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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

FIGURE – 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 Ti the increasing sequence of arrival times of the jumps of N. Then, fix t > 0 conditional on Nt, the vector (T1, T2, . . . , TNt) has the same law as (U(1), U(2), . . . , U(Nt)), i.e. the order statistics built from uniform IID random variables with density

µ(s)Is∈[0,t] 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

FIGURE – 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

• f the factors Xi(t) :

Y(t) =

• Xi(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, Xx+y

t

is equal in law to the independent sum of Xx

t and Xy 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 X0 = X(1) + X(2) , then the process is said a branching process. 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 σ2 q2 + ∞ (e−q ζ − 1 + q ζ) π(dζ), with σ ≥ 0, β ∈ R and π being a L´ evy measure such that ∞ (ζ ∧ ζ2) π(dζ) < ∞ . The CBI process X has as generator the operator L acting on C2

0(R+) as

Lf(x) = σ2 2 x f ′′(x) − β x f ′(x) + x ∞

• f(x + ζ) − f(x) − ζ f ′(x)
• π(dζ).

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). Xt = − t Xs a du ds + σ t Xs W(ds, du) + t Xs−

• R+ ζ

N(ds, du, dζ), W(ds, du) : white noise on R2

+ with intensity ds dv,

• N(ds, du, dζ) : compensated Poisson random measure on R3

+ with intensity ds du π(dζ),

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

Xt = t a(b − Xs) ds + σ t Xs W(ds, du) +γ t Xs−

• R+ ζ

N(ds, du, dζ) + γ t

• R+ ζ M(ds, dζ),

M(ds, dζ) : compensated Poisson random measure on R2

+ with intensity dsπ(dζ),

The process will be exponential ergodic if a > 0. Lf(x) = a(b − x) f ′(x) σ2 2 x f ′′(x) + x ∞

• f(x + ζ) − f(x) − ζ f ′(x)
• π(dζ)

+ ∞

• f(x + ζ) − f(x)
• π(dζ)

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 Ex e−p Xt = exp

• −x v(t, p) −

t Φ

• v(s, p)
• ds
• ,

where v : R+ × R+ → R satisfies ∂v(t, p) ∂t = −Ψ(v(t, p)), v(0, p) = p , and Ψ and Φ are functions on R+ given by Ψ(q) = a q + 1 2 σ2 q2 + ∞ (e−q ζ − 1 + q ζ) π(dζ), Φ(q) = ab q + ∞ (1 − e−q ζ) π(dζ), with σ, ab ≥ 0, a ∈ R and π, π being two L´ evy measures such that ∞ (ζ ∧ ζ2) π(dζ) < ∞ and ∞ (1 ∧ ζ) π(dζ) < ∞.

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(dζ) = δ1(dζ), then X is given by Xt = X0 − t

• a + π(R+)
• Xs ds +

t Xs− N(ds, du) (1) which is the intensity of Hawkes process t Xs− N(ds, du), 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 Xt = X0 + t a (b − Xs) ds + σ t Xs W(ds, du) + σZ t Xs−

• R+ ζ

N(ds, du, dζ), (2) W(ds, du) : white noise on R2

+ with intensity dsdu,

• N(ds, du, dζ) : compensated Poisson random measure on R3

+ with intensity ds du µ(dζ),

µ(dζ) : a L´ evy measure satisfying ∞ (ζ ∧ ζ2) µ(dζ) < ∞. We choose the L´ evy measure to be µ(dζ) = − 1{ζ>0} dζ cos(πα/2) Γ(−α) ζ1+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 Xt = X0 + t a (b − Xs) ds + σ t √ Xs dBs + σZ t X1/α

s− dZs

(3) B = (Bt, t ≥ 0) a Brownian motion Z = (Zt, t ≥ 0) a spectrally positive α-stable compensate L´ evy process with parameter α ∈ (1, 2] with E

• e−q Zt

= exp

• −

t qα cos(π α/2)

• ,

q ≥ 0. B and Z are independent. Zt follows the α-stable distribution Sα(t1/α, 1, 0) with scale parameter t1/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

0.2 0.4 0.6 0.8 1 −3 −2 −1 1 2 3 4 t Zt α−stable process Z: r0=0.1, a=0.1, b=0.3, σ=0.1, σZ=0.3 α=2 α=1.5 α=1.2 0.2 0.4 0.6 0.8 1 0.05 0.1 0.15 0.2 0.25 0.3 0.35 t rt α−CIR process: r0=0.1, a=0.1, b=0.3, σ=0.1, σZ=0.3 α=2 α=1.5 α=1.2

FIGURE – Three parameters of α : 2 (blue), 1.5 (green) and 1.2 (black).

