Arbitrage Pricing in Electricity Markets

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Motivation A structural model of electricity prices Model and notations No Arbitrage of the 2nd_kind _Results

Arbitrage Pricing in electricity Markets

Adrien Nguyen Huu Universit´e Dauphine, CEREMADE

and FiME- EDF R&D Paris, France

GT FiME IHP, 05.11.10

Joint work with alumni.

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Phenomenology of Electricity markets

• Non Storability of the underlying asset

Arbitrage pricing theory is no longer valid :

Ft(T ) 6= EQ[ST | Ft] = Ster (T −t) ⇒ no pure link between spot and future prices

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Phenomenology of Electricity markets

• Price formation

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Phenomenology of Electricity markets

• Price formation

Prices as physical supply demand equilibrium + inflexible demand ⇒ price as a production cost (Barlow)

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Phenomenology of Electricity markets

• Price formation

Prices as physical supply demand equilibrium

Non storability + market rules ⇒ possible specific prices (negative prices, bounded, spikes, seasonality).

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Phenomenology of Electricity markets

• Price formation

• Granularity of term structure

Availability of assets on the French future market :

⇒ no hedging without a more subtile term structure (incomplete market)

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Phenomenology of Electricity markets

• Price formation

• Illiquidity andtransaction costs

• Specific market (and minor against OTC)

• Localized selling (national, regional)

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Phenomenology of Electricity markets

• Price formation

⇒ How can we link spot prices and future prices with classical financial paradigm ?

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Phenomenology of Electricity markets

• Price formation

⇒ How can we link spot prices and future prices with classical financial paradigm ?

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Open problems

• Future contract pricing

• Options on non available granularity

• Spread options on electricity and combustible

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Open problems

• Spread options on electricity and combustible

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Open problems

• Spread options on electricity and combustible • Production/Plant pricing

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Open problems

• Spread options on electricity and combustible • Production/Plant pricing

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Previous work and literature

• Plethora of spot models (Barlow, Cartea, Benth,

Hepperger,...) : levy processes, regime switching models, processes in Hilbert space.

• Plethora of Future models : two factor model, integrals of spot prices.

• Statistical models : link with commodities, weather, production capacities, demand.

• Calibration on future prices, option prices, spot prices, specific knowledge.

Still: F (t, T ) 6= EQ_[S T | Ft]

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Previous work and literature

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Previous work and literature

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Previous work and literature

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Previous work and literature

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A structural model of electricity prices

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Idea

How prices are computed ?

Consider

• N production means depending with respective costs (Sk t),

k ≤ N.

• their N capacities of production ∆k_t, k ≤ N.

• the permutationπt(k)s.t. Stπt(1)≤ · · · ≤ S πt(N)

t

• an independent positive demand processDt. Then the electricity spot price/cost is :

Pt= X k≤N S_tk1_D_t_∈Ik t where I k t := "_k−1 X i =1 ∆πt(i )_, k X i =1 ∆πt(i ) #

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Idea

How prices are computed ? Consider

• N production means depending with respective costs (Sk t), k ≤ N.

• their N capacities of production ∆k_t, k ≤ N.

• the permutationπt(k)s.t. Stπt(1)≤ · · · ≤ S πt(N)

t

• an independent positive demand processDt.

Then the electricity spot price/cost is :

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Idea

• their N capacities of production ∆k_t, k ≤ N. • the permutationπt(k)s.t. Stπt(1)≤ · · · ≤ S

πt(N)

t

• an independent positive demand processDt.

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Idea

πt(N)

t • an independent positive demand processDt.

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Idea

πt(N)

t • an independent positive demand processDt.

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Idea

πt(N)

t • an independent positive demand processDt. Then the electricity spot price/cost is :

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Pricing methodology

Situation :

• Commodities are storable : classical (yet difficult) arbitrage pricing

• Parameters ∆k_t are supposed to be known (by the producer).

• D is non-tradable asset (⇒ incomplete market)

How we choose the EMM ?

