Modelling electricity market data: the CARMA spot model, forward prices and the risk premium

Transcription

1 Modelling electricity market data: the CARMA spot model, forward prices and the risk premium Formatvorlage des Untertitelmasters Claudia Klüppelberg durch Klicken bearbeiten Technische Universität München Zentrum Mathematik Institute for Advanced Study Focus Group Risk Analysis and Stochastic Modelling March 2011

2 Finding a model for electricity spot prices Stylized facts of electricity spot prices Seasonal behaviour in yearly, weekly and daily cycles More or less stationary behaviour Non-Gaussianity Extreme spikes

3 Linear time series analysis X t = Λ t + Y ( t τ1 + 2πt Λ t = β 0 + β 1 cos 365 t = 1,..., n ) ( τ2 + 2πt + β 2 cos 7 ) + β 3 t Figure: Singapore data (January April 2007) (X t ) t=1,...,n for n = 831 data points with estimated seasonality function Λ t (in red)

4 Figure: Plot of the autocorrelation functions of the estimated residuals (ε t ) t=1,...,n and (ε 2 t ) t=1,...,n after fitting an ARMA(1,2) process.

5 Unconditional quantiles For q (0, 1) and t Z the unconditional q-quantile of X t is given by x q (t) := inf{x R : P(X t x) q}. Figure: Data (X t ) t=1,...,n with 95%- and 99% quantile

6 Conditional quantiles: the one-step predicted quantiles For q (0, 1), t Z and h N the conditional q-quantile of (X t ) t Z up to time t h is given by: x h q(t) := inf { x R : P ( X t x X s, s t h ) q }. Figure: Conditional quantiles for q = 0.95 and q = 0.99 for the original data. There are 57 exceedances for the 95%-quantile and 11 for the 99%-quantile. Thus both quantiles are slightly underestimated.

7 Figure: German EEX daily spot data together with the forwards.

8 CARMA Lévy models [Brockwell, since 1990] Let L be a Lévy-process. The CARMA(p, q) process is defined as stationary solution of the state space equations Y(t) = b X(t) (1) dx(t) = AX(t)dt + e p dl(t), (2) with X(t) = (X(t), X (1) (t),..., X (p 2) (t), X (p 1) (t)) b 0 b 1 b =., e p = b p 2 b p , A = a p a p 1 a p 2. a 1. a 1,..., a p, b 0,..., b p 1 are such that b q = 1 and b j = 0 for j q.

9 (1) and (2) are the state-space representation of the p-th order SDE a(d)y(t) = b(d)dl(t) t 0 (3) where D denotes differentiation with respect to t, and a(z) := z p + a 1 z p a p b(z) := b 0 + b 1 z + + b q z q. Eq. (3) is a natural continuous-time analogue of the linear difference equations, which define an ARMA process [Brockwell & Davis (1991)].

10 For L we take an α-stable Lévy process with characteristic function log E[e izl(t) ] = tφ L (z) for z R, where, γ α z α (1 iβ(sign z) tan ( ) πα 2 ) + iµz for α 1, φ L (z) = γ z (1 + iβ 2 π (sign z) log z ) + iµz for α = 1. α (0, 2) shape parameter γ > 0 scale parameter β [ 1, 1] skewness parameter µ location parameter. [García, Klüppelberg & Müller (2010)]

11 Solution of (2) is a p-dimensional Ornstein-Uhlenbeck process: X(t) = e A(t s) X(s) + t From (1) we find that Y is given by Y(t) = b e A(t s) X(s) + s t s e A(t u) e p dl(u), 0 s < t. b e A(t u) e p dl(u), 0 s < t. The stable integral is defined as in Samorodnitsky and Taqqu (1994).

12 Assumptions: [Brockwell & Lindner (2009)] (a) a( ) and b( ) have no common zeros (b) E [ log + L(1) ] < (c) eigenvalues of A are distinct and have strictly negative real parts Then Y and X have strictly stationary and causal versions. Spot price dynamics: S(t) = Λ(t) + Z(t) + Y(t) t 0 Λ trend/seasonality function: deterministic Z zero mean Lévy process: low frequency non-stationary dynamics Y stationary α-stable CARMA process: short term variations.

