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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
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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 |
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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?
Anna Nazarova | Chair for Energy Trading and Finance, University of Duisburg-Essen |
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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].
Anna Nazarova | Chair for Energy Trading and Finance, University of Duisburg-Essen |
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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 ]).
Anna Nazarova | Chair for Energy Trading and Finance, University of Duisburg-Essen |
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Seite 6/30 Model Risk and Power Plant Valuation | Theoretical Aspects
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]
Anna Nazarova | Chair for Energy Trading and Finance, University of Duisburg-Essen |
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Seite 7/30 Model Risk and Power Plant Valuation | Theoretical Aspects
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 .
Anna Nazarova | Chair for Energy Trading and Finance, University of Duisburg-Essen |
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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.
Anna Nazarova | Chair for Energy Trading and Finance, University of Duisburg-Essen |
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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 .
Anna Nazarova | Chair for Energy Trading and Finance, University of Duisburg-Essen |
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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.
Anna Nazarova | Chair for Energy Trading and Finance, University of Duisburg-Essen |
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Seite 11/30 Model Risk and Power Plant Valuation | Empirical Investigation of the Model Risk
Power, Gas, and Carbon Prices, 25.09.2009 - 08.06.2012
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Anna Nazarova | Chair for Energy Trading and Finance, University of Duisburg-Essen |
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Seite 12/30 Model Risk and Power Plant Valuation | Empirical Investigation of the Model Risk
Clean Spark Spread, 25.09.2009 - 08.06.2012
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Spark Spread
Value
Date
Spark Spread
Anna Nazarova | Chair for Energy Trading and Finance, University of Duisburg-Essen |
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Seite 13/30 Model Risk and Power Plant Valuation | Empirical Investigation of the Model Risk
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.
Anna Nazarova | Chair for Energy Trading and Finance, University of Duisburg-Essen |
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Seite 14/30 Model Risk and Power Plant Valuation | Empirical Investigation of the Model Risk
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).
Anna Nazarova | Chair for Energy Trading and Finance, University of Duisburg-Essen |
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Seite 15/30 Model Risk and Power Plant Valuation | Empirical Investigation of the Model Risk
General Procedure: Step 2
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).
Anna Nazarova | Chair for Energy Trading and Finance, University of Duisburg-Essen |
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Seite 16/30 Model Risk and Power Plant Valuation | Empirical Investigation of the Model Risk
General Procedure: Step 3
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].
Anna Nazarova | Chair for Energy Trading and Finance, University of Duisburg-Essen |
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Seite 17/30 Model Risk and Power Plant Valuation | Empirical Investigation of the Model Risk
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).
Anna Nazarova | Chair for Energy Trading and Finance, University of Duisburg-Essen |
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Seite 18/30 Model Risk and Power Plant Valuation | Empirical Investigation of the Model Risk
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%
Anna Nazarova | Chair for Energy Trading and Finance, University of Duisburg-Essen |
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Seite 19/30 Model Risk and Power Plant Valuation | Empirical Investigation of the Model Risk
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
Anna Nazarova | Chair for Energy Trading and Finance, University of Duisburg-Essen |
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Seite 20/30 Model Risk and Power Plant Valuation | Empirical Investigation of the Model Risk
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
0.01 Bid−Ask Delta
AVaR0.1
Bid−Ask Delta
AVaR0.5
Bid−Ask Delta
Anna Nazarova | Chair for Energy Trading and Finance, University of Duisburg-Essen |
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Seite 21/30 Model Risk and Power Plant Valuation | Empirical Investigation of the Model Risk
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
0.01 Bid−Ask Delta
AVaR0.1
Bid−Ask Delta
AVaR0.5
Bid−Ask Delta
Anna Nazarova | Chair for Energy Trading and Finance, University of Duisburg-Essen |
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Seite 22/30 Model Risk and Power Plant Valuation | Empirical Investigation of the Model Risk
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
0.01 Bid−Ask Delta
AVaR0.1
Bid−Ask Delta
AVaR0.5
Bid−Ask Delta
Anna Nazarova | Chair for Energy Trading and Finance, University of Duisburg-Essen |
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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.
Anna Nazarova | Chair for Energy Trading and Finance, University of Duisburg-Essen |
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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!
Anna Nazarova | Chair for Energy Trading and Finance, University of Duisburg-Essen |
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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
0.01 Bid−Ask Delta
AVaR0.1
Bid−Ask Delta
AVaR0.5
Bid−Ask Delta
Anna Nazarova | Chair for Energy Trading and Finance, University of Duisburg-Essen |
![Page 26: 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](https://reader033.vdocuments.us/reader033/viewer/2022042220/5ec5f09384ab9661de78685b/html5/thumbnails/26.jpg)
Seite 26/30 Model Risk and Power Plant Valuation | Appendix
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
0.01 Bid−Ask Delta
AVaR0.1
Bid−Ask Delta
AVaR0.5
Bid−Ask Delta
Anna Nazarova | Chair for Energy Trading and Finance, University of Duisburg-Essen |
![Page 27: 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](https://reader033.vdocuments.us/reader033/viewer/2022042220/5ec5f09384ab9661de78685b/html5/thumbnails/27.jpg)
Seite 27/30 Model Risk and Power Plant Valuation | Appendix
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
0.01 Bid−Ask Delta
AVaR0.1
Bid−Ask Delta
AVaR0.5
Bid−Ask Delta
Anna Nazarova | Chair for Energy Trading and Finance, University of Duisburg-Essen |
![Page 28: 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](https://reader033.vdocuments.us/reader033/viewer/2022042220/5ec5f09384ab9661de78685b/html5/thumbnails/28.jpg)
Seite 28/30 Model Risk and Power Plant Valuation | Appendix
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
0.01 Bid−Ask Delta
AVaR0.1
Bid−Ask Delta
AVaR0.5
Bid−Ask Delta
Anna Nazarova | Chair for Energy Trading and Finance, University of Duisburg-Essen |
![Page 29: 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](https://reader033.vdocuments.us/reader033/viewer/2022042220/5ec5f09384ab9661de78685b/html5/thumbnails/29.jpg)
Seite 29/30 Model Risk and Power Plant Valuation | Appendix
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
0.01 Bid−Ask Delta
AVaR0.1
Bid−Ask Delta
AVaR0.5
Bid−Ask Delta
Anna Nazarova | Chair for Energy Trading and Finance, University of Duisburg-Essen |
![Page 30: 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](https://reader033.vdocuments.us/reader033/viewer/2022042220/5ec5f09384ab9661de78685b/html5/thumbnails/30.jpg)
Seite 30/30 Model Risk and Power Plant Valuation | Appendix
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
0.01 Bid−Ask Delta
AVaR0.1
Bid−Ask Delta
AVaR0.5
Bid−Ask Delta
Anna Nazarova | Chair for Energy Trading and Finance, University of Duisburg-Essen |