Model Risk for Energy Markets

Transcription

1 Seite 1 Model Risk for Energy Markets Rüdiger Kiesel, Karl Bannör, Anna Nazarova, Matthias Scherer R. Kiesel Centre of Mathematics for Applications, Oslo University Chair for Energy Trading, University of Duisburg-Essen

2 Seite 2 Motivation Spread Options Risk-Capturing Functionals General Setup AVaR α induced risk capturing functional Models and Empirics Commodity Models Data Estimation Procedures Results Correlation Risk Base Price Risk Jump Risks Risk Table Political Risk

3 Seite 3 Motivation Motivation Model risk has been recognized as one of the fundamental reasons for financial distress for banks and insurance companies. Recently, a number of authors addressed this issue: Schoutens et. al. (2004): A perfect calibration - now what? Cont (2006): Model uncertainty and its impact on the pricing of derivative instruments. Bannör, Scherer (2011): Quantifying the degree of parameter uncertainty in complex stochastic models Important questions: How sensitive is the value of a given derivative to the choice of the pricing model (parametric setting)? Can one quantify a provision for model risk (as for market and credit risk)?

4 Seite 4 Motivation Problem Setting Model risk has not been discussed in the context of energy markets (to our knowledge). A topical question is the need for reinvestment (replacement investments and building more capacity) in the power plant park. The financial streams of such an investment can be generated on the market for energy derivatives in terms of spread options. We use the Bannör, Scherer (2011) approach to discuss the model risk in such a valuation problem.

5 Seite 5 Spread Options Spread Options Market participants are exposed to the difference of commodity prices. Examples are the dark spread between power and coal (model for a coal-fired power plant) the spark spread between power and gas (model for a gas-fired power plant) In countries covered by the European Union Emissions Trading Scheme, utilities have to consider also the cost of carbon dioxide emission allowances. Emission trading has started in the EU in January 2005.

6 Seite 6 Spread Options Clean Spark Spread CSS τ = P τ h G τ c E E τ, (1) where P τ is the power price, G τ is the gas price, E τ is the carbon certificate price at maturity τ, h is the heat rate, c E emission conversion rate. The clean spark spread reflects the profit/loss of generating power from gas after taking into account gas and carbon allowance costs. A positive spread effectively means that it is profitable to generate electricity, while a negative spread means that generation would be a loss-making activity. Note that the clean spark spreads do not take into account additional generating charges beyond gas and carbon.

7 Seite 7 Spread Options Present Value of a Power Plant The operator acts on the spot market. The specific daily configuration of the power plant is not traded, so we use historical probabilities. We don t consider any further restrictions. The plant runs for another few years, so future values will be discounted.

8 Seite 8 Spread Options Spread Options to Manage Market Risk Spread options can be used by owners of corresponding plants to manage the market risk. Instead of spread trading with futures the owner of a power plant can directly purchase/sell a spread option. The payoff of a typical spread option is C (τ) spread = max(s 1(τ) S 2 (τ) K, 0) with S i the underlyings, K the strike.

9 Seite 9 Spread Options Valuation of Spread Options In the Black Scholes world there is an analytic formula for K = 0 (exchange option) due to Margrabe (1978). C spread (t) = (S 1 (t)φ(d 1 ) S 2 (t)φ(d 2 )) P spread (t) = (S 2 (t)φ( d 2 ) S 1 (t)φ( d 1 )) where d 1 = log(s 1(t)/S 2 (t))+σ 2 (τ t)/2, σ 2 (τ t) d 2 = d 1 σ 2 (τ t) and σ = σ1 2 2ρσ 1σ 2 + σ2 2 where ρ is the correlation between the two underlyings. For K 0 no easy analytic formula is available.

10 Seite 10 Spread Options Spread Option Value and Correlation The value of a spread option depends strongly on the correlation between the two underlyings. S 1 = S 2 = 100, τ = 3, r = 0.02, σ 1 = 0.6, σ 2 = 0.4. The higher the correlation between the two underlyings the lower is the volatility of the spread and hence the value of the spread option.

