On the pricing of emission allowances

On the pricing of emission allowances Umut Çetin Department of Statistics London School of Economics Umut Çetin (LSE) Pricing carbon 1 / 30

Kyoto protocol The Kyoto protocol opened for signature at the 1997 conference in Kyoto, Japan. Signatory nations must reduce their emissions for carbon dioxide and five other gases in 2008-2012 by 5% with respect to 1990 levels in order to comply with the protocol. Emissions Trading, the Clean Development Mechanism (CDM) and Joint Implementation (JI) are mechanisms defined under the Kyoto Protocol intended to lower the overall costs of achieving its emissions targets. Through the JI, any Annex I country can invest in emission reduction projects in any other Annex I country as an alternative to reducing emissions domestically. Through the CDM, countries can meet their domestic emission reduction targets by buying greenhouse gas reduction units from (projects in) non Annex I countries. Umut Çetin (LSE) Pricing carbon 2 / 30

EU ETS and carbon credits The European Commission (EC) launched the European Climate Change Programme (ECCP) in June 2000 with the objective to identify, develop and implement the essential elements of an EU strategy to implement the Kyoto Protocol. The European Union Emission Trading System (EU ETS) is a significant part of the ECCP and currently constitutes the largest emissions trading scheme in the world. The ETS now operates in 30 countries and covers carbon emissions from installations such as power stations, combustion plants, oil refineries and iron and steel works, as well as factories making cement, glass, lime, bricks, ceramics, pulp, paper and board. It covers some 11,000 power stations and industrial plants. Airlines will join the scheme in 2012. The EU ETS will be further expanded to the petrochemicals, ammonia and aluminium industries and to additional gases in 2013, when the third trading period will start. Umut Çetin (LSE) Pricing carbon 3 / 30

To participate in the ETS, EU member states had to first submit a National Allocation Plan (NAP) for approval to the EC in Phase I (2005-2007) and Phase II (2008-2012). For the third trading period, the allocation will be directly made at the EU level. The EU ETS works on the cap and trade principle. Within this cap, every year companies receive emission allowances which they can trade as needed. Selected carbon intensive installations receive (mostly) free emission credits under the terms of this NAP, enabling them to emit greenhouse gases up to the assigned tonnage. From 2013 free allocation of the allowances will be mostly abandoned and replaced by an auction mechanism. The number of allowances is reduced over time so that total emissions fall. Umut Çetin (LSE) Pricing carbon 4 / 30

Trading within ETS Actual trading with EU ETS emission allowances began January 1st, 2005. By the end of the same year, almost 400 million tonnes of carbon equivalent had been traded. At the end of each year (not the trading period) each company must surrender enough allowances to cover all its excess emissions, otherwise a fine of e100 per tonne is paid. Moreover, the company will need to surrender the missing allowances in the next year. Umut Çetin (LSE) Pricing carbon 5 / 30

Borrowing in ETS Although not explicitly mentioned in the EU Directive on ETS, there is also a limited borrowing opportunity. Umut Çetin (LSE) Pricing carbon 6 / 30

Borrowing in ETS Although not explicitly mentioned in the EU Directive on ETS, there is also a limited borrowing opportunity. The reason for this implicit opportunity is that when the installations are required to surrender allowances to cover emissions at the end of the year, they are already given the allowances for the next year which can be used to cover the excess emissions. However, one cannot barrow allowances from the next trading phase. This limited borrowing makes the pricing allowances more complicated. In case of unlimited borrowing the price of the carbon in the final trading period would be Penalty P [non-compliance F t ], where the underlying probability measure can be determined via an equilibrium argument, see e,g. Seifert, et al. (2007) and Carmona et al. (2009). Umut Çetin (LSE) Pricing carbon 6 / 30

Details of the EU ETS market, how it works, and further documentation can be found at http://ec.europa.eu/clima/policies/ets/index en.htm. Umut Çetin (LSE) Pricing carbon 7 / 30

Details of the EU ETS market, how it works, and further documentation can be found at http://ec.europa.eu/clima/policies/ets/index en.htm. In the subsequent analysis, we will ignore this limited borrowing and work as if borrowing is not possible. Our goal is to obtain a relationship between the spot and forward prices of emission allowances under no-banking and banking options. Clearly, the price of allowances will heavily depend on whether the EU zone is net short carbon credits. The next plot will also show that the impact of the release of sensitive information regarding the ETS net position in carbon emission allowances can be dramatic and the market does not have a perfect information on the net position of the zone. Umut Çetin (LSE) Pricing carbon 7 / 30

