AVaR

0.01AskPrice

AVaR0.01

BidPrice

AVaR0.1

AskPrice

AVaR0.1

BidPrice

AVaR0.5

AskPrice

AVaR0.5

BidPrice

500 1000 1500 2000 2500 3000 3500 4000 4500 50000.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Simulations

Bid

−A

sk D

elta

Val

ue

Relative bid−ask spread width accounting for the parameter risk in diffusion components with Laplace jumps

AVaR


AVaR0.1

Bid−Ask Delta

AVaR0.5

Bid−Ask Delta


Model Risk and Power Plant Valuation - uni-due.de€¦ · Seite 3/30Model Risk and Power Plant Valuation j Motivation and Introduction Motivation and Introduction: Questions I How

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