11 Seite 11 Spread Options Approximative Spread Option Valuation A very good reference is Carmona, Durrleman (2003), Siam Review 45 (4), There is also a survey by Krekel, de Kok, Korn, Man in Wilmott Magazine (2004) available.

12 Seite 12 Spread Options Clean Spread Option Valuation R.Carmona, M. Coulon, D. Schwarz (2012) present a valuation approach using a full structural model the difference between reduced form models (which we use) and the structural model is relatively small for high-efficiency gas plants, but reduced-form overprices for low-efficiency plants we also define the power price exogeneously An accurate approximation formula for the three asset case is also given in E.Alos, A.Eydeland and P.Laurence, Energy Risk, (2011).

13 Seite 13 Risk-Capturing Functionals Parameter Uncertainty To use models we need to specify the parameters estimation some estimator ˆϑ is used instead the true parameter ϑ bias and volatility of the estimator have to be considered calibration search for parameter that minimizes some pricing error condition, e.g. ϑ c = argmin ϑ set of derivatives model price(ϑ) market price parameters may not be uniquely identified Both approaches produce parameter uncertainty, may disregard information.

14 Seite 14 Risk-Capturing Functionals Parameter uncertainty set-up (Ω, F, F) filtered measurable space S = (S t ) basic instruments, contingent claim X = F(S) parametrized family of (martingale) measures (Q θ ) θ Θ on (Ω, F). parameter θ Θ, (risk neutral) value of contingent claim is θ E θ (X) := E Qθ (X).

15 Seite 15 Risk-Capturing Functionals Bannör-Scherer Approach distribution R for likelihood of parameter on parameter space Θ available convex risk measures gauge extent of parameter risk this allows to calculate parameter risk-induced spreads Advantages parameter s distribution is exploited risk aversion can be incorporated without being maximally conservative Cont s (2006, Math. Finance, 16(3), ) suggestion is an extreme points

16 Seite 16 Risk-Capturing Functionals Convex Risk Measures Let (Ω, F) be a measurable space and X L 0 (Ω) a vector space. Y X be a sub-vector space and π Y. ρ : X R (2) is a convex risk measure with π translation invariance iff ρ is monotone: ρ is convex: X Y = ρ(x) ρ(y ). λ [0, 1] : ρ(λx + (1 λ)y ) λρ(x) + (1 λ)ρ(y ). ρ is π-translation invariant: Y Y : ρ(x + Y ) = ρ(x) + π(y ).

17 Seite 17 Risk-Capturing Functionals Convex Risk Measures Properties ρ is coherent ρ(cx) = cρ(x), c > 0. ρ is normalized ρ(0) = 0. Let P be a probability measure on (Ω, F). ρ is P-law invariant P X = P Y implies ρ(x) = ρ(y ).

18 Seite 18 Risk-Capturing Functionals Risk Capturing Functionals We denote the space of all derivatives by D := L 1 (Q θ ) (3) θ Θ We call Γ : D R a risk-capturing functional with properties order preservation X Y Γ(X) Γ(Y ) diversification λ [0, 1] : Γ(λX + (1 λ)y ) λγ(x) + (1 λ)γ(y ). parameter independence consistency θ E θ (X) constant Γ(X) = E θ (X).

19 Seite 19 Risk-Capturing Functionals Model Risk Cont s Suggestion For X a derivative we associate with Γ(X) the ask price and with Γ( X) its bid price. Cont s suggestion Γ u (X) = sup E Q Q Q and Γ l (X) = Γ u ( X) = inf Q Q E Q. This approach produces typically a wide bid-ask spread.