A price history 35 EUA prices: Dec 07 Grey vs. Dec 08 Black 30 25 Price EUR t 20 15 10 5 0 100 200 300 400 Trading day since June 7, 2005 Figure: EUA price history between June 7th, 2005 and May 5th, 2007 Umut Çetin (LSE) Pricing carbon 8 / 30

Some related literature Equilibrium models: Seifert, et al. (2007), Chesney and Taschini (2008), Carmona et al. (2009), Grüll and Kiesel (2009),.... Reduced-form and hybrid models: Grüll and taschni (2009), Çetin and Verschuere (2009), Carmona and Hinz (2010), Howison and Schwarz (2010),.... Umut Çetin (LSE) Pricing carbon 9 / 30

Model with no banking (Çetin and Verschuere (2009)) We aim to obtain a relationship between the spot (the permit that matures at the end of the current year) and the forward (the permit that matures at the end of the next year) prices of carbon permits. It is apparent from the earlier figure that the EUA prices before and after the information release have different drifts. Moreover, starting from the last quarter of 2006 Dec-07 prices and Dec-08 prices show opposite trends. The reason for this behaviour is the shift in the demand from worthless spot (henceforth called EUA0) contracts to forward (henceforth called EUA1) contracts. In view of these observations we model the forward prices process, denoted with S, as follows: with S 0 = s. ds t = S t (µ + αθ t )dt + σs t dw t, (1) Umut Çetin (LSE) Pricing carbon 10 / 30

Here, θ is a Markov chain modeling the net position of the market, and W is a Brownian motion independent of θ. θ is a càdlàg Markov chain in continuous time taking values in E := { 1, 1}. θ t = 1 (resp. θ t = 1) corresponds to market being long (resp. short) at time t. The assumption that θ takes only two values is for simplicity and our theory can be extended to the case when E is any finite set. We suppose the Markov chain θ stays in state i for an exponential amount of time with parameter λ(i). The initial distribution for θ is denoted with p. We are working on the time interval [0, T ] which corresponds to the current trading year. Umut Çetin (LSE) Pricing carbon 11 / 30

Process for the zone s net position If R is the matrix of transition probabilities and Q is the generator matrix defined by ( ) λ(1) λ(1) Q =, λ(2) λ(2) then the transition probabilities solves the forward equation R t = R t Q, R 0 = I, (2) where I is the identity matrix and R t (i, j) = d dt R t(i, j). It is well-known that θ t = θ 0 2 t for each t, where N is a martingale. 0 θ s λ(θ s )ds + N t, (3) Umut Çetin (LSE) Pricing carbon 12 / 30

Pricing with no banking of the unused permits The following assumption is to simplify the computations and the exposition. We stress here that our approach still works without the next assumption. Assumption 1 λ(1) = λ(2) = λ. Letting P denote the price process of EUA0 contracts and assuming 0 interest rates, we have the following relation between S and P at time T { ST + K, if θ P T = T 0; (4) 0, otherwise, Next we will discuss the local risk minimisation approach to the pricing and hedging in EU ETS market under complete and incomplete information regarding the net position of the zone. Umut Çetin (LSE) Pricing carbon 13 / 30

Pricing under complete information For the time being let s suppose the market has full information on θ. Note that the market is still incomplete since θ is not tradable. Following Föllmer and Schweizer (1991) we define the optimal hedging strategy for a given contingent claim in an incomplete market as follows. Definition 1 Let C L 2 (Ω, F T, P) be a contingent claim. A predictable trading strategy ξ C is said to be optimal if there exists a square integrable F-martingale, L C, orthogonal to W such that T C = c + ξt C ds t + L C T. (5) 0 Existence of (5) is intimately linked to the so-called minimal martingale measure. Umut Çetin (LSE) Pricing carbon 14 / 30

Minimal martingale measure Definition 2 Let X be a continuous semimartingale with the canonical decomposition X = X 0 + M + A with M a martingale and A is adapted, continuous and of finite variation. A probability measure P P is called minimal martingale measure if X follows a martingale under P, P = P on F 0 and any square integrable martingale orthogonal to M remains a martingale under P. The minimal martingale measure is uniquely determined and in our case is defined by d P ( T dp = exp 0 µ + αθ s dw s 1 T σ 2 0 ( ) µ + 2 αθs ds). (6) σ Umut Çetin (LSE) Pricing carbon 15 / 30

