A Structural Model for Carbon Cap-and-Trade Schemes

A Structural Model for Carbon Cap-and-Trade Schemes Sam Howison and Daniel Schwarz University of Oxford, Oxford-Man Institute The New Commodity Markets Oxford-Man Institute, 15 June 2011

Introduction

The Need for Emission Reduction Which Policy Options Should we Use? Aim is to reduce emission of atmospheric gases such as carbon dioxide, methane, ozone and water vapour. Different policy options: Emission norm (direct regulation), Emission tax (market based), Emissions trading (market based).

Emissions Trading How Does it Work? For each country that participates in an emissions trading scheme Impose binding limit (cap) on the cumulative emissions during one year (compliance period) and penalise excess emissions (monetary penalty). Divide cap into equal amounts, which define one unit (AAU). Print paper certificates (allowances) representing one AAU and distribute to firms (initial allocation). Trade allowances.

Full equilibrium models: Modelling Approaches 1. R. Carmona et al., Market design for emission trading schemes, 2009 2. R. Carmona et al., Optimal stochastic control and carbon price formation, 2009 Stylized risk-neutral models: 1. R. Carmona et al., Risk-neutral models for emission allowance prices and option valuation, 2010. 2. K. Borovkov et al., Jump-diffusion modelling in emission markets, 2010. 3. J. Hinz et al., On fair pricing of emission-related derivatives, 2010. Structural approach: 1. M. Coulon, Modelling Price Dynamics Through Fundamental Drivers in Electricity and Other Energy Markets, 2009 2. S. Howison and D. Schwarz, Risk-neutral pricing of financial instruments in emission markets, 2010.

From Electricity to CO 2 Emissions

[0, T ] Market Setup Introducing the Key Drivers of the Pricing Model finite time interval (Ω, F, (F t ), P) (F t ) generated by (W t ) R 2 ξ max market s capacity to produce electricity (ξ t ) supply as a proportion of capacity (0 ξ t 1) (D t ) demand as a proportion of capacity (0 D t 1) Walrasian equilibrium assumption, D t = ξ t. (E t ) cumulative emissions up to time t (A t ) allowance certificate price

The Bid- and Emissions Stack Price Setting and Emission Measurement in Electricity Markets Assumption The market administrator arranges generators bids in increasing order of bid price (merit order).

The Bid- and Emissions Stack Price Setting and Emission Measurement in Electricity Markets Assumption The market administrator arranges generators bids in increasing order of bid price (merit order). Definition The business-as-usual bid stack is given by b BAU (ξ) a map from the marginal production unit to the corresponding price of electricity (e per MWh), db BAU dξ (ξ) > 0. Bid Stack

The Bid- and Emissions Stack Price Setting and Emission Measurement in Electricity Markets Definition The marginal emissions stack is given by e(ξ) a map from the marginal production unit to its marginal emissions rate (tco 2 per MWh). Emissions Stack To obtain the business-as-usual market emissions rate, µ BAU e (ξ), integrate the marginal emissions stack up to current level of demand: D µ BAU e (D) := e(ξ) dξ. 0

Load Shifting The Effects of Emissions Trading on the BAU Economy With emissions trading, bids increase by (cost of carbon) (marginal emissions rate). Carbon Impact

Load Shifting The Effects of Emissions Trading on the BAU Economy With emissions trading, bids increase by (cost of carbon) (marginal emissions rate). Carbon Impact Given an allowance price A 0, the bid stack now becomes b(ξ; A) := b BAU (ξ) + Ae(ξ). For A > 0 this function may no longer be monotonic!

Load Shifting The Effects of Emissions Trading on the BAU Economy Under the previous assumptions, given electricity price P and allowance price A, active generators are uniquely identified by { } S(A, P) := ξ [0, 1] : b BAU (ξ) + Ae(ξ) P. Further, given demand D, the market price of electricity P is P(A, D) = inf {P 0 : λ(s(a, P)) D}, where λ( ) denotes the Lebesgue measure.