20 Seite 20 Risk-Capturing Functionals Construction of Risk Capturing Functionals R a probability measure on Θ Let A L 0 (R) be a vector space of measurable functions containing the constants { D A := X } L 1 (Q θ ) : θ E θ (X) A (4) θ Θ ρ : A R be convex risk measure (normalized, law-invariant) Define the parameter risk capturing function Γ : D A R, Γ(X) = ρ (θ E θ (X)) (5)

21 Seite 21 Risk-Capturing Functionals Parameter Risk-Capturing Valuation Model: complex financial market Parameter space Θ Discounted derivative payout X Derivative price E θ [X] Probability measure R on Θ Pricing function θ E θ [X] Derivative price distribution induced by R and θ E θ [X] Risk measure ρ Ask price: Г(X)= ρ(θ E θ [X]) Bid price: -Г(-X) Quantifies parameter risk of derivative price

22 Seite 22 Risk-Capturing Functionals Definition AVaR general probability space (Ω, F, P), β (0, 1], X L 1 (P), then VaR β (X) = q X P (1 β). the average value at risk at level α (0, 1] is AVaR α (X) = 1 α α 0 VaR β (X)dβ. AVaR α is a convex risk measure (coherent and law-invariant).

23 Seite 23 Risk-Capturing Functionals Definition AVaR risk capturing functional Assume a parametrized family of (martingale) measures Q Θ = (Q θ ) θ Θ. Let R be a distribution on Θ. Consider the L 1 (R) admissible functionals, so AVaR α : L 1 (R) R. Define the AVaR α risk-capturing functional R AVaR α : C L1 (R) R as R AVaR α (X) := AVaR α (θ E θ (X)).

24 Seite 24 Risk-Capturing Functionals Convergence Property of AVaR Assume R N R 0, (N ) weakly on Q Θ ; ρ N a sequence of convex risk measures with ρ N is R N invariant; A sequence Γ N with Γ N = ρ N (Q Θ E θ (X)) has the convergence property (CP) if and only if lim Γ N(X) = Γ 0 (X) = ρ 0 (Q Θ E θ (X)) X C A. N AVaR -induced risk-capturing functionals fulfill (CP) for Θ compact.

25 Seite 25 Risk-Capturing Functionals Using asymptotic distributions (CP) allows us, if the parameter distribution R is complicated to calculate or even unknown, to use a parameter distribution R which is close to the original distribution R (in the sense of weak convergence, like, e.g., some asymptotic distribution) and calculate the risk-captured price with the parameter distribution R instead. In particular, if the distribution R is propagated from an estimator ˆθ N and the asymptotic distribution of the estimator ˆθ N is known (let us, e.g., denote the asymptotic distribution by R ), we can use the distribution R instead, if the sample size N N is large enough.

26 Seite 26 Risk-Capturing Functionals Calculating AVaR Assume (θ N ) N N is an asymptotically normal sequence of estimators for the true parameter θ 0 Θ R m with positive definite covariance matrix Σ, so N (θn θ 0 ) N m (0, Σ). If θ E θ (X) is continuously differentiable and E θ0 0, then ( N EθN (X) E θ0 (X) ) N (0, ( ) ) E θ0 Σ Eθ0 For θ N AVaR α (X) we calculate the AVaR as for a normally distributed variable θ N AVaR α (X) E θ0 (X) + ϕ ( Φ 1 (α) ) α ( Eθ0) Σ Eθ0, N

27 Seite 27 Models and Empirics Emission Certificates We model the emission price as a geometric Brownian motion de t = α E E t dt + σ E E t dw E t, (6)

28 Seite 28 Models and Empirics Gas Price We model the gas price as a mean-reverting process G t = e g(t)+z t, dz t = α G Z t dt + σ G dw G t, (7) α G is the speed of mean-reversion for gas prices.

29 Seite 29 Models and Empirics Power Price We model the power price as a sum of two mean-reverting processes P t = e f (t)+x t +Y t, dx t = α P X t dt + σ P dwt P, dy t = β Y t dt + J t dn t, (8) α P and β are speeds of mean-reversion for the smooth and the jump component of power prices. N is a Poisson process with intensity λ. J t are independent identically distributed (i.i.d) random variables representing the jump size.