Minimal Martingale Measure Q(Follmer and Schweizer) ⇒ S is a Q-martingale, D is the same under Q and under P. ⇒ ”Some” risk neutral pricing in Electricity markets :

Ft(T ) = n X i =1 X π∈Π F_tπ(i )(T )Q[DT ∈ ITi |Ft]Qπ(i )[πT = π|Ft]

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Pricing methodology

Situation :

• Parameters ∆k_t are supposed to be known (by the producer). • D is non-tradable asset (⇒ incomplete market)

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Pricing methodology

Situation :

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Pricing methodology

Situation :

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Pricing methodology

Situation :

Minimal Martingale Measure Q(Follmer and Schweizer)

⇒ S is a Q-martingale, D is the same under Q and under P. ⇒ ”Some” risk neutral pricing in Electricity markets :

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Pricing methodology

Situation :

Minimal Martingale Measure Q(Follmer and Schweizer) ⇒ S is a Q-martingale, D is the same under Q and under P.

⇒ ”Some” risk neutral pricing in Electricity markets :

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Pricing methodology

Situation :

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Novelty and limits of the model

• Reproduce price stylized facts.

• Allows pricing and hedging of claims (Aid, Campi, Langrenet)

but

• A structural approach to the optimal behaviour of the producer

• The production function is a (tractable) approximation

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Novelty and limits of the model

but

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Novelty and limits of the model

but

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Novelty and limits of the model

but

• The production function is a (tractable) approximation • Production issues and Financial issues are separated

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Novelty and limits of the model

but

• The production function is a (tractable) approximation • Production issues and Financial issues are separated

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No Marginal Arbitrage for High Production Regime in discrete time investment-production models

With proportional transaction costs.

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Motivation

From an electricity producer point of view :

• how to optimize the production with a general production function ?

• how to consider both production and financial strategies ?

• What is then a No Arbitrage Condition ?

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Motivation

• how to consider both production and financial strategies ?

• What is then a No Arbitrage Condition ?

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Motivation

• how to consider both production and financial strategies ? • What is then a No Arbitrage Condition ?

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Motivation

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Motivation

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Model description - The financial market

• _{Probability space : (Ω, F , P), F := {F}_t}t=0,··· ,T.

• Bid-ask matrix : π := (πt)t≤0⊂ L0(Md, F)

• πjit ∈ L0(Ft) = number of units of asset i needed to obtain 1

unit of asset j .

• πijtπ jk

t ≥ πikt > 0, πiit = 1

• Financial position : V ∈ L0(Rd) with Vi = number of units of asset i held in portfolio.

• Solvency cone process : K := (Kt)t≤T with

Kt(ω) :=    x ∈ Rd : ∃aij ≥ 0 s.t. xi+X j 6=i aji −aijπ_tij(ω) ≥ 0 ∀i   

aij = number of units of i obtained against units of j . • Set of self-financed exchanges at time t : −Kt(ω).

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Model description - The financial market

• _{Probability space : (Ω, F , P), F := {F}_t}t=0,··· ,T. • Bid-ask matrix : π := (πt)t≤0⊂ L0(Md, F)

• πjit ∈ L0(Ft) = number of units of asset i needed to obtain 1

unit of asset j .

• πijtπ jk

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Model description - The financial market

• πtji ∈ L0(Ft) = number of units of asset i needed to obtain 1

unit of asset j .

• πijtπ jk

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Model description - The financial market

unit of asset j .

• πtijπ jk

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Model description - The financial market

unit of asset j .

• πtijπ jk

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Model description - The financial market

unit of asset j .

• πtijπ jk

aij = number of units of i obtained against units of j .

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Model description - The financial market

unit of asset j .

• πtijπ jk

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A comprehensive geometrical interpretation

-K

K

K*

transaction cost impact :orders cone

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Model description - The Production

• Family of random maps (Rt)t≤T

• Rt+1: β ∈ L0(Rd+, Ft) 7→ Rt+1(β) ∈ L0(Rd, Ft+1)

• βi _{= number of asset i consumed and sent into the production}

system at time t.

• R_t+1j (β) = number of asset j obtained at time t + 1 from the production regime β.

Example :

• Asset 1 = cash, Asset 2 = Future on Electricity (for a given maturity), Asset 3 = Fuel.

• Rt+1(β) depends only on β3 • R_t+1i (β) = 0 for i = 2, 3.

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Model description - The Production

system at time t.

