Energy Markets III: Emissions Cap-and-Trade …...Energy Markets III: Emissions Cap-and-Trade Market Models Rene Carmona´ Bendheim Center for Finance Department of Operations Research

Energy Markets III:Emissions Cap-and-Trade Market Models

Rene Carmona

Bendheim Center for FinanceDepartment of Operations Research & Financial Engineering

Princeton University

Munich, February 13, 2009

Carmona Energy Markets, Munich

Emission Trading

SOx and NOx TradingHave existed in the US for a long timeLiquidity and Price Collapse Issues

Cap & Trade for Green House Gases (Kyoto)Carbon Markets (RGGI started Sept. 25 2008)Lessons learned from the EU Experience

Mathematical (Equilibrium) ModelsFor emission credits only (RC-Fehr-Hinz)Joint for Electricity and Emission credits (RC-Fehr-Hinz-Porchet)Calibration & Option Pricing (RC-Fehr-Hinz)

Computer ImplementationsSeveral case studies (Texas, Japan)Practical Tools for Regulators and Policy Makers


(Simplified) Cap-and-Trade Scheme: Data

Regulator Input at inception of program (i.e. time t = 0)INITIAL DISTRIBUTION of allowance certificates θ0

Set PENALTY π per ton of CO2 equivalent emitted and NOT offsetby allowance certificate at time of compliance

Given exogenously{Dt}t=0,1,·,T daily demand for electricity

{Cnt }t=0,1,·,T production cost for 1MWh of electricity from nuclear

plant{Cg

t }t=0,1,·,T production cost for 1MWh of electricity from gas plant{Cc

t }t=0,1,·,T production cost for 1MWh of electricity from coal plant

Known physical characteristicsen emission (in CO2 ton-equivalent) for 1MWh from nuclear planteg emission (in CO2 ton-equivalent) for 1MWh from gas plantec emission (in CO2 ton-equivalent) for 1MWh from coal plant


(Simplified) Cap-and-Trade Scheme: Outcome

{St}t=0,1,·,T daily price of electricity{At}t=0,1,·,T daily price of a credit allowanceProduction schedules

{ξnt }t=0,1,·,T daily production of electricity from nuclear plant{ξg

t }t=0,1,·,T production of electricity from gas plant{ξc

t }t=0,1,·,T production of electricity from coal plant

Inelasticity constraint

ξnt + ξ

gt + ξc

t = Dt t = 0,1, · · · ,T

Daily Production Profits & Losses

ξnt (St−cn

t )+ξgt (St−cg

t )+ξct (St−cc

t ) =(DtSt − (ξn

t cnt + ξ

gt cg

t + ξct cc

t ))

(possible) Pollution Penalty

π

(T∑

t=0

(ξnt en + ξ

gt eg + ξc

t eg)− θ0

)+


EU ETS First Phase: Main Criticism

No (Significant) Emissions ReductionDID Emissions go down?Yes, but as part of an existing trend

Significant Increase in PricesCost of Pollution passed along to the ”end-consumer”Small proportion (40%) of polluters involved in EU ETS

Windfall ProfitsCannot be avoidedProposed Remedies

Stop Giving Allowance Certificates Away for Free !Auctioning


What Happened? Falling Carbon Prices

Figure A: M arket Price of the EUA from Decem ber 2004 through Decem ber 2007

(Pointcarbon)


More Historical Prices: CDM?

26

24

22

20

18

16

14

12

10

8

6

4

2

0

Price Spot - Powernext Carbon Price Futures Dec 08 - ECX CER Price Dec 08 - Reuters

€ /

ton

of C

O2

Spot Price 0.05 €

Nov07

Oct07

Sep07

Aug07

Jul07

Jun07

May07

Apr07

Mar07

Feb07

Jan07

Dec06

Nov06

Futures Price Dec 08

22.35 €

CER Price Dec 08

17.78 €

Carbon prices: spot price – 1st period 2005-2007

futures price Dec.08 – 2nd period 2008-2012

and CER price Dec. 08


Description of the Economy

Finite set I of risk neutral agents/firmsProducing a finite set K of goodsFirm i ∈ I can use technology j ∈ J i,k to produce good k ∈ KDiscrete time {0,1, · · · ,T}Inelastic Demand

{Dk (t); t = 0,1, · · · ,T − 1, k ∈ K}.