600 b BAU ( ) The Bid Stack under BAU Price (Euro/MWh) 100 0 D 1 Supply (as a fraction of the market capacity) 1 The Emissions Stack under BAU e( ) Marginal Emissions (tco 2 /MWh) 0.4 0 D 1 Supply (as a fraction of the market capacity)

600 b BAU ( ) b( ;A), A=100 The Bid Stack under Cap and Trade Price (Euro/MWh) 100 0 ξ 1 D ξ 2 1 Supply (as a fraction of the market capacity) 1 The Emissions Stack under Cap and Trade Marginal Emissions (tco 2 /MWh) 0.4 0 ξ 1 D ξ 2 1 Supply (as a fraction of the market capacity)

Allowance Pricing

Market Assumptions Applying the Risk-Neutral Pricing Methodology Traded assets in the market are Allowance certificates Derivatives written on the certificate Riskless money market account Assumption There exists an equivalent (risk-neutral) martingale measure P, under which, for 0 t T, the discounted price of any tradable asset is a martingale.

Market Assumptions Concretising the Demand and Emissions Process Demand (as a fraction of capacity ξ max ) evolves according to the following Itô diffusion; for 0 t T, dd t = η(d t D(t))dt+ 2ησ d D t (1 D t )d W 1 t, D 0 = d (0, 1). With this definition D t (0, 1) t [0, T ]. Cumulative emissions have drift µ e and we allow for uncertainty by adding a volatility term σ e. Then, for 0 t T, de t = µ e (A t, D t )dt + σ e d W 2 t, E 0 = 0.

Γ cap 0 π 0 Market with One Compliance Period Characterising the Allowance Price at t = T inital allocation of certificates monetary penalty {E T Γ cap } non-compliance event

Γ cap 0 π 0 Market with One Compliance Period Characterising the Allowance Price at t = T inital allocation of certificates monetary penalty {E T Γ cap } non-compliance event Terminal value of the allowance certificate is A T = πi {ET Γ cap}. As a traded asset, A t is given by A t = e r(t t) πẽ [ ] I {ET Γ cap} F t, for 0 t T. Martingale Representation Theorem: d ( e rt A t ) = Z 1 t d W 1 t + Z 2 t d W 2 t, for 0 t T and some F t -adapted process (Z t ) := ( Zt 1, Zt 2 ).

Market with One Compliance Period FBSDE Formulation of the Pricing Problem Combining the processes for demand, cumulative emissions and the allowance certificate leads to the FBSDE dd t = µ d (t, D t )dt + σ d (D t )d W t 1, D 0 = d (0, 1), de t = µ e (A t, D t )dt + σ e (E t )d W t 2, E 0 = 0, da t = ra t dt + Zt 1 d W t 1 + Zt 2 d W t 2, A T = πi {ET Γ cap}. (D t, E t ) forward part (A t ) (Z t ) backward part generator

Market with One Compliance Period PDE Representation of the FBSDE Solution Let A t = α(t, D t, E t ), then α t +1 2 σ2 d (D) 2 α D 2 +1 2 σ2 e(e) 2 α E 2 +µ d(t, D) α D +µ e(α, D) α rα = 0, E with terminal condition α(t, D, E) = πi {E Γcap}, 0 D 1, E 0. The solution α also satisfies lim α(t, D, E) = E e r(t t) π, 0 t T, 0 D 1.

Allowance Certificate Price at t=0.5 T (market with one compliance period) Allowance Certificate Price at t=t (market with one compliance period) 100 100 Price (Euro/MWh) Price (Euro/MWh) 0 1 1 0 1 1 Cumulative Emissions (as a fraction of the initial allocation) 0 Demand (as a fraction of the market capacity) Cumulative Emissions (as a fraction of the initial allocation) 0 Demand (as a fraction of the market capacity)

Market with Multiple Compliance Periods Banking Banking: an additional incentive to reduce emissions. Banking 1 1 Γ cap E T 1 1 Γ E 1 2 cap T Γ 1 cap E 2 T 2 0 T 1 T 2 In the event of compliance, a number ( Γ 1 cap E 1) of certificates with price A 1 T 1 are exchanged for certificates valid during the next compliance period, with price A 2 T 1.