Example :

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Model description - The Production

system at time t.

Example :

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Model description - The Production

system at time t.

Example :

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Model description - The Production

system at time t.

Example :

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Model description - The Production

system at time t.

Example :

• Rt+1(β) depends only on β3

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Model description - The Production

system at time t.

Example :

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Model description - Wealth process

• Strategies

(ξ, β) ∈ A0 := L0((−K ) × Rd+, F),

i.e. s.t. (ξt, βt) ∈ L0((−Kt) × Rd+, Ft) for all 0 ≤ t ≤ T .

• Set of portfolio holdings that are attainable at time T by trading and producing from time t with zero initial holding

AR_t(T ) := ( _T X s=t ξs− βs+ Rs(βs−1)1s≥t+1, (ξ, β) ∈ A0 ) .

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Model description - Wealth process

• Strategies

(ξ, β) ∈ A0 := L0((−K ) × Rd+, F),

i.e. s.t. (ξt, βt) ∈ L0((−Kt) × Rd+, Ft) for all 0 ≤ t ≤ T . • Set of portfolio holdings that are attainable at time T by

trading and producing from time t with zero initial holding

AR_t(T ) := ( _T X s=t ξs− βs+ Rs(βs−1)1s≥t+1, (ξ, β) ∈ A0 ) .

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Back to the structural model

• Let πt be the bid-ask prices of assets : 1 =cash, 2 . . . n =commodities.

• c_ti the conversion factor from 1 unit of asset i to 1 MWh.

• πe

t the spot price of electricity in cash.

• ∆i_t, i = 2 . . . n the maximum capacity of production from the i th commodity.

Then we can write :

R_t+11 (β) = π_t+1e (β) × X i >2 c_t+1i min(βi, ∆i_t+1) ! with π_t+1e (β) = max i (π 1i t+1ct+1i 1βi_>0)

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Back to the structural model

• c_ti the conversion factor from 1 unit of asset i to 1 MWh. • πe

Then we can write :

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Back to the structural model

Then we can write :

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Back to the structural model

Then we can write :

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Back to the structural model

Then we can write :

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Beyond the structural model

Some advantages...

• Additional features on the production function (starting costs, various conversion factors for different plants).

• Possibility to chose another electricity spot price πe. ... and difficulties :

• an additional optimization problem (with possible no solution) • Possible no explicit solutions for pricing claims.

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No-arbitrage of the second kind with R ≡ 0

• NA2 (Rasonyi, 2009) : for ζ ∈ L0(Rd, Ft) and t < T , (ζ + AK ,0_t (T )) ∩ L0(KT, FT) 6= {0} ⇒ ζ ∈ L0(Kt, Ft) .

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No-arbitrage of the second kind with R ≡ 0

• NA2

• EF: there is efficient frictionif

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No-arbitrage of the second kind with R ≡ 0

• NA2

• EF: there is efficient frictionif

πijπji > 1, ∀i 6= j, t ≤ T .

Under EF,

NA2 : ζ ∈ L0(Kt+1, F ) ⇒ ζ ∈ L0(Kt, F ), t < T ,

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No-arbitrage of the second kind with R ≡ 0

• NA2 : ζ ∈ Kt+1⇒ ζ ∈ Kt , ∀ζ ∈ L0(Rd, Ft) • EF

• MT

t : set of martingale selectors Z on [t, T ] of the random setsintK_s∗, for t ≤ s ≤ T , with

K_s∗(ω) = n

z ∈ Rd : 0 ≤ zj ≤ ziπ_sij(ω), i , j ≤ d o

the positive dual of Ks(ω).

• Strictly consistent price system: Z ∈ intK_s∗ : Zj

s/Zsi < πijs. • Z is a martingale fictitious price better than the market.

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No-arbitrage of the second kind with R ≡ 0

• NA2 : ζ ∈ Kt+1⇒ ζ ∈ Kt , ∀ζ ∈ L0(Rd, Ft) • EF: intK_s∗ 6= ∅

• MT

t : set of Strictly consistent price systems • PCE: Prices are consistently extendable if

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No-arbitrage of the second kind with R ≡ 0

• MT

t : set of Strictly consistent price systems

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No-arbitrage of the second kind with R ≡ 0

• MT

t : set of Strictly consistent price systems

• PCE: ∃Z ∈ MT_t s.t. Zt= X , ∀X ∈ L1(intKt∗, Ft).