· · · · · · · · · · · ·


Regulator Input (EU ETS)

At inception of program (i.e. time t = 0)INITIAL DISTRIBUTION of allowance certificates

θi0 to agent i ∈ I

Set PENALTY π for emission unit NOT offset by allowancecertificate at end of compliance period

Variations (not discussed in this talk)

Risk aversion and agent preferences (existence theory easy)

Auctioning of allowances (redistribution of P&L’s)

Distributionover time of allowances (stochastic game theory)

Elastic demand (e.g. smart meters)

Multi-period period lending and borrowing (more realistic)

· · · · · · · · · · · ·


Goal of Equilibrium Analysis

Find two stochastic processesPrice of one allowance

A = {At}t≥0

Prices of goodsS = {Sk

t }k∈K , t≥0

satisfying the usual conditions for the existence of a

competitive equilibrium

(to be spelled out below).


Individual Firm ProblemDuring each time period [t , t + 1)

Firm i ∈ I produces ξi,j,kt of good k ∈ K with technology j ∈ J i,k

Firm i ∈ I holds a position θit in emission credits

LA,S,i (θi , ξi ) :=Xk∈K

Xj∈J i,k

T−1Xt=0

(Skt − C i,j,k

t )ξi,j,kt

+ θi0A0 +

T−1Xt=0

θit+1(At+1 − At )− θi

T +1AT

− π(Γi + Πi (ξi )− θiT +1)+

where

Γi random, Πi (ξi ) :=Xk∈K

Xj∈J i,k

T−1Xt=0

ei,j,kξi,j,kt

Problem for (risk neutral) firm i ∈ I

max(θi ,ξi )

E{LA,S,i (θi , ξi )}


In the Absence of Cap-and-Trade Scheme (i.e. π = 0)If (A∗,S∗) is an equilibrium, the optimization problem of firm i is

sup(θi ,ξi )

E

24Xk∈K

Xj∈J i,k

T−1Xt=0

(Skt − C i,j,k

t )ξi,j,kt + θi

0A0 +

T−1Xt=0

θit+1(At+1 − At )− θi

T +1AT

35We have A∗t = Et [A∗t+1] for all t and A∗T = 0 (hence A∗t ≡ 0!)

Classical competitive equilibrium problem where each agent maximizes

supξi∈U i

E

24Xk∈K

Xj∈J i,k

T−1Xt=0

(Skt − C i,j,k

t )ξi,j,kt

35 , (1)

and the equilibrium prices S∗ are set so that supply meets demand. For each time t

((ξ∗i,j,kt )j,k )i = arg max((ξ

i,j,kt )J i,k )i∈I

Xi∈I

Xj∈J i,k

−C i,j,kt ξ

i,j,kt

Xi∈I

Xj∈J i,k

ξi,j,kt = Dk

t

ξi,j,kt ≤ κi,j,k for i ∈ I, j ∈ J i,k

ξi,j,kt ≥ 0 for i ∈ I, j ∈ J i,k


Business As Usual (cont.)

The corresponding prices of the goods are

S∗kt = maxi∈I, j∈J i,k

C i,j,kt 1{ξ∗i,j,k

t >0},

Classical MERIT ORDERAt each time t and for each good k

Production technologies ranked by increasing production costs C i,j,kt

Demand Dkt met by producing from the cheapest technology first

Equilibrium spot price is the marginal cost of production of the mostexpansive production technoligy used to meet demand

Business As Usual(typical scenario in Deregulated electricity markets)


Equilibrium Definition for Emissions Market

The processes A∗ = {A∗t }t=0,1,··· ,T and S∗ = {S∗t }t=0,1,··· ,T form anequilibrium if for each agent i ∈ I there exist strategiesθ∗i = {θ∗it }t=0,1,··· ,T (trading) and ξ∗i = {ξ∗it }t=0,1,··· ,T (production)