Simone Scotti (Paris Diderot) Paris, March 16, 2017

A Branching Process Approach to Power Markets α-CIR model

Similar properties with CIR model I

Boundary condition :

The point 0 is an inaccessible boundary if and only if 2 a b ≥ σ2. In particular, a pure jump α-CIR process with ab > 0 never reaches 0 since σ = 0.

Ergodic law :

The process is exponentially ergodic, the limit distribution denoted by r∞ satisfies E

• e−pX∞

= exp   − p abq aq + σ2

2 q2 − σα

Z

cos(πα/2) qα dq

   .

Simone Scotti (Paris Diderot) Paris, March 16, 2017

A Branching Process Approach to Power Markets α-CIR model

Similar properties with CIR model II

Branching property :

r can be decomposed as X = X(1) + X(2) where for i = 1, 2, X(i) is an α-CIR(a, b(i), σ, σZ, α) process such that X0 = X(1) + X(2) and b = b(1) + b(2). See Dawson and Li (2006). This property is a direct consequence of linearity of integrals, homogeneity of measures.

Simone Scotti (Paris Diderot) Paris, March 16, 2017

A Branching Process Approach to Power Markets α-CIR model

Equivalence of two representations

Then the root representation (3) and the integral representation (2) are equivalent in the following sense : The solutions of the two equations have the same probability law. On an extended probability space, they are equal almost surely. See Theorem 9.32 in Li (2011). The equivalence is useful since we have two ways to study the properties of the model.

Simone Scotti (Paris Diderot) Paris, March 16, 2017

A Branching Process Approach to Power Markets α-CIR model

Power Price Modeling

We will assume then the spot price process S(t) to evolve according to the basic dynamics : S(t) = B(t) +

• Xi(t),

Xi(t) = Xi(0) + t a (bi − Xi(s)) ds + σi t Xi(s) W(ds, du) +γi t Xi(s−)

• R+ ζ

Ni(ds, du, dζ). This kind of dynamics extends that proposed by F.E. Benth, J. Kallsen and T. Meyer-Brandis, by keeping the basic features of an Ornstein-Uhlenbeck process driven by a subordinator, but it introduces the self exciting properties in a direct and natural way.

Simone Scotti (Paris Diderot) Paris, March 16, 2017

A Branching Process Approach to Power Markets α-CIR model

Finite activity case

Xi(t) = Xi(0) + t [abi − ˜ aiXi(s)] ds + σi t Xi(s) W(ds, du) +γi t Xi(s−)

• R+ ζ Ni(ds, du, dζ),

for t ≥ 0, where : ˜ ai(t) = a + σi

• R+ ζ πi(dζ).

The last expression shows how a different mean-reversion speed for each factor Yi can arise even if a common mean-reversion speed is assigned.

Simone Scotti (Paris Diderot) Paris, March 16, 2017

A Branching Process Approach to Power Markets α-CIR model

FIGURE – The Power Spot Price Dynamics.

Simone Scotti (Paris Diderot) Paris, March 16, 2017

A Branching Process Approach to Power Markets α-CIR model

Locally equivalent CIR process with jumps

Consider the α-CIR process with initial value X0 and introduce Pt = X0 + t a (b − Ps) ds + σ t Ps W(ds, du) + γ t X0

• R+ ζ

N(ds, du, dζ), where the processes W and N are (almost) the same as in (3). the above CIR process with jumps can be written as dPt = X0 + a (b − Pt) dt + σ √ Pt dBt + σZ

α

• X0 dZt,

The implicit negative drifts lead to a linear decay for Pt while an exponential decay for Xt : when γ increases, the decreasing drift plays a more important role in α-CIR than in equivalent CIR with jumps.

Simone Scotti (Paris Diderot) Paris, March 16, 2017

A Branching Process Approach to Power Markets α-CIR model

Comparison between α-CIR and CIR with α-stable jumps (continued)

Separating small and large jumps in CIR with jumps, we get

Pt = X0 + t a

• b − γ X0 Θ(α, y)

a − Ps

• ds + σ

t Ps W(ds, du) + γ t X0 y ζ N(ds, du, dζ) + γ t X0 ∞

y

ζ N(ds, du, dζ),

where Θ(α, y) = 2 π α Γ(α − 1) sin(πα/2) yα−1 . In a similar way, the α-CIR process can be written as

Xt = X0 + t

• a(α, y)
• b(α, y) − Xs
• ds + σ

t Xs W(ds, du) + γ t Xs− y ζ N(ds, du, dζ) + γ t Xs− ∞

y

ζ N(ds, du, dζ),

where

• a(α, y) = a + γ Θ(α, y),
• b(α, y) =

ab a + γ Θ(α, y) .

Simone Scotti (Paris Diderot) Paris, March 16, 2017

A Branching Process Approach to Power Markets Derivatives Pricing

Change of Probability

Proposition :

Let X be a CBI(a, b, σ, γ, π) process under the probability measure P and assume that the filtration F is generated by the random fields W and

• N. Fix η ∈ R and θ ∈ R+, and define

Ut := η t Xs W(ds, du) + t Xs− ∞ (e−θ ζ − 1) N(ds, du, dζ). Then the Dol´ eans-Dade exponential E(U) is a martingale and the probability measure Q defined by dQ dP

• Ft

= E(U)t, is equivalent to P. Moreover, under Q, r is a CBI-Levy type process with the parameters (a′, b′, σ′, γ′, π′), where a′ = a − σ η − γ ∞ ζ (e−θζ − 1) π(dζ), b′ = ab/a′, σ′ = σ, γ′ = γ π′(dζ) = e−θ ζ π(dζ).

Simone Scotti (Paris Diderot) Paris, March 16, 2017

A Branching Process Approach to Power Markets Derivatives Pricing

Forward Pricing

In all the present section the model parameters are assumed to be those defined by the risk-neutral dynamics. F(τ, T) = EQ S(T)

• Fτ
• ,

F(τ, T) = B(T) +

• i

EQ

• Xi(0) +

t ai (bi − Xi(s)) ds + σi t Xi(s) Wi(ds, du) +γi t Xi(s−)

• R+ ζ

N(ds, du, dζ)

• Fτ
• .

= B(T) − B(τ) + S(τ) +

• i

e−ai(T−τ) − 1 ai (Xi(τ) − bi)

Simone Scotti (Paris Diderot) Paris, March 16, 2017

A Branching Process Approach to Power Markets Derivatives Pricing

Flow Forwards

If we denote by [T1, T2] the delivery period, the value of the contract F(τ; T1, T2), τ < T1, is given by the following formulas : F(τ, T1, T2) = EQ

• 1

T2 − T1 T2

T1

S(u)du

• Fτ
• =

1 T2 − T1 T2

T1

EQ [ S(u)| Fτ] du = S(τ) − α(τ) −

• i

e−ai(T2−τ) − e−ai(T1−τ) + ai(T2 − T1) a2

i (T2 − T1)

(Xi(τ) − bi) + 1 T2 − T1 T2

T1

α(u)du

Simone Scotti (Paris Diderot) Paris, March 16, 2017

A Branching Process Approach to Power Markets Derivatives Pricing

The Risk Premium

The risk premium is a relevant quantity in power markets description. We want then to provide an explicit representation formula for this quantity in the present modeling framework. The risk premium can be defined as the difference between conditional expectations of the underlying price computed with respect to the risk-neutral measure Q and the historical measure P : R(τ, T) = EQ [S(T)|Fτ] − EP [S(T)|Fτ] .

Simone Scotti (Paris Diderot) Paris, March 16, 2017

A Branching Process Approach to Power Markets Derivatives Pricing

The Risk Premium

The risk premium is a relevant quantity in power markets description. We want then to provide an explicit representation formula for this quantity in the present modeling framework. The risk premium can be defined as the difference between conditional expectations of the underlying price computed with respect to the risk-neutral measure Q and the historical measure P : R(τ, T) = EQ [S(T)|Fτ] − EP [S(T)|Fτ] . According to the results obtained in the previous section we can write : R(τ, T) =

• i

e−Ai(T−τ) − 1 Ai (Xi(τ) − Bi) −

• i

e−ai(T−τ) − 1 ai (Yi(τ) − bi) , where the parameters Ai, Bi are related to the parameters ai and bi by the relations describing the measure change : Ai = ai − σi ηi − γi ∞ ζ (e−θi ζ − 1) µi(dζ), Bi = ai bi Ai

Simone Scotti (Paris Diderot) Paris, March 16, 2017

A Branching Process Approach to Power Markets Derivatives Pricing

0.5 1 1.5 2 −0.4 −0.2 0.2 0.4 0.6 0.8 1 t Risk premium Risk premium α=1.3 α=1.5 α=1.8

FIGURE – The Risk Premium Term Structure.

