Modeling the dependence between a Poisson process and a continuous semimartingale

Transcription

1 1 / 28 Modeling the dependence between a Poisson process and a continuous semimartingale Application to electricity spot prices and wind production modeling Thomas Deschatre 1,2 1 CEREMADE, University of Paris Dauphine and PSL 2 EDF Lab and FiME Lab March 218

2 2 / 28 Motivation Dependence between S and Q can be modeled by a correlation but it does not take into account dependence in extreme events. However, spikes can be caused by high renewable production. This dependence can have an impact on the distribution of the incomes. General problem : modeling and estimating the dependency between extreme events and exogenous variable.

3 3 / 28 Outline 1 Dependence between a jump process and a continuous time exogenous factor 2 A joint model for the German electricity spot price and wind penetration index (in collaboration with Almut Veraart

4 4 / 28 Outline 1 Dependence between a jump process and a continuous time exogenous factor 2 A joint model for the German electricity spot price and wind penetration index (in collaboration with Almut Veraart

5 5 / 28 Statistical setting We observe an Itô continuous semimartingale on [, T ] X t = t t µ s ds + σ s dw s. We observe a Poisson process (N t t T with stochastic intensity (λ t t T. We assume that λ t = nq (X t, t [, T ] ; n corresponds to the asymptotic. Objectives Estimate q on a given interval I and evaluate its performance relatively to the L 2 (I norm, Develop a test in order to determine if q belongs to some parametric family on I.

6 Local time of X Theorem (from Barlow and Yor (1982; Revuz and Yor (213; Yen and Yor (213 Let us assume that : (i σ s σ > a.s. and E (ii X t = t. ( T µs ds + sup t σudwu t [,T ] < or Then, there exists a function defined on R [, T ] and denoted (x, t lt x verifying (i an occupation time formula t f (Xs ds = R f (x lx t dx for any measurable function f on Ω R, ( l x T (ii E sup lt x x R < and (iii x lt x is continuous on R under (i and has one point of discontinuity under (ii. measures the time spent by X around the point x before time T = must be large enough to estimate q at point x. We work on D (I, ν = {ω Ω, inf x I lx T (ω νt I }, ν (, 1]. 6 / 28

7 Local polynomial estimator Let K a positive kernel, K h ( = 1 h K ( T h, U (x = (1, x, x 2 2!,.., x m m! and h H (,. We define the local polynomial estimator ˆq h = U T ( ˆθ h with Theorem ˆθ h = argmin θ R m+1 2 n θt T T ( + θ T Xs x U h ( Xs x U h ( U T Xs x h K h (X s x 1 Xs IdN s K h (X s x 1 Xs Ids θ. On D (I, ν, if there exists > such that K is bounded from below by a constant > on [, ], ˆq h is well defined on I for h 2 3 I. Kernel estimator case (m = : ˆq h (x = 1 n T K h (X s x 1 Xs IdN s T K h (X s x 1 Xs Ids. 7 / 28

8 8 / 28 Bandwidth selection We want to minimize E ( q ˆq h 2 I D (I, ν w.r.t. h. Use of the method of Lacour et al. (216 in the context of i.i.d observations and density estimation. When h min = min H, the bias q h q 2 I q h q hmin 2 I. = An unbiased estimator of the bias is ˆB h = ˆq h ˆq hmin 2 I + bias correction term. An unbiased estimator of the variance (kernel estimator case is ˆV h = 1 T n 2 I K h (X s x 1 Xs I ( T 2 dxdn s. K h (X u x 1 Xu Idu We choose the bandwidth ĥ = argmin h H (ˆBh + κ ˆV h, κ >.

9 9 / 28 Oracle inequality Use of concentration inequalities from Reynaud-Bouret (214 and Houdré and Reynaud-Bouret (23. Theorem Let us suppose that min H K K 1 I n ( E ( ˆqĥ q 2 I D (I, ν κ 1 κ + C 1 log (n + C 2 (κ log (ne ( q hmin q 2 I D (I, ν + C 3 (K, κ I ν 2 T 2 + C 4 q 2 I n 4 and max H 2 3 I. ( min E ˆq h q 2 I D (I, ν h H ( ( 1 + q I, E ( l T I, D (I, ν I log (n H 5 + n + C 4 ( 5 (K I i=1 q,i T i ν 2 T 2. n log (n H 6 n