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No-arbitrage of the second kind with linear production

• Assume that Rt(β) = Ltβ, ∀β ∈ Rd, t ≤ T , with L ∈ L0(Md, F).

(i) ζ−β + Lt+1(β)∈ L0(Kt+1, Ft+1) ⇒ ζ ∈ Kt,

(ii) −β + Lt+1(β) ∈ L0(Kt+1, Ft+1) ⇒ β = 0.

• LT

t : set of martingales Z s.t. for t ≤ s < T

EZs+10 (Ls+1− I ) | Fs ∈ L0(intRd−, Fs) • PCEL :

∃Z ∈ MT

t ∩ LTt s.t. Zt= X , ∀t ≤ T , X ∈ L1(intKt∗, Ft).

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No-arbitrage of the second kind with linear production

• There is no arbitrage of the second kind for L (NA2L) : ∀(ζ, β) ∈ L0_(Rd_{× R}d +, Ft) (i) ζ −β + Lt+1(β) ∈ L0_(K t+1, Ft+1) ⇒ ζ ∈ Kt, (ii) −β + Lt+1(β) ∈ L0(Kt+1, Ft+1) ⇒ β = 0. • LT

∃Z ∈ MT

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No-arbitrage of the second kind with linear production

• There is no arbitrage of the second kind for L (NA2L) : ∀(ζ, β) ∈ L0_(Rd_{× R}d

+, Ft)

(i) ζ−β + Lt+1(β)∈ L0(Kt+1, Ft+1) ⇒ ζ ∈ Kt,

(ii) −β + Lt+1(β) ∈ L0(Kt+1, Ft+1) ⇒ β = 0.

• LT

∃Z ∈ MT

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No-arbitrage of the second kind with linear production

• NA2L : ∀(ζ, β) ∈ L0(Rd× Rd +, Ft)

(i) ζ−β + Lt+1(β)∈ L0(Kt+1, Ft+1) ⇒ ζ ∈ Kt,

(ii) −β + Lt+1(β) ∈ L0(Kt+1, Ft+1) ⇒ β = 0.

• What is the position (Ls+1− I )β in the price system Z ?

• LT

∃Z ∈ MT

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No-arbitrage of the second kind with linear production

• NA2L : ∀(ζ, β) ∈ L0(Rd× Rd +, Ft)

(i) ζ−β + Lt+1(β)∈ L0(Kt+1, Ft+1) ⇒ ζ ∈ Kt,

(ii) −β + Lt+1(β) ∈ L0(Kt+1, Ft+1) ⇒ β = 0.

• What is the position (Ls+1− I )β in the price system Z ?

• EZs+10 (Ls+1− I )βs | Fs < 0 if Z is strictly more favorable

than π.

• or production arbitrage.

• LT

∃Z ∈ MT

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No-arbitrage of the second kind with linear production

• Assume that Rt(β) = Ltβ, ∀β ∈ Rd, t ≤ T , with L ∈ L0(Md, F). • NA2L : ∀(ζ, β) ∈ L0(Rd× Rd +, Ft) (i) ζ−β + Lt+1(β)∈ L0(Kt+1, Ft+1) ⇒ ζ ∈ Kt, (ii) −β + Lt+1(β) ∈ L0(Kt+1, Ft+1) ⇒ β = 0. • LT

EZs+10 (Ls+1− I ) | Fs ∈ L0(intRd−, Fs)

• PCEL : ∃Z ∈ MT

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No-arbitrage of the second kind with linear production

∃Z ∈ MT

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No-arbitrage of the second kind with linear production

∃Z ∈ MT

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No-arbitrage of the second kind with general production

• No need to prove the closedness of AL_t(T ) first.