(i) All financial positions are in constant net supply∑i∈I

θ∗it =∑i∈I

θi0, ∀ t = 0, . . . ,T + 1

(ii) Supply of each good meets demand∑i∈I

∑j∈J i,k

ξ∗i,j,kt = Dkt , ∀ k ∈ K, t = 0, . . . ,T − 1

(iii) Each agent i ∈ I is satisfied by its own strategy

E[LA∗,S∗,i (θ∗i , ξ∗i )] ≥ E[LA∗,S∗,i (θi , ξi )] for all (θi , ξi )


Necessary Conditions

Assume(A∗,S∗) is an equilibrium(θ∗i , ξ∗i ) optimal strategy of agent i ∈ I

thenThe allowance price A∗ is a bounded martingale in [0, π]

Its terminal value is given by

A∗T = π1{Γi +Π(ξ∗i )−θ∗iT +1≥0} = π1{Pi∈I(Γi +Π(ξ∗i )−θ∗i

0 )≥0}

The spot prices S∗k of the goods and the optimal productionstrategies ξ∗i are given by the merit order for the equilibriumwith adjusted costs

C i,j,kt = C i,j,k

t + ei,j,k A∗t


Social Cost Minimization Problem

Overall production costs

C(ξ) :=

T−1Xt=0

X(i,j,k)

ξi,j,kt C i,j,k

t .

Overall cumulative emissions

Γ :=Xi∈I

Γi Π(ξ) :=

T−1Xt=0

X(i,j,k)

ei,j,kξi,j,kt ,

Total allowancesθ0 :=

Xi∈I

θi0

The total social costs from production and penalty payments

G(ξ) := C(ξ) + π(Γ + Π(ξ)− θ0)+

We introduce the global optimization problem

ξ∗ = arg infξmeets demands

E[G(ξ)],


Social Cost Minimization Problem (cont.)

First Theoretical ResultThere exists a set ξ∗ = (ξ∗i )i∈I realizing the minimum social cost

Second Theoretical Result(i) If ξ minimizes the social cost, then the processes (A,S) defined by

At = πPt{Γ + Π(ξ)− θ0 ≥ 0}, t = 0, . . . ,T

and

Skt = max

i∈I, j∈J i,k(C i,j,k

t +ei,j,kt At )1{ξi,j,k

t >0}, t = 0, . . . ,T−1 k ∈ K ,

form a market equilibrium with associated production strategy ξ(ii) If (A∗,S∗) is an equilibrium with corresponding strategies (θ∗, ξ∗),

then ξ∗ solves the social cost minimization problem(iii) The equilibrium allowance price is unique.


Effect of the Penalty on Emissions

��

��

��

��

��

��

��

��

��

��

��

��

��


Equilibrium Sample Paths

��

��

��

��

��

��

��

��

��

��

��


Costs in a Cap-and-Trade

Consumer Burden Xt

Xk

(Sk,∗t − Sk,BAU∗

t )Dkt .

Reduction Costs (producers’ burden)Xt

Xi,j,k

(ξi,j,k∗t − ξBAU,i,j,k∗

t )C i,j,kt

Excess ProfitXt

Xk

(Sk,∗t −Sk,BAU∗

t )Dkt −

Xt

Xi,j,k

(ξi,j,k∗t −ξBAU,i,j,k∗

t )C i,j,kt −π(

Xt

Xijk

ξijkt eijk

t −θ0)+

Windfall Profits

WP =T−1Xt=0

Xk∈K

(S∗kt − Skt )Dk

t

whereSk

t := maxi∈I,j∈J i,k

C i,j,kt 1{ξ∗i,j,k

t >0}.


Costs in a Cap-and-Trade Scheme

��

��

��

��

��

��

��

� ��

��

��

��

��

��

�

�

�

��

Histograms of the difference between the consumer cost, social cost, windfallprofits and penalty payments of a standard cap-and-trade scheme calibratedto reach the emissions target with 95% probability and BAU.


One of many Possible GeneralizationsIntroduction of Taxes / Subsidies

LA,S,i (θi , ξi ) = −T−1∑t=0

Git +∑k∈K

∑j∈J i,k

T−1∑t=0

(Skt − C i,j,k

t − Hkt )ξi,j,k

t

+T−1∑t=0

θit (At+1 − At )− θi

T AT

− π(Γi + Πi (ξi )− θiT )+.