10 1 / 28 Minimax optimality Let Σ (β, I is the Hölder class of order β on I, Λ ρ,β = {f : I R : f (x ρ, f I, < } Σ (β, L, I and ( R ( q n, Λ ρ,β, ϕ n = sup ϕ 2 n I ( q n q (x 2 dx D (I, ν the minimax risk. q Λ ρ,β E Theorem (Minimax optimality If H = {h > h K K 1 I n lim sup n lim sup n lim inf n inf q n, h 2 3 I and I h 1 N}, R ( q n, Λ ρ,β, n β 2β+1 >. R (ˆqĥ, Λ ρ,β, n β 2β+1 < if m β, R (ˆqĥ, Λ ρ,β, n m 2m+1 < if m β,

11 11 / 28 Parametric versus non parametric test (kernel estimator case Test { H : θ Θ R d, q ( = g θ ( against H 1 : q g θ θ Θ. T Let M n (θ = ˆqĥ ( Under H, ˆθ n with ˆV n = C (K I T + bias correction term, ˆθ n = arginf θ Θ K hn (X s K h n (X u 1 Xu Idu 1 X s Ig θ (X s ds 2 I M n (θ. p θ at the speed n and we reject H if n ĥmn (ˆθ n ˆV n Φ 1 (1 α ( g ˆθn (y T K ĥ (y Xs1 Xs Ids 2 dy.

12 12 / 28 Resume We propose : a model taking into account the dependence of the extreme events with an exogenous variable, a local polynomial estimator of the intensity of jumps as a function of the exogenous covariate which is optimal in the minimax sense over the Hölder classes, a way to choose optimally the bandwidth of the estimator in a non asymptotic framework with an oracle inequality, a test to determine if the intensity as a function of the exogenous covariate belongs to a parametric family, which can be useful for operational purposes.

13 13 / 28 Outline 1 Dependence between a jump process and a continuous time exogenous factor 2 A joint model for the German electricity spot price and wind penetration index (in collaboration with Almut Veraart

14 14 / 28 Recall of the problem Spikes of S can be caused by high renewable production Q. Need to model this dependence to capture all the risks. Objectives Use of Wind penetration index = wind energy production electricity load Propose a joint model of electricity spot price and wind penetration index with dependence in the extremes.

15 15 / 28 Data The data sets are considered from the 1/1/212 to the 31/12/216 and are the following ones : The German and Austrian hourly electricity spot prices (from the day-ahead auction, the German and Austrian hourly load data, the German and Austrian hourly wind energy production data. The data has been aggregated over the four German transmission system operators 5 Hertz Transmission, Amprion, Tennet TSO and EnBW Transportnetze and the Austrian transmission system operator APG.

16 16 / 28 Model S t = Γ 1,t + X 1,t + K i e β(t T i N t i=1 WP t = expit ( Γ 2,t + X 2,t observed on a regular grid with steps n where expit (x = 1 1+e x, Γ 1 and Γ 2 are seasonality functions, X 1 and X 2 are centered continuous stochastic processes, N t is a Poisson process with stochastic intensity λ t, K i and T i are the sizes and the times of the jumps. β is the speed of the mean reversion of the spikes and is large. Two structures of dependence : a constant correlation between X 1 and X 2, the intensity λ t = q (WP t.

17 17 / 28 Jump detection We consider the method of Deschatre et al. (218 to detect spikes when β is large. There is a spike in ((i 1 n, i n ] iif n i S = S i n S (i 1 n > 5ˆσ.49 n n i S n i+1 S < and where ˆσ 2 is the multipower variation of order 2. We find 84 positive jumps and 3 negative jumps. They propose a consistent estimator for β : we find year 1. The size of the jumps are estimated with n i S at a time of a jump and the empirical distribution is considered in our model.

18 Jump detection versus wind penetration Positive spot jumps Negative spot jumps Spot price Positive spot jumps Negative spot jumps Wind penetration Time Time Jumps in German spot price (left and wind penetration index (right. 18 / 28

19 Intensity process modeling Positive and negative jumps are distinguished with one intensity function for each, λ t = q (WP t and λ + t = q + (WP t. Testing negative jumps intensity function as constant is rejected = Model : λ t = λ low 1 WP t WP thre + λ up1 WPt WP thre but is not for the positive jumps intensity = Model : λ + t = λ +. Kernel estimator Kernel estimator Intensity of negative spikes Parametric estimator Intensity of positive spikes Parametric estimator Wind penetration Wind penetration Kernel estimators for q (left and q + (right. 19 / 28

20 Estimating the continuous part of the spot model (1/2 with S c t = Γ 1,t + X 1,t ( ( Γ 1,t = c,1 + c 1,1 t + c 2,1 t 2 τ,1 +2πt τ1,1 +2πt + c 3,1 cos c 4,1 cos c 5,1 cos a seasonality function estimated after filtering ( τ2,1 +2πt 24 the spikes using a L 2 minimization and X 1 a continuous autoregressive process (CAR of order 24 estimated using the method of Benth and Benth ( Spot price Spot seasonality Time Seasonality function of the filtered spot price. 2 / 28

21 Estimating the continuous part of the spot model (2/2 Autocorrelation Partial Autocorrelation Autocorrelation and partial autocorrelation of the deseasonalised and filtered spot price. Autocorrelation Partial Autocorrelation Autocorrelation and partial autocorrelation of the spot price residuals. 21 / 28

22 Estimating the wind penetration model (1/2 WP t = expit ( Γ 2,t + X 2,t with Γ 2 is a seasonality function parametrized in a same way than Γ 1, X 2 is a CAR process of order 24 correlated to X 1. 6 Logit wind penetration Logit wind penetration seasonality Time Seasonality function of the logit wind penetration. 22 / 28

23 Estimating the wind penetration model (2/2 Autocorrelation Partial Autocorrelation Autocorrelation and partial autocorrelation of the deseasonalised logit wind penetration. Autocorrelation Partial Autocorrelation Autocorrelation and partial autocorrelation of the logit wind penetration residuals. 23 / 28

24 24 / 28 Numerical results on the incomes of a distributor Incomes = T QWP t C t (S t K dt where C t is the total load assumed to be deterministic, Q = 1% and K = 3 e/mwh. Two models are considered. Model Simple correlation Negative spikes dependence Expectation [742.5, 754.8] [642.1, 654.5] VaR 95% [ 96.6, 88.9] [ 19.7, 978.9] VaR 99% [ , ] [ , ] ES 95% [ 1324., ] [ , ] ES 99% [ , ] [ 236.1, ] Different risk measures (ke for the portfolio simulated between the 1 st of January 217 and the 31 st of December 217 1, times. Confidence intervals are computed with classical bootstrap methods.

25 25 / 28 Conclusion We proposed a model of dependence, simple enough to understand and to simulate, and a method to estimate it in an optimal way. This model is adapted for the modeling of the electricity spikes and allows to add a dependency component in the model of Deschatre et al. (218. It confirms statistically that high wind penetration level has an impact on the negative spikes frequency. Taking into account this dependence has a strong impact on the different risk measures of a simple contract, and can not be neglected.

26 26 / 28 Thank you for your attention. Do you have any questions?

27 27 / 28 Bibliography I Barlow, M. T. and Yor, M. (1982. Semi-martingale inequalities via the garsia-rodemich-rumsey lemma, and applications to local times. Journal of functional Analysis, 49(2 : Benth, F. E. and Benth, J. Š. (212. Modeling and pricing in financial markets for weather derivatives, volume 17. World Scientific. Deschatre, T., Féron, O., and Hoffmann, M. (218. Estimating fast mean-reverting jumps in electricity market models. arxiv preprint arxiv : Houdré, C. and Reynaud-Bouret, P. (23. Exponential inequalities, with constants, for u-statistics of order two. In Stochastic inequalities and applications, pages Springer. Lacour, C., Massart, P., and Rivoirard, V. (216. Estimator selection : a new method with applications to kernel density estimation. Sankhya A, pages Revuz, D. and Yor, M. (213. Continuous martingales and Brownian motion, volume 293. Springer Science & Business Media.

28 28 / 28 Bibliography II Reynaud-Bouret, P. (214. Concentration inequalities, counting processes and adaptive statistics. In ESAIM : Proceedings, volume 44, pages EDP Sciences. Yen, J.-Y. and Yor, M. (213. Local times and excursion theory for brownian motion a tale of wiener and ito measures preface.