• Imagine that there exists L ∈ L0(Md, F) s.t. NAL and ∀t < T , β ∈ L0(Rd+, Ft),

lim

η→∞Rt+1(ηβ)/η = Lt+1β. Then there is no marginal arbitrage asymptotically. • NMA2 : ∃(c, L) ∈ L0(Rd × Md_{, F) s.t. NA2}L _and ct+1+ Lt+1β − Rt+1(β) ∈ L0(Kt+1, Ft+1),

∀β ∈ L0_(Rd

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No-arbitrage of the second kind with general production

• No need to prove the closedness of AL_t(T ) first. • Imagine that there exists L ∈ L0(Md, F) s.t. NAL

and ∀t < T , β ∈ L0(Rd+, Ft), lim

η→∞Rt+1(ηβ)/η = Lt+1β.

Then there is no marginal arbitrage asymptotically. • NMA2 : ∃(c, L) ∈ L0(Rd × Md_{, F) s.t. NA2}L _and ct+1+ Lt+1β − Rt+1(β) ∈ L0(Kt+1, Ft+1),

∀β ∈ L0_(Rd

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No-arbitrage of the second kind with general production

η→∞Rt+1(ηβ)/η = Lt+1β. Then there is no marginal arbitrage asymptotically.

• NMA2 : ∃(c, L) ∈ L0(Rd × Md_{, F) s.t. NA2}L _and ct+1+ Lt+1β − Rt+1(β) ∈ L0(Kt+1, Ft+1),

∀β ∈ L0_(Rd

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No-arbitrage of the second kind with general production

η→∞Rt+1(ηβ)/η = Lt+1β. Then there is no marginal arbitrage asymptotically. • NMA2 : ∃(c, L) ∈ L0(Rd× Md_{, F) s.t. NA2}L _and ct+1+ Lt+1β − Rt+1(β) ∈ L0(Kt+1, Ft+1),

∀β ∈ L0_(Rd

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The Closedness Property

Theorem :AL₀(T ) is closed in probability under NA2L. The same holds for AR₀(T ) under NMA2 and (USC), where

(USC) : lim sup β∈Rd

+,β→β0

Rt(β) − Rt(β0) ∈ −Kt for all β0∈ Rd+.

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The Closedness Property

Theorem :AL₀(T ) is closed in probability under NA2L.

The same holds for AR₀(T ) under NMA2 and (USC), where

+,β→β0

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The Closedness Property

Theorem :AL₀(T ) is closed in probability under NA2L. The same holds for AR₀(T ) under NMA2 and (USC), where

+,β→β0

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Application - Super Replication Theorem

Under some additional assumptions

• λRt(β1) + (1 − λ)Rt(β2) − Rt(λβ1+ (1 − α)β2) ∈ −Kt, • Rt(β)−∈ L∞(Rd, F ) for β ∈ L∞(Rd+, F ).

Proposition :Assume that NMA2 holds. Let V ∈ L0(Rd, F ) be such that V + κ ∈ L0(KT, F ) for some κ ∈ Rd. Then the following are equivalent :

(i) V ∈ AR 0(T ),

(ii) E [ZT0 V ] ≤ αR(Z ) for all Z ∈ MT0.

If moreover limη→∞Rt+1(ηβ)/η = Lt+1β then the following are equivalent :

(i) V ∈ AR₀(T ),

(ii) E [ZT0 V ] ≤ αR(Z ) for all Z ∈ MT0∩LT0.

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Application - Super Replication Theorem

(i) V ∈ AR₀(T ),

(ii) E [Z_T0 V ] ≤ αR(Z ) for all Z ∈ MT0∩LT0.

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Application - Super Replication Theorem

(i) V ∈ AR₀(T ),

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Application - Super Replication Theorem

(i) V ∈ AR₀(T ),

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Application - Portfolio optimization

Setting

• _{U a P − a.s. upper continuous, concave, random map from} Rd to ] − ∞, 1],

• U(V ) = −∞ on {V /∈ KT}, • U (x₀) :=V ∈ AR

0(T ) : E [|U(x0+ V )|] < ∞ 6= ∅.

Proposition :If NMA2 , (USC) hold and AR₀(T ) is convex, then ∃V (x0) ∈ AR0(T ) such that

E [U(x0+ V (x0))] = sup V ∈U (x0)

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Next steps

• Extension to continuous time ;

• Specification of a realistic production function ; • Caracterization of MT₀ ∩ LT

0 ; • Numerical implementation...