In this caseIn equilibrium, production and trading strategies remain thesame (θ†, ξ†) = (θ∗, ξ∗)

Abatement costs and Emissions reductions are also the sameNew equilibrium prices (A†,S†) given by

A†t = A∗t for all t = 0, . . . ,T (2)

S†kt = S∗kt + Hkt for all k ∈ K , t = 0, . . . ,T − 1 (3)

Cost of the tax passed along to the end consumerCarmona Energy Markets, Munich

Alternative Market Design

Currently Regulator SpecifiesPenalty πOverall Certificate Allocation θ0 (=

Pi∈I θ

i0)

Alternative Scheme (Still) Controlled by Regulator(i) Sets penalty level π(ii) Allocates allowances

θ′0 at inception of program t = 0then proportionally to production

yξi,j,kt to agent i for producing ξi,j,k

t of good k with technology j

(iii) Calibrates y , e.g. in expectation.

y =θ0 − θ′0PT−1

t=0

Pk∈K E{Dk

t }

So total number of credit allowance is the same in expectation, i.e.θ0 = E{θ′0 + y

PT−1t=0

Pk∈K Dk

t }


Yearly Emissions Equilibrium Distributions

��

��

��

��

��

��

��

��

��

�� !� ��

��

��

�

�

��

��

��

Yearly emissions from electricity production for the Standard Scheme, theRelative Scheme, a Tax Scheme and BAU.


Abatement Costs

��

��

��

��

��

��

��

��

��

��

��

��

��

��

�

�

��

Yearly abatement costs for the Standard Scheme, the Relative Scheme and aTax Scheme.


Windfall Profits

��

��

��

��

��

��

� ��

��

��

��

�� !��"#��

��

�

�

�

��

Histograms of the yearly distribution of windfall profits for the StandardScheme, a Relative Scheme, a Standard Scheme with 100% Auction and aTax Scheme


Japan Case Study: Windfall Profits

��

��

��

��

��

��

��

� ��

��

��

��

��

��

�

�

�

��

Histograms of the difference of consumer cost, social cost, windfall profitsand penalty payments between BAU and a standard trading scheme scenariowith a cap of 300Mt CO2. Notice that taking into account fuel switching evena reduction to 1990 emission levels is not very expensive (below2Dollar/MWh). (Rene: Japan is discussing to change their reduction target toa reduction relative to their 2005 emission level. Due to extra coal firedproduction they had a huge increase in emissions since 1990 and are afraidthat their target which means a 20percent reduction from todays emissionlevel is too expensive). The low reduction costs and windfall profits comparedto Texas are due to a downsloping linear trend of fuel switch prices. Today theprice is 85$ per MWh in average. With todays down-sloping trend it will be 50$ per MWh in 2012.


Japan Case Study: More Windfall Profits

��

��

��

��

��

��

��

� ��

��

��

��

��

��

�

�

�

��

Histograms of the consumer cost, social cost, windfall profits and penaltypayments under a standard trading scheme scenario with a cap of330MtCO2.


Japan Case Study: Consumer Costs

��

��

��

��

��

��

��

��

� ��

��

��

��

�� ! "��

��

�

�

�

��

Histogram of the yearly distribution of consumer costs for the StandardScheme, a Relative Scheme and a Tax Scheme. Notice that the StandardScheme with Auction possesses the same consumer costs as the StandardScheme. Carmona Energy Markets, Munich

Numerical Results: Windfall Profits

1.451.51.551.61.651.71.751.8

x 108

00.1

0.20.3

0.40.5

−2

−1

0

1

2

3

4

5

6

7

x 109

ye

Windfall Profits

θ+E(ye D)

w

1.451.51.551.61.651.71.751.8

x 108

00.1

0.20.3

0.40.5

1.45

1.5

1.55

1.6

1.65

1.7

1.75

x 108

ye

95% Quantile of Emissions

θ+E(ye D)

q

Windfall profits (left) and 95% percentile of total emissions (right) as functions of therelative allocation parameter and the expected allocation