Pricing under the minimal martingale measure Definition 3 Let C L 2 (Ω, F T, P) be a contingent claim and let P be the unique minimal martingale measure for S given by (6). The fair price Pt C at time t for C is given by := Ê[C F t]. P C t For the problem under consideration C = (S T + K )1 {θt = 1} = S T +K 2 (1 θ T ). Recall that θ = θ 0 + N + A where N is a martingale and A is predictable process. Since θ and W are independent, it follows that N and W are orthogonal martingales under P; thus, N remains a martingale under P and is orthogonal to S. Umut Çetin (LSE) Pricing carbon 16 / 30

Pricing carbon permits under full information Proposition 1 Let Then M is a P-martingale and Ê t [ ST + K 2 M t := θ t exp( 2λ(T t)), t [0, T ]. A T ] = θ t S t + K 2 exp( 2λ(T t)) (θ 0 + N t ) S t + K 2. Umut Çetin (LSE) Pricing carbon 17 / 30

Theorem 4 The fair price for EUA0 contracts is given by P t = (S t + K ) 1 θ t exp( 2λ(T t)). 2 The optimal hedging strategy, ξ 0 associated with EUA0 contracts is given by for each t [0, T ]. ξ 0 t := 1 M t, 2 In other words, part of the risk at time t, corresponding to the term t 0 1 Ms 2 ds s, can be hedged if one follows the locally-risk minimising strategy which consist of holding (1 M)/2 shares of the traded underlying, whose price process is given by S. Umut Çetin (LSE) Pricing carbon 18 / 30

Pricing under incomplete information Now we study the pricing of EUA0 contracts under incomplete information. We suppose the only information available to the market is the usual right-continous and complete augmentation of S, denoted with F S and the one-time announcement of the true value of θ at time T. If G denotes the filtration modelling the information structure of the market, then { F S G t = t, for t < T ; FT S σ(θ T ), for t = T. Let θ denote the optional projection of θ to F S which gives θ t = E[θ t F S t ], for each t 0. We now apply the aforementioned local-risk minimisation approach to the pricing and hedging of EUA0 under incomplete information, i.e. when the available information is modelled by G. Umut Çetin (LSE) Pricing carbon 19 / 30

Theorem 5 Define W by W t = t 0 ds s (µ + αθ s )S s ds σs s, and Z by Z t = 1 [t=t ] (θ t θ t ) for each t 0. Then, W and Z are orthogonal G-martingales. Moreover, W is a G-Brownian motion. Definition 6 The fair price of the EUA0 contracts at time t under incomplete information is defined to be P t := E [(S T + K )(1 θ T )/2 G t ], where E is the expectation operator under P, the unique minimal martingale measure associated to S with respect to G. Umut Çetin (LSE) Pricing carbon 20 / 30

It is easy to see that P t := E [(S T + K )(1 θ T )/2 G t ]. However, no explicit formulae can be found for pricing or hedging. One can do Monte-Carlo simulation using the conditional probabilities obtained by filtering methods, or obtain a PDE for the pricing function. Umut Çetin (LSE) Pricing carbon 21 / 30

Calculation of short-probabilities It is possible to calculate the probabilities related to the zone s net position. Let π i (t) := P[θ t = i G t ] for each i { 1, 1}. Clearly, θ t = π 1 (t) π 1 (t) = 2π 1 (t) 1. Therefore, t π i (t) = P[θ 0 = i G 0 ] + + t 0 0 ( λπj (s) λπ i (s) ) ds π i (s) (µ + αi)s s S s ({µ + αi}π i (s) + {µ + αj}π j (s)) σs s dw s, where W is the G-Brownian motion defined in Theorem 5, and {j} = E\{i}. Umut Çetin (LSE) Pricing carbon 22 / 30

Since π i (s) + π j (s) = 1 for all s, the above reduces to In particular, for i = 1, t π i (t) = P[θ 0 = i G 0 ] + t + 0 t t π 1 (t) = p + λ (1 2π 1 (s)) ds + 0 0 0 λ (1 2π i (s)) ds π i (s) α(i j)(1 π i(s)) dw s. σ π 1 (s) 2α(1 π 1(s)) dw s ; σ and t t α θ t = 2p 1 2 λθ s ds + 0 0 σ (1 θ 2 s )dw s. (7) Umut Çetin (LSE) Pricing carbon 23 / 30

Effect of intermediate announcements Every year, the European union aggregates submitted emission data and compares this to the quantity of allowances surrendered. The processing of emissions data for the entire zone usually takes a couple of months time, and announcements on the zone s net position are not released until mid April every year. As the EU announces net results every year, it is a good idea to look a bit deeper into the effects of intermediate announcements on net positