A Branching Process Approach to Power Markets Simone Scotti - PowerPoint PPT Presentation

A Branching Process Approach to Power Markets A Branching Process Approach to Power Markets Simone Scotti Universit´ e Paris-Diderot Joint work with : Ying Jiao , ISFA, University of Lyon Chunhua Ma , Nankai University Carlo Sgarra , Politecnico di Milano Thematic Semester on Statistics for Energy Markets Workshop 2. The Modelling of Energy Markets Research supported by Institute Europlace de Finance Research program “Clusters and Information Flow : Modelling, Analysis and Implications” Paris, March 16, 2017 Simone Scotti (Paris Diderot) Paris, March 16, 2017

A Branching Process Approach to Power Markets Spikes in electricity prices F IGURE – Single National Price (PUN at 7PM only working days) of electricity in Italy between April 2004 and December 2015 and large positive fluctuations We observe 96 positive jumps over 140 months, that is one jump each 30 working days. However, the distribution is far to be homogeneous. Simone Scotti (Paris Diderot) Paris, March 16, 2017

A Branching Process Approach to Power Markets Poisson versus self-exciting structure Rejection of Poisson framework We first test the goodness of fit of a pure Poisson distribution using the Kolmogorov Smirnov statistics. The value of the test is 1 . 821 that is larger than the critical value 1 . 628 for a significance level 1 % . Simone Scotti (Paris Diderot) Paris, March 16, 2017

A Branching Process Approach to Power Markets Poisson versus self-exciting structure Rejection of Poisson framework We first test the goodness of fit of a pure Poisson distribution using the Kolmogorov Smirnov statistics. The value of the test is 1 . 821 that is larger than the critical value 1 . 628 for a significance level 1 % . Lemma Let N be a non-homogeneous Poisson process with rate µ . Denote by T i the increasing sequence of arrival times of the jumps of N . Then, fix t > 0 conditional on N t , the vector ( T 1 , T 2 , . . . , T N t ) has the same law as ( U ( 1 ) , U ( 2 ) , . . . , U ( N t ) ) , i.e. the order statistics built µ ( s ) I s ∈ [ 0 , t ] from uniform IID random variables with density � t 0 µ ( u ) du . Acceptance of self-exciting framework We test the goodness of fit of a pure self-exciting jumps framework, that is the intensity µ is proportional to the price of electricity itself. The value of the Kolmogorov-Smirnov test is 1 , 06 that is lower than the critical value 1 . 224 for a significance level 10 % . Simone Scotti (Paris Diderot) Paris, March 16, 2017

A Branching Process Approach to Power Markets Kolmogorov-Smirnov plot F IGURE – QQ-plot of jumps arrival. Pure Poisson case in orange. Self-exciting case in blue. Simone Scotti (Paris Diderot) Paris, March 16, 2017

A Branching Process Approach to Power Markets Power Price Modeling We will assume then the spot price process S ( t ) to evolve according to the basic dynamics : S ( t ) = B ( t ) + Y ( t ) , where B ( t ) is a seasonality function of deterministic type and the process Y ( t ) is a superposition of the factors X i ( t ) : � Y ( t ) = X i ( t ) , The main objective is to propose new candidates for the evolution of the factors X including self-exciting structure. We propose to look at the class of continuous state branching processes with immigration. Simone Scotti (Paris Diderot) Paris, March 16, 2017

A Branching Process Approach to Power Markets Branching Processes in continuous time Branching property Branching property : A process X has the Branching Property if for any t and x , y in the state space of X , X x + y is t t and X y equal in law to the independent sum of X x t . If a process X can be decomposed as X = X ( 1 ) + X ( 2 ) where for i = 1 , 2, X ( i ) satisfying the same SDE with X 0 = X ( 1 ) + X ( 2 ) , then the process is said a branching process . 0 0 We have the following result, see Kawazu and Watanabe (1971). Generator Markov process X with state space R + with Branching mechanism : � ∞ Ψ( q ) = β q + 1 2 σ 2 q 2 + ( e − q ζ − 1 + q ζ ) π ( d ζ ) , 0 � ∞ ( ζ ∧ ζ 2 ) π ( d ζ ) < ∞ . with σ ≥ 0, β ∈ R and π being a L´ evy measure such that 0 The CBI process X has as generator the operator L acting on C 2 0 ( R + ) as � ∞ L f ( x ) = σ 2 � � 2 x f ′′ ( x ) − β x f ′ ( x ) + x f ( x + ζ ) − f ( x ) − ζ f ′ ( x ) π ( d ζ ) . 0 Simone Scotti (Paris Diderot) Paris, March 16, 2017

A Branching Process Approach to Power Markets Branching Processes in continuous time Dawson Li (2006) representation Integral representation The previous generator admits the following semigroup (Hille-Yosida theorem). � t � X s � t � X s X t = − a du ds + σ W ( ds , du ) 0 0 0 0 � t � X s − � R + ζ � + N ( ds , du , d ζ ) , 0 0 W ( ds , du ) : white noise on R 2 + with intensity ds dv , � N ( ds , du , d ζ ) : compensated Poisson random measure on R 3 + with intensity ds du π ( d ζ ) , Besides, W and N are independent of each other. Main problem : the process converges to 0 if a > 0 or to ∞ otherwise. As a consequence it is not ergodic. Simone Scotti (Paris Diderot) Paris, March 16, 2017

A Branching Process Approach to Power Markets Branching Processes in continuous time Continuous state branching process with immigration Integral representation � t � t � X s X t = a ( b − X s ) ds + σ W ( ds , du ) 0 0 0 � t � X s − � � t � R + ζ � + γ N ( ds , du , d ζ ) + γ R + ζ M ( ds , d ζ ) , 0 0 0 M ( ds , d ζ ) : compensated Poisson random measure on R 2 + with intensity ds π ( d ζ ) , The process will be exponential ergodic if a > 0. � ∞ a ( b − x ) f ′ ( x ) σ 2 � � 2 x f ′′ ( x ) + x f ( x + ζ ) − f ( x ) − ζ f ′ ( x ) L f ( x ) = π ( d ζ ) 0 � ∞ � � + f ( x + ζ ) − f ( x ) π ( d ζ ) 0 Simone Scotti (Paris Diderot) Paris, March 16, 2017

A Branching Process Approach to Power Markets Branching Processes in continuous time Continuous state branching process with immigration (CBI) CBI (Kawazu & Watanabe 1971) of branching mechanism Ψ( · ) and immigration rate Φ( · ) : Markov process X with state space R + verifying � � t � E x � e − p X t � � � = exp − x v ( t , p ) − Φ v ( s , p ) , ds 0 where v : R + × R + → R satisfies ∂ v ( t , p ) = − Ψ( v ( t , p )) , v ( 0 , p ) = p ∂ t , and Ψ and Φ are functions on R + given by � ∞ Ψ( q ) = a q + 1 2 σ 2 q 2 + ( e − q ζ − 1 + q ζ ) π ( d ζ ) , 0 � ∞ ( 1 − e − q ζ ) π ( d ζ ) , Φ( q ) = ab q + 0 � ∞ ( ζ ∧ ζ 2 ) π ( d ζ ) < ∞ with σ, ab ≥ 0, a ∈ R and π , π being two L´ evy measures such that � ∞ 0 and ( 1 ∧ ζ ) π ( d ζ ) < ∞ . 0 Simone Scotti (Paris Diderot) Paris, March 16, 2017

A Branching Process Approach to Power Markets Branching Processes in continuous time Link to Hawkes process When σ = γ = 0 and π ( d ζ ) = δ 1 ( d ζ ) , then X is given by � t � t � X s − � � a + π ( R + ) X t = X 0 − X s ds + N ( ds , du ) (1) 0 0 0 which is the intensity of Hawkes process � t � X s − N ( ds , du ) , 0 0 N being the Poisson random measure with intensity ds du . Simone Scotti (Paris Diderot) Paris, March 16, 2017

A Branching Process Approach to Power Markets α -CIR model The α -CIR model setup : Integral representation (Dawson-Li) Integral form by using the random fields � t � t � X s X t = X 0 + a ( b − X s ) ds + σ W ( ds , du ) 0 0 0 (2) � t � X s − � R + ζ � + σ Z N ( ds , du , d ζ ) , 0 0 W ( ds , du ) : white noise on R 2 + with intensity dsdu , � N ( ds , du , d ζ ) : compensated Poisson random measure on R 3 + with intensity ds du µ ( d ζ ) , � ∞ ( ζ ∧ ζ 2 ) µ ( d ζ ) < ∞ . µ ( d ζ ) : a L´ evy measure satisfying 0 We choose the L´ evy measure to be 1 { ζ> 0 } d ζ µ ( d ζ ) = − cos( πα/ 2 ) Γ( − α ) ζ 1 + α , 1 < α < 2 , For existence and uniqueness of the solution see Dawson and Li (2012), Theorem 3.1 and Li and Ma (2015) Theorem 2.1. Simone Scotti (Paris Diderot) Paris, March 16, 2017

A Branching Process Approach to Power Markets α -CIR model The α -CIR model setup We consider the following usual SDE � t � t � t √ X 1 /α X t = X 0 + a ( b − X s ) ds + σ X s dB s + σ Z s − dZ s (3) 0 0 0 B = ( B t , t ≥ 0 ) a Brownian motion Z = ( Z t , t ≥ 0 ) a spectrally positive α -stable compensate L´ evy process with parameter α ∈ ( 1 , 2 ] with � � t q α � e − q Z t � E = exp − , q ≥ 0 . cos( π α/ 2 ) B and Z are independent. Z t follows the α -stable distribution S α ( t 1 /α , 1 , 0 ) with scale parameter t 1 /α , skewness parameter 1 and zero drift. The existence of a unique strong solution for the SDE (3) follows from Fu and Li (Theorem 5.3, 2010). Simone Scotti (Paris Diderot) Paris, March 16, 2017

A Branching Process Approach to Power Markets α -CIR model Simulation of processes Z and X with different α α −stable process Z: r 0 =0.1, a=0.1, b=0.3, σ =0.1, σ Z =0.3 α −CIR process: r 0 =0.1, a=0.1, b=0.3, σ =0.1, σ Z =0.3 4 0.35 α =2 α =2 α =1.5 α =1.5 3 0.3 α =1.2 α =1.2 2 0.25 1 0.2 Z t r t 0 0.15 −1 0.1 −2 0.05 −3 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 t t F IGURE – Three parameters of α : 2 (blue), 1.5 (green) and 1.2 (black). Simone Scotti (Paris Diderot) Paris, March 16, 2017