30 Seite 30 Models and Empirics Seasonal components g(t) and f (t) are seasonal trend components for gas and power, respectively, defined as f (t) = a 1 + a 2 t + a 3 cos(a 5 + 2πt) + a 4 cos(a 6 + 4πt), g(t) = b 1 + b 2 t + b 3 cos(b 5 + 2πt) + b 4 cos(b 6 + 4πt), (9) where a 1 and b 1 may be viewed as production expenses, a 2 and b 2 are the slopes of increase in these costs. The rest of the parameters are responsible for two seasonal changes in summer and winter respectively.

31 Seite 31 Models and Empirics Dependence Structure In the current setting we also assume that W E, W G and N are mutually independent processes, but there is some correlation between W P and W G dw P t dw G t = ρ dt. (10)

32 Seite 32 Models and Empirics Parameter Uncertainty The total set of parameters includes {α E, σ E, g(t), α G, σ G, f (t), α P, β, σ P, λ, E[J], E[J 2 ], ρ}. Hence, the hybrid model we have chosen for modelling the clean spark spread is not parsimonious and allows for several degrees of freedom. Consequently, the risk of determining parameters in a wrong way is considerable.

33 Seite 33 Models and Empirics Data sources Phelix Day Base: It is the average price of the hours 1 to 24 for electricity traded on the spot market. It is calculated for all calendar days of the year as the simple average of the auction prices for the hours 1 to 24 in the market area Germany/Austria. (EUR/MWh), NCG: Delivery is possible at the virtual trading hub in the market areas of NetConnect Germany GmbH & Co KG. daily price (EUR/MWh), Emission certificate daily price: One EU emission allowance confers the right to emit one tonne of carbon dioxide or one tonne of carbon dioxide equivalent. (EUR/EUA). We cover the last three years:

34 Seite 34 Models and Empirics Price Paths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

35 Seite 35 Models and Empirics Clean Spark Spread, Spark Spread Spark Spread Value Date

36 Seite 36 Models and Empirics Emissions and Gas Apply a standard procedure to de-seasonalize gas (don t change notation). log E t and log G t are normally distributed. Thus, we can use standard Maximum Likelihood Methods.

37 Seite 37 Models and Empirics Power I The estimation procedure for the power price includes several steps: Estimation of the seasonal trend and deseasonalisation. With an iterative procedure we filter out returns with absolute values greater than three times the standard deviation of the returns of the series at the current iteration. The process is repeated until no further outliers can be found. As a result we obtain a standard deviation of the jumps, σ j, and a cumulative frequency of jumps, l. The latter is defined as the total number of filtered jumps divided by the annualised number of observations.

38 Seite 38 Models and Empirics Power II Once we have filtered the X t process, we can identify it as a first order autoregressive model in continuous time, i.e. so-called AR(1) process. Discretizing the process and estimating it by maximum likelihood method (MLE) yields the estimates.

39 Seite 39 Models and Empirics Estimation Results Estimation Step Product Estimates Method GBM Emissions α E = , σ E = MLE Seasonal trend Power a 1 = , a 2 = , a 3 = OLS a 4 = , a 5 = , a 6 = Seasonal trend Gas b 1 = , b 2 = , b 3 = OLS b 4 = , b 5 = , b 6 = Filtering Power 3 Std.Dev rule Base process Gas α G = , σ G = Multivariate Base process Power α P = , σ P = , ρ = normal regression Spike mean-reversion Power β = Spike intensity Power λ = Annual frequency Spike size (Laplace) Power µ s(median) = , σ s(scale) = MLE Spike size (normal) Power µ s(mean) = , σ s(variance) = MLE Heat rate Gas h = 2.5 Interest rate r = 3%

40 Seite 40 Results We will be capturing model risk in Jump size distribution; Correlation; Gas alone; Gas and power base signal; Gas, power and emissions (all the parameters, except of jump size).