Simone Scotti (Paris Diderot) Paris, March 16, 2017

A Branching Process Approach to Power Markets Calibration

Calibration : Two-Factor Model

The first factor is continuous and corresponds to a standard CIR model and the second one is with jumps. Our objective is to make a thorough analysis of the jump behavior, in particular, for large jumps and spikes. Let the first factor Y1 be driven by a Gaussian random measure as Y1(t) = Y1(0) + t a1 (b1 − Y1(s)) ds + σ1 t Y1(s) W1(ds, du) (4) and the second factor Y2 be driven by a pure jump Poisson random measure as Y2(t) = Y2(0) + t a2 (b2 − Y2(s)) ds + γ2 t Y2(s−)

• R+ ζ

N2(ds, du, dζ) (5)

Simone Scotti (Paris Diderot) Paris, March 16, 2017

A Branching Process Approach to Power Markets Calibration Jump behaviour

Spike factor

We are interested in the evolution of process Y2 between two jump times, that is for any t ∈ [τk, τk+1), Y2(t) = Y2(τk) + t

τk

a2 (b2 − Y2(s)) ds + γ2 t

τk

Y2(s−)

• R+ ζ

N2(ds, du, dζ). With these notations, the time τk+1 is the arrival time of the first jump after τk larger than z0 for the measure N2 or equivalently, larger than γ2z0 for Y2. By the following result, we can separate small and large jumps and move the compensation inside the speed and mean coefficients.

Simone Scotti (Paris Diderot) Paris, March 16, 2017

A Branching Process Approach to Power Markets Calibration Jump behaviour

The process Y2 can be written, for all t ∈ [τk, τk+1), as Y2(t) = Y2(τk) + t

τk

A2

• B2 − Y(z0)

2

(s)

• ds + γ2

t

τk

Y(z0)

2

(s−)

z0 ζ N2(ds, du, dζ) +γ2 t

τk

Y(z0)

2

(s−)

∞

z0

ζN2(ds, du, dζ), where z0 > 0 is a fixed constant, A2 := a2 + γ2 ∞

z0

ζµ(dz), B2 := a2b2 A2 and Y(z0)

2

is the truncated process, for all t ∈ [τk, τk+1), defined by Y(z0)

2

(t) = Y2(τk) + t

τk

A2

• B2 − Y(z0)

2

(s)

• ds + γ2

t

τk

Y(z0)

2

(s−)

z0 ζ N2(ds, du, dζ).

Simone Scotti (Paris Diderot) Paris, March 16, 2017

A Branching Process Approach to Power Markets Calibration Jump behaviour

We stated that the truncated process Y(z0)

2

is linked to the intensity of the large jumps of Y2. The following proposition explains in detail this link. Up to a constant, the process Y(z0)

2

is the stochastic intensity or hazard rate of the random time of the next big jump. Let {τk}k∈N be defined by : τ (z0)

k

= inf{t > τ (z0)

k−1 : ∆Y2(t) > γ2z0},

τ (z0) = 0. Then we have P (τk+1 − τk > t) = E

• exp
• −K(z0)

Y

τk+t

τk

Y(z0)

2

(s)ds

• ,

where the renormalisation term K(z0)

Y

= γ2 ∞

z0 ζµ2(dζ) which is the proper truncated mass of

the jumps distribution, and the frequency process Y(z0)

2

is given by the previous formula.

Simone Scotti (Paris Diderot) Paris, March 16, 2017

A Branching Process Approach to Power Markets Calibration Jump behaviour

Finally, we deal with the asymptotic behaviour of Y(z0)

2

when the mean reverting speed a2

• diverges. Let us introduce the process

Y(z0)

2

defined as

• Y(z0)

2

(t) = b2 + e−A2t [Y2(0) − b2] + γ2 t

Y(z0)

2

(s−)

∞

z0

e−A2(t−s)ζN2(ds, du, dζ). The next proposition shows that the two processes Y2 and Y(z0)

2

have the same behaviour when a2 goes to infinity. As a consequence, we can approximate the frequency of large jumps by the

• ne of the Hawkes process as soon as a2 is large. Then, if a2 is large enough, the intensity of the

jumps exhibits two behaviours. It is quite stable around b2 but it jumps at all jumps times {τk}k∈N and exhibits an fast exponential decay to b2 with speed A2 ≥ a2. We have the following

Proposition

Consider Y2 with E[Y2(0)] < ∞. As a2 → +∞, we have that for each t > 0, Y2(t) − Y2(t) goes to zero in probability.

Simone Scotti (Paris Diderot) Paris, March 16, 2017

A Branching Process Approach to Power Markets Calibration Statistical Analysis

First steps

Following the ideas presented in previous papers, the first step to perform is to de-seasonalise the data. The second step is to split the components Y1 and Y2 emerging from the data. This issue is well analyzed in the papers by Beth, Kiesel and Nazarova (EE 2012) and their approach is directly applicable to our framework. Then, we first focus on the process Y1, sometimes called the base signal. We look for the ergodic distribution of Y1 fitting the data. By recalling that the ergodic distribution of a CIR diffusion is of Gamma type, our model is in agreement with the previous literature and we obtain in a similar way the estimated parameters for Y1.

Simone Scotti (Paris Diderot) Paris, March 16, 2017

A Branching Process Approach to Power Markets Calibration Statistical Analysis

Spike process

The estimation of the parameters of the spike process Y2 is then our following main issue. Unfortunately, we cannot apply the techniques proposed since their model does not include the clustering effect that is crucial in our framework. We remark that the process Y2 is not directly observable since the data are given by the sum of three components, i.e. the seasonality function, the base signal and the spike process itself. Moreover, the great variance of the base signal covers the spike process far from the times of spikes. That is the observation is reduced to the sequence (τk, ∆S(τk))k∈N, ∆S(τk) = S(τk) − S(τ −

k ),

where τk is the time of the kth spike and ∆Sτk is its jump size. Due to the continuity of the seasonality function and the base signal, we have ∆S(τk) = ∆Y2(τk). As a consequence, we can assume that we observe the jump times and the jump sizes of Y2.

Simone Scotti (Paris Diderot) Paris, March 16, 2017

A Branching Process Approach to Power Markets Calibration Statistical Analysis

Limit for large mean reverting speed

Our idea is then to estimate the intensity process Y(z0)

2

rather than Y2 itself. The mean-reverting speed is very high with respect to the one of the base signal as it has been pointed out in literature. We may then consider that the limit distribution expressed as the approximate distribution of the jump frequency and we can then neglect small jumps. In looking then at the sequence (τk, ∆S(τk))k∈N, it can be considered as the realization of a marked Hawkes process N2 with intensity Y(z0)

2

. We remark that the parameters B2, γ2 and A2 can then be estimated by the maximum likelihood estimator.

Simone Scotti (Paris Diderot) Paris, March 16, 2017

A Branching Process Approach to Power Markets Calibration Statistical Analysis

Likelihood function

Proposition

Given the observations (τk, ∆S(τk))k=1...N, we have the following Likelihood function log L (τ1, ∆S(τ1), . . . τN, ∆S(τN)|B2, γ2, A2) := −B2τN +

N

• i=1

γ2∆S(τi) A2

• e−A2(τN−τi) − 1
• +

N

• i=1

log   B2 + γ2

i−1

• j=1

∆S(τj)e−A2(τi−τj)   

Simone Scotti (Paris Diderot) Paris, March 16, 2017

A Branching Process Approach to Power Markets Calibration Statistical Analysis

Moreover, the MLE estimators are : ∂ log L ∂B2 = −τN +

N

• i=1

  B2 + γ2

i−1

• j=1

∆S(τj)e−A2(τi−τj)   

−1

∂ log L ∂γ2 =

N

• i=1

∆S(τi) A2

• e−A2(τn−τi) − 1
• +

N

• i=1

i−1

j=1 ∆S(τj)e−A2(τi−τj)

B2 + γ2 i−1

j=1 ∆S(τj)e−A2(τi−τj)

∂ log L ∂A2 =

N

• i=1

γ2∆S(τi) A2

2

• 1 −
• A2(τn − τi) + 1
• e−A2(τn−τi)
• −

N

• i=1

γ2 i−1

j=1 ∆S(τj)(τi − τj)e−A2(τi−τj)

B2 + γ2 i−1

j=1 ∆S(τj)e−A2(τi−τj)

Simone Scotti (Paris Diderot) Paris, March 16, 2017

A Branching Process Approach to Power Markets Calibration Statistical Analysis

Thank you for your attention!

Simone Scotti (Paris Diderot) Paris, March 16, 2017