More Numerical Results: Windfall Profits

1.45 1.5 1.55 1.6 1.65 1.7 1.75 1.8

x 108

0

0.1

0.2

0.3

0.4

0.5

150000000

160000000

170000000

−2000000000

0

2000000000

4000000000

6000000000

ye

θ+E(ye D)

Level Sets

60 70 80 90 100 110 120 130 1400

2

4

6

8

10

12

14

16x 10

8 Social Cost

Do

lla

r

Penalty

Standard SchemeRelative Scheme

(left) Level sets of previous plots. (right) Production costs for electricity for one year asfunction of the penalty level for both the absolute and relative schemes.


Equilibrium Models: (Temporary) Conclusions

Market Mechanisms CANNOT solve all the pollution problemsCap-and-Trade Schemes CAN Work!

Given the right emission targetUsing the appropriate tool to allocate emissions creditsSignificant Windfall Profits for Standard Schemes

TaxesPolitically unpopularCannot reach emissions targets

AuctioningFairness is Smoke Screen: Re-distribution of the cost

Relative SchemesCan Reach Emissions TargetPossible Control of Windfall ProfitsMinimize Social Costs

Extensions of the Present Work (Sharpening the ToolsIncluding Risk Averse Agents and Inelastic DemandsStatistical Analysis of Equilibrium PricesExogenous Prices and Large Scale Case StudiesOther Schemes (e.e. California Low Emissions Fuel Standards)


Reduced Form Models & Option Pricing

Emissions Cap-and-Trade Markets SOON to exist in the USOption Market SOON to develop

Underlying {At}t non-negative martingale with binary terminalvalueCan think of At as of a binary optionUnderlying of binary option should be Emissions

Need for Formulae (closed or computable)for Pricesfor Hedges

Reduced Form Models


Reduced Form Model for Emissions Abatement

{Xt}t actual emissions at time tdXt = σ(t ,Xt )dWt − ξtdt

ξt abatement (in ton of CO2) at time tXt = Et −

R t0 ξsds

cumulative emissions in BAU minus abatement up to time t

π(XT − K )+ penaltyT maturity (end of compliance period)K regulator emissions’ targetπ penalty (40 EURO) per ton of CO2 not offset by an allowancecertificate

Social Cost E{∫ T

0 C(ξs)ds + π(XT − K )+}C(ξ) cost of abatement of ξ ton of CO2


Representative Agent Stochastic Control Problem

Informed Planner Problem

infξ={ξt}0≤t≤T

E{∫ T

0C(ξs)ds + π(XT − K )+}

Value Function

V (t , x) = inf{ξs}t≤s≤T

E{∫ T

tC(ξs)ds + π(XT − K )+|Xt = x}

HJB equation (e.g. C(ξ) = ξ2)

Vt +12σ(t , x)2Vxx −

12

V 2x


Calibration

Emission Allowance Price

At = Vx (t ,Xt )

Emission Allowance Volatility

σA(t) = σ(t ,Xt )Vxx (t ,Xt )

Calibration (σ(t) deterministic)Multiperiod (Cetin. et al)Close Form Formulae for PricesClose Form Formulae for Hedges


References (personal) Others in the Text

Emissions MarketsR.C., M. Fehr and J. Hinz: Mathematical Equilibrium and MarketDesign for Emissions Markets Trading Schemes. QuantitativeFinance (2009)R.C., M. Fehr, J. Hinz and A. Porchet: Mathematical Equilibriumand Market Design for Emissions Markets Trading Schemes. SIAMReview (2009)R.C., M. Fehr and J. Hinz: Mathematical Properly DesignedEmissions Trading Schemes do Work! (working paper)R.C., M. Fehr and J. Hinz: Calibration and Risk Neutral Dynamicsof Carbon Emission Allowances (working paper)R.C. & M. Fehr: Mathematical Models for the Clean DevelopmentMechanism and Emissions Markets. (in preparation)


Documents

Energy Markets III: Emissions Cap-and-Trade …...Energy Markets III: Emissions Cap-and-Trade Market Models Rene Carmona´ Bendheim Center for Finance Department of Operations Research