41 Seite 41 Results General Procedure We reduce the problem here by considering the distributions of the single parameters separately (e.g. the correlation coefficient, the jump size distribution parameters). Hence, we do some kind of sensitivity analysis w.r.t. different parameters, disregarding the remaining parameter risk. Each parameter θ j is to be estimated by an estimator ˆθ j (X 1,..., X N ) under the real-world measure and we assume the other parameters θ 1,..., θ j 1, θ j+1, θ N to be known. We use plug-in estimators as the true values and figure out the asymptotic distribution of the estimators. We calculate the parameter risk-captured prices which are generated by the Average-Value-at-Risk (AVaR) w.r.t. different significance levels α (0, 1].

42 Seite 42 Results Spark Spread Analysis I In our investigation we will focus on the clean spark spread to model the value of virtual gas power plant. We will use spot price processes in order to assess the day-by-day risk position of such a plant. Thus, we will model the daily profit (or loss) of a power plant as V t = max{p t h G t c E E t, 0}, (11) where P t is the power price, G t is the gas price, E t is the carbon certificate price, h is the heat rate, c E emission conversion rate.

43 Seite 43 Results Spark Spread Analysis II We compute the spark spread value V t given in (11) for every day t for a time period of three years. Then, by fixing all the parameters except of one (e.g. correlation) and setting the shift value (e.g. 1%), we compute shifted up and down spark spread values, i.e. V up t and Vt down.

44 Seite 44 Results Power Plant Analysis I We compute the value of the power plant (VPP) by means of Monte Carlo simulations. For a fixed large number N and a fixed period T = 3 years we have VPP(t, T ) = 1 N N VPP i (t, T ), i=1 where VPP i (t, T ) = T e r(t s) V i (s). s=t

45 Seite 45 Results Power Plant Analysis II We also compute shifted both up and down power plant values, i.e. VPP up (t, T ) and VPP down (t, T ) (i.e. w.r.t. shifted spark spread values) and calculate the sensitivity svpp(θ 0 ) = VPPup (t, T ) VPP down (t, T ). 2 shift Finally, we compute the bid and ask prices, i.e. we use the closed formula for AVaR to get the risk-captured prices by subtracting and adding risk-adjustment value to VPP(t, T ) respectively. For a specified significance level α (0, 1) this risk-adjustment value is computed as ϕ(φ 1 (α)) α svpp(θ0 ) Σ svpp(θ 0 ). N

46 Seite 46 Results Correlation: the Estimator and its Distribution We have correlation between the base signal X t of power price and the log gas price logg t implied by the driving Brownian motions Let x i and y i, 1 = 1,... n the corresponding discrete observations, then we use Pearson s sample coefficient ρ (n) n n i=1 = x iy i ( n i=1 x ( n i) i=1 y ) i n i=1 x i 2 ( n i=1 x ) 2 n i i=1 y i 2 ( n i=1 y ). 2 i In our bivariate normal setting we can apply Fisher s transformation and have ( artanh ρ (n)) ( ) 1 N artanh(ρ 0 ), n 3

47 Seite 47 Results Parameter-risk implied bid-ask spread w.r.t. correlation parameter, Gaussian jumps Bid and ask prices accounting for the parameter risk in correlation with normal jumps AVaR AskPrice Relative bid ask spread width accounting for the parameter risk in correlation with normal jumps AVaR Bid Ask Delta AVaR 0.01 BidPrice AVaR AskPrice 0.1 AVaR BidPrice AVaR 0.1 Bid Ask Delta AVaR 0.5 Bid Ask Delta 3.35 AVaR 0.5 AskPrice AVaR 0.5 BidPrice 0.06 Price Value Bid Ask Delta Value Simulations Simulations

48 Seite 48 Results Parameter-risk implied bid-ask spread w.r.t. correlation parameter, Laplace jumps. 7.7 Bid and ask prices accounting for the parameter risk in correlation with Laplace jumps AVaR 0.01 AskPrice Relative bid ask spread width accounting for the parameter risk in correlation with Laplace jumps AVaR 0.01 Bid Ask Delta AVaR BidPrice 0.01 AVaR AskPrice 0.1 AVaR BidPrice 0.1 AVaR 0.5 AskPrice 0.03 AVaR 0.1 Bid Ask Delta AVaR Bid Ask Delta AVaR 0.5 BidPrice Price Value Bid Ask Delta Value Simulations Simulations

49 Seite 49 Results Parameter-risk implied bid-ask spread w.r.t. the gas price process, Gaussian jumps Bid and ask prices accounting for the parameter risk in gas signals with normal jumps AVaR 0.01 AskPrice Relative bid ask spread width accounting for the parameter risk in gas signals with normal jumps AVaR Bid Ask Delta AVaR BidPrice 0.01 AVaR AskPrice 0.1 AVaR BidPrice 0.1 AVaR 0.5 AskPrice AVaR 0.1 Bid Ask Delta AVaR 0.5 Bid Ask Delta 3.35 AVaR 0.5 BidPrice 0.05 Price Value Bid Ask Delta Value Simulations Simulations

50 Seite 50 Results Parameter-risk implied bid-ask spread w.r.t. the gas price process, Laplace jumps. 7.7 Bid and ask prices accounting for the parameter risk in gas signals with Laplace jumps AVaR 0.01 AskPrice 0.03 Relative bid ask spread width accounting for the parameter risk in gas signals with Laplace jumps AVaR Bid Ask Delta AVaR BidPrice 0.01 AVaR AskPrice 0.1 AVaR 0.1 Bid Ask Delta AVaR 0.5 Bid Ask Delta 7.5 AVaR BidPrice 0.1 AVaR 0.5 AskPrice AVaR 0.5 BidPrice Price Value Bid Ask Delta Value Simulations Simulations

51 Seite 51 Results Parameter-risk implied bid-ask spread w.r.t. the gas and power base processes, Gaussian jumps. Bid and ask prices accounting for the parameter risk in base power and gas signals with normal jumps 3.45 AVaR 0.01 AskPrice Relative bid ask spread width accounting for the parameter risk in base power and gas signals with normal jumps 0.07 AVaR 0.01 Bid Ask Delta AVaR 0.01 BidPrice AVaR 0.1 Bid Ask Delta 3.4 AVaR AskPrice 0.1 AVaR BidPrice AVaR 0.5 Bid Ask Delta AVaR 0.5 AskPrice 3.35 AVaR 0.5 BidPrice 0.05 Price Value Bid Ask Delta Value Simulations Simulations

52 Seite 52 Results Parameter-risk implied bid-ask spread w.r.t. the gas and power base processes, Laplace jumps. Price Value Bid and ask prices accounting for the parameter risk in base power and gas signals with Laplace jumps 7.7 AVaR 0.01 AskPrice AVaR BidPrice AVaR AskPrice AVaR BidPrice 0.1 AVaR 0.5 AskPrice 7.4 AVaR 0.5 BidPrice Bid Ask Delta Value 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 AVaR 0.1 Bid Ask Delta AVaR Bid Ask Delta Simulations Simulations

53 Seite 53 Results Parameter-risk implied bid-ask spread w.r.t. all the parameters, except of the Gaussian jump size Bid and ask prices accounting for the parameter risk in diffusion components with normal jumps AVaR AskPrice 0.01 AVaR 0.01 BidPrice AVaR AskPrice 0.1 AVaR BidPrice 0.1 AVaR 0.5 AskPrice AVaR 0.5 BidPrice Relative bid ask spread width accounting for the parameter risk in diffusion components with normal jumps 0.09 AVaR Bid Ask Delta 0.01 AVaR 0.1 Bid Ask Delta 0.08 AVaR 0.5 Bid Ask Delta 0.07 Price Value Bid Ask Delta Value Simulations Simulations