13 Forward price dynamics I From general arbitrage theory, the forward price at time t 0 for a contract maturing at time τ t is f (t, τ) = E Q [S(τ) F t ], where Q is a risk neutral probability measure, and we assume that S(τ) L 1 (Q). By non-storability spot cannot be traded: buy-and-hold hedging argument fails. Hence every Q P may be chosen as pricing measure. Choose Q = P, then S(τ) has estimate ˆα > 1, hence S(τ) L 1 (Q).

14 Forward price dynamics II Theorem Let S = Λ + Z + Y be the spot dynamics. Assume that Q P is such that Z is a Lévy process under Q. Assume that E Q [Z(1)] < and E Q [ L(1) c ] < for some c > 1. Then the forward price dynamics for 0 t τ is given by f (t, τ) =Λ(τ) + Z(t) + b e A(τ t) X(t) + (τ t)e Q [Z(1)] + b A 1( I e A(τ t)) e p E Q [L(1)].

15 Forward price dynamics III The forward price F(t, T 1, T 2 ) at time 0 t T 1 < T 2 for a contract with delivery period [T 1, T 2 ] is defined as [ 1 T2 ] F(t, T 1, T 2 ) = E Q S(τ)dτ 1 F t = T 2 T 1 T 2 T 1 T 1 T2 where we have assumed that settlement takes place at T 2. T 1 f (t, τ)dτ,

16 Forward price dynamics IV Theorem F(t, T 1, T 2 ) = = where 1 T 2 T 1 T 1 +Γ Q (t, T 1, T 2 ), T2 1 f (t, τ)dτ T 2 T 1 T 1 T2 Λ(τ)dτ + Z(t) + b A 1 ( ) e AT 2 e AT 1 e At X(t) T 2 T 1 Γ Q (t, T 1, T 2 ) = ( 1 2 (T 2 + T 1 ) t)e Q [Z(1)] b A 2 ( ) e AT 2 e AT 1 e At e p E Q [L(1)] + b A 1 e p E Q [L(1)]. T 2 T 1

17 Risk Premium Difference of forward price and predicted spot: [ 1 R pr (t, T 1, T 2 ) = F(t, T 1, T 2 ) E T 2 T 1 T2 For our model we obtain for given pricing measure Q R pr (t, T 1, T 2 ) = Γ Q (t, T 1, T 2 ) Γ(t, T 1, T 2 ) = b A 2 ( ) ( e AT 2 e AT 1 e At e p EQ [L(1)] E[L(1)] ) T 2 T 1 T 1 ] S(u) du F t +b A 1 e p ( EQ [L(1)] E[L(1)] ) + ( 1 2 (T 2 + T 1 ) t ) E Q [Z(1)].

18 Measure transformation Assumptions: We consider Q = Q L Q Z, such that: L, Z preserve independence under P and Q L, Z preserve Lévy property under Q. We Esscher transform Z: ν QZ (dx) = e θ Zx ν(dx), θ Z R. We temper L (Rosinski (2007)): ν QL (dx) = c +e θ Lx 1 (0, )(x) dx + c e θl x x 1+α Here, θ L < 0 and c, c + 0. Then x 1+α 1 (,0)(x) dx. E Q [L(1)] E[L(1)] = Γ(1 α) θ L α 1 (c + c ).

19 Fitting the model to German electricity data I Data: Spot and monthly delivery forwards (42 contracts) from 01-Jan-2002 to 27-Apr Model: S(t) = Λ(t) + Z(t) + Y(t) t 0 (1) Estimate deterministic trend/seasonality Λ and subtract from S. (2) Estimate Z and Y from deseasonalised spot and forward prices. Use that forward prices far from delivery have Z dynamic, only close to delivery large fluctuations Y occur by market inelasticity. In our model this manifests in lim T 1,T 2 b A 2 ( ) e AT 2 e AT 1 e At e p E Q [L(1)] = 0 T 2 T 1 lim T 1,T 2 b A 1 ( ) e AT 2 e AT 1 e At X(t) = 0 T 2 T 1

20 Fitting the model to German electricity data II Find optimal separation point w such that for large time to maturity u we can approximate these two terms by 0: Denote time to maturity by u := 1 2 (T 1 + T 2 ) t. For each threshold 1 w 200 (longest time to maturity in our data) and u w: Set F(u) := F(t, T 1, T 2 ) and approximate µ F (u) = E[F(u)] E[Z(t)] + b A 1 e p E Q [L(1)] + u E Q [Z(1)] =: C + ue Q [Z(1)] estimate C and E Q [Z(1)] by robust linear regression. Estimate Z(t) = Z (1 2 (T 1 + T 2 ) u ) := F(u) C u E Q [Z(1)]. A good model (marginals and acf) for Z is a NIG Lévy process.