Market with Multiple Compliance Periods Withdrawal Withdrawal: additional punishment for excess emissions. Withdrawal 1 Γ cap E 1 T 1 1 Γ E 1 2 cap T Γ 1 cap E 2 T 2 0 T 1 T 2 In the event of non-compliance, a number min ( E 1 Γ 1 cap, Γ 2 cap) of certificates with price A 2 T 1 are subtracted from Γ 2 cap.

Market with Multiple Compliance Periods Banking and Withdrawal Aggregate supply of certificates during the second compliance period: (initial allocation) + (banked or withdrawn certificates). ˆΓ 2 cap = ( Γ 2 cap + Γ 1 cap E 1) +. The terminal condition for the allowance price becomes A 1 T 1 = A 2 T 1 I {ET1 <Γ 1 cap} + ( π 1 + A 2 T 1 ) I{Γ 1 cap E T1 <Γ 1 cap+γ 2 cap} + ( π 1 + π 2) I {ET1 Γ 1 cap+γ 2 cap}.

Allowance Certificate Price at t=t (market with one compliance period) Allowance Certificate Price at t=t1 (market with two compliance periods; banking and withdrawal) 100 Price (Euro/MWh) Price (Euro) 100 0 1 1 0 1 1 Cumulative Emissions (as a fraction of the initial allocation) 0 Demand (as a fraction of the market capacity) Cumulative Emissions (as a fraction of the initial allocation) 0 Demand (as a fraction of the market capacity)

Market with Multiple Compliance Periods Borrowing Borrowing: decreases the probability of non-compliance. Borrowing E 1 1 Γ T cap 1 1 1 Γ cap E T 1 1 Γ cap E 1 T 1 2 Γ cap E 2 T 2 0 T 1 T 2 If the emissions exceed the current compliance period s cap, a number min ( E 1 Γ 1 cap, Γ 2 cap) of certificates with price A 2 T1 are borrowed from Γ 2 cap.

Market with Multiple Compliance Periods Banking, Borrowing and Withdrawal Aggregate supply of certificates: as before. ˆΓ 2 cap = ( Γ 2 cap + Γ 1 cap E 1) +.

Market with Multiple Compliance Periods Banking, Borrowing and Withdrawal Aggregate supply of certificates: as before. ˆΓ 2 cap = ( Γ 2 cap + Γ 1 cap E 1) +. The terminal condition for the allowance price becomes A 1 T 1 = A 2 T 1 I {ET1 <Γ 1 cap +Γ2 cap} + ( π 1 + π 2) I {ET1 Γ 1 cap +Γ2 cap}.

Allowance Certificate Price at t=t1 (market with two compliance periods; banking and withdrawal) Allowance Certificate Price at t=t1 (market with two compliance periods; borrowing, banking and withdrawal) 100 Price (Euro) 100 Price (Euro) 0 1 1 0 1 1 Cumulative Emissions (as a fraction of the initial allocation) 0 Demand (as a fraction of the market capacity) Cumulative Emissions (as a fraction of the initial allocation) 0 Demand (as a fraction of the market capacity)

Option Pricing

The European Call on the Allowance Certificate Formulating the Pricing Problem Our example of choice is a European call (C t (τ)) t [0,τ] with maturity τ, where 0 τ T, and strike K 0. Its payoff is C τ (τ) := (A τ K) +. Require knowledge of A t need to solve problem for A t and for C t in parallel. The option price does not affect the rate at which firms emit expect the option pricing problem to be linear.

The European Call on the Allowance Certificate The Pricing PDE Letting C t = v(t, D t, E t ), v t +1 2 σ2 d(d) 2 v D 2 +1 2 σ2 e (E) 2 v E 2 +µ d(t, D) v D +µ e(α(t, D, E), D) v rv = 0, E with terminal condition Further, v satisfies v(t, D, E) = (A τ K) +, 0 D 1, E 0. ( ) + lim v(t, D, E) = E e r(t t) π e r(t τ) K, 0 t τ, 0 D 1.