54 Seite 54 Results Parameter-risk implied bid-ask spread w.r.t. all the parameters, except of the Laplace jump size Bid and ask prices accounting for the parameter risk in diffusion components with Laplace jumps AVaR 0.01 AskPrice AVaR BidPrice 0.01 AVaR AskPrice 0.1 AVaR BidPrice 0.1 AVaR 0.5 AskPrice AVaR 0.5 BidPrice Relative bid ask spread width accounting for the parameter risk in diffusion components with Laplace jumps 0.04 AVaR 0.01 Bid Ask Delta AVaR 0.1 Bid Ask Delta AVaR Bid Ask Delta Price Value Bid Ask Delta Value Simulations Simulations

55 Seite 55 Results Parameter-risk implied bid-ask spread w.r.t. jump size distribution: Gaussian Bid and ask prices accounting for the parameter risk in jump distribution with normal jumps AVaR 0.01 AskPrice AVaR BidPrice 0.01 AVaR AskPrice 0.1 AVaR BidPrice 0.1 AVaR 0.5 AskPrice AVaR 0.5 BidPrice Relative bid ask spread width accounting for the parameter risk in jump distribution with normal jumps 1.3 AVaR Bid Ask Delta AVaR 0.1 Bid Ask Delta AVaR 0.5 Bid Ask Delta Price Value Bid Ask Delta Value Simulations Simulations

56 Seite 56 Results Parameter-risk implied bid-ask spread w.r.t. jump size distribution: Laplace Bid and ask prices accounting for the parameter risk in jump distribution with Laplace jumps AVaR 0.01 AskPrice AVaR BidPrice 0.01 AVaR AskPrice 0.1 AVaR BidPrice 0.1 AVaR 0.5 AskPrice AVaR 0.5 BidPrice Relative bid ask spread width accounting for the parameter risk in jump distribution with Laplace jumps 2.4 AVaR 0.01 Bid Ask Delta AVaR Bid Ask Delta AVaR Bid Ask Delta Price Value 8 6 Bid Ask Delta Value Simulations Simulations

57 Seite 57 Results Resulting values for the relative width of the bid-ask spread for various model risk sources. α 1 = 0.01, α 2 = 0.1, α 3 = 0.5. Gaussian Jumps size distribution Laplace Model Risk α 1 α 2 α 3 α 1 α 2 α 3 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%

58 Seite 58 Results Gas Power Plant

59 MW Seite 59 Results A day in august Solar Wind Konventionell Stunden

60 Seite 60 Results Wind, sun and electricity

61 Seite 61 Results RWE Response 14.August 2013 Decision on capacity measures Measure Plant MW 1 Fuel Location Date Decommissioning Long-term mothballing Amer Hard coal NL Q Moerdijk Gas NL Q Gersteinwerk F 355 Gas steam turbine DE Q Gersteinwerk G 355 Gas steam turbine DE Q Weisweiler H 270 Topping gas turbine 3 DE Q Weisweiler G 270 Topping gas turbine 3 DE Q mid-size units 85 Gas NL Q Summer mothballing Termination of 3 contracts Total Emsland B 360 Gas steam turbine DE Q Emsland C 360 Gas steam turbine DE Q Confidential 1,170 Hard coal DE Q Q ,265 MW 1 Net nominal capacity 2 Depending on the final decision on the Dutch Energieakkoord, with a decision expected by the end of August At a lignite plant RWE AG H Conference Call 14 August

62 Seite 62 Results Conclusions What we did We suggested a methodology to quantify model risk in power plant valuation approaches (spread options) We studied correlation and spike risk What we still need/want to do Perform more and better model analysis: estimation methods, approximation of quantities Improve simulation method: use analytic approaches as benchmarks Discuss multi-variate parameter model risk Study more realistic examples of power plants and valuation methodology Consider other energy derivatives

63 Seite 63 Results Energy & Finance Essen Energy & Finance Conference in Essen, October 9-11, 2013

64 Seite 64 Results Contact Chair for Energy Trading and Finance University Duisburg-Essen Universitätsstrße Essen, Germany phone +49 (0) fax +49 (0) web: Thank you for your attention...