21 Fitting the model to German electricity data II Model Y = S Λ Z as stable CARMA(2,1) process: estimate the coefficients a 1, a 2, b 0 (recall that b 1 = 1) and estimate stable parameters (α, γ, β, µ) of L. Estimate E Q [L(1)] = C (b A 1 e p ) 1. Recall the risk premium for fixed v = T 2 T 1 and u 1 2 (T 2 T 1 ) R pr (u, v) = Γ Q (t, T 1, T 2 ) Γ P (t, T 1, T 2 ) = 1 ( ) ( v b A 2 e 1 2 Av e 2 1 Av e Au e p EQ [L(1)] E[L(1)] ) +b A 1 e p ( EQ [L(1)] E[L(1)] ) + u E Q [Z(1)]. Estimate R pr (u, v) by the estimated CARMA parameters and the estimated Z.

22 Figure: The observed forward prices (red) with the estimated forward prices (black) for optimal threshold w.

23 Empirical risk premium Compare this model based estimated R pr (u, v) with the empirical version based on all available data U(u, v) := { t, T 1, T 2 : 1 2 (T 2 T 1 ) t = u, T 2 T 1 = v and F(t, T 1, T 2 ) exists }. The empirical risk premium is given by R pr (u, v) := 1 v b A 2( e 1 2 Av e 1 2 Av) e Au e p E[L(1)] b A 1 e p E[L(1)] + 1 card U(u, v) t,t 1,T 2 U(u,v) [ 1 F(t, T1, T 2 ) T 2 T 1 b A 1 ( ) e AT 2 e AT 1 e At X(t) Z(t) ]. T 2 T 1 Need states X = (X 1, X 2 ) : we estimate them by an L 1 -filter. T2 T 1 Λ(τ)dτ

24 Figure: True state (blue) and estimated states (red) of a simulated stable CARMA(2,1) process.

25 Figure: The estimated risk premium R pr (green), together with the empirical risk premium R pr (red).

26 Now define for fixed v = T 2 T 1 = 1 and all u 1 2 (T 2 T 1 ) the MSE of the model based estimate R pr (u, v) and the emprical R pr (u, v) for all different thresholds w. Choose an optimal threshold (recall that all estimates depend on w): 180 w = argmin 1 w 180 R pr (u, v) R pr (u, v) 2 du. v/2

27 Conclusion We find a negative risk premium, which indicates that electricity producers are price takers, willing to accept a lower price to hedge their production. Seems realistic!!

28 [1] Bernhardt, C., Klüppelberg, C. and Meyer-Brandis, T. (2008) Estimating high quantiles for electricity prices by stable linear models. Journal of Energy Markets 1 (1), [2] Klüppelberg, C., Meyer-Brandis, T. and Schmidt, A. (2010) Electricity spot price modelling with a view towards extreme spike risk. Quantitative Finance 10(9), [3] García, I., Klüppelberg, C. and Müller, G. (2010) Estimation of stable CARMA models with an application to electricity spot prices. Statistical Modelling. To appear. [4] Benth, F., Klüppelberg, C. and Vos, L. (2010) Forward pricing in electricity markets based on stable CARMA spot models. In preparation. [5] Hepperger, P. (2010) Option pricing in Hilbert space valued jump-diffusion models using partial integro-differential equations. SIAM Journal on Financial Mathematics 1, [6] Hepperger, P. (2010) Numerical hedging of electricity swaptions with a portfolio of swaps. Submitted for publication.

29 [7] Hepperger, P. (2010) Hedging electricity swaptions using partial integro-differential equations. Submitted for publication. [8] Sen, R. and Klüppelberg, C. (2010) Time series of functional data analysis. Submitted for publication.