Call Option Price at t=τ 100 K Price (Euro) 0 1 1 Cumulative Emissions (as a fraction of the initial allocation) 0 Demand (as a fraction of the market capacity)

Work in Progress

Work in Progress Analysis of Emissions Uncertainty Characterise the allowance price A t := α(t, D t, E t ), as σ e 0 and the pricing problem develops an additional singularity.

Work in Progress Analysis of Emissions Uncertainty Characterise the allowance price A t := α(t, D t, E t ), as σ e 0 and the pricing problem develops an additional singularity. Solution of corresponding first order PDE α t + µ d(t, D) α D + µ e(α, D) α E rα = 0 with terminal condition α(t, D, E) = πi {E Γcap} exhibits a rarefaction wave. Characteristics meet in one line (T, D, Γ cap ).

A ɛ,δ t Work in Progress Asymptotic Analysis of Pricing Problem (c.f. Howison, Schwarz 2011) := α ɛ,δ (t, D ɛ t, E δ t ): value of the allowance price assuming that Demand is fast mean-reverting; The impact of load shifting is small. Formal expansion α ɛ,δ = α 0,0 + ɛα 1,0 + δα 0,1 + O(ɛ) + O(δ 2 ), we find that α 0,0 (t, E) satisfies α 0,0 t + 1 2 σ2 e 2 α ( 0,0 E 2 + µ BAU e ) α0,0 (D), Φ E rα 0,0 = 0, with terminal condition α 0,0 (T, E) := πi {E Γcap}.

Work in Progress Valuation of Power Plants in a Cap-and-Trade market (c.f. Carmona, Coulon, Schwarz, 2011) Let (S c t ) and (S g t ) denote the price processes of coal and gas. Extend the suggested model to a stochastic bid stack model, in which the electricity price P t is given by and the allowance price A t by P t := b(a t, D t, S c t, S g t ) A t := α(t, D t, E t, S c t, S g t ). For maturities T 1, T 2,..., T n, closed form approximation to the price of spread options and hence value a gas plant v g, where v g (t, A t, D t, S c t, S g t ) := T n ( PT h g S g T e g A T ) + T =T 1 and h g, e g denote the heat and emissions rate of the plant under consideration.

Future Work Real Option Problem: When is it optimal to invest in a large abatement project, which changes the BAU emissions stack forever? Market Design: Analysis of alternatives to standard cap-and-trade schemes; e.g. allow the regulator to adjust the cap at specific times throughout the trading period.

Conclusion

Conclusion In this talk we showed: Bid stack contains information about active generators (in principal). Allowances can be regarded as derivatives on demand for polluting goods and cumulative emissions. Borrowing, banking and withdrawal can be analysed in multi-period setting. Options on allowances can be priced in this structural modelling framework.

Thank you for your attention. Questions?

600 b BAU ( ) The Bid Stack under BAU Price (Euro/MWh) 100 0 D 1 Supply (as a fraction of the market capacity)

1 The Emissions Stack under BAU e( ) Marginal Emissions (tco 2 /MWh) 0.4 0 D 1 Supply (as a fraction of the market capacity)

Market Bids, Low Carbon Cost Market Bids, High Carbon Cost Market Bids, Rearranged Price (Euro/MWh) Coal Gas Price (Euro/MWh) Coal Gas Price (Euro/MWh) Gas Coal Marginal Emissions (tco 2 /MWh) 0 Supply (MW) Emission Stack, Low Carbon Cost Coal Gas 0 Supply (MW) Marginal Emissions (tco 2 /MWh) 0 Supply (MW) Emission Stack, High Carbon Cost Coal Gas 0 Supply (MW) Marginal Emissions (tco 2 /MWh) 0 Supply (MW) Emission Stack, Rearranged Gas Coal 0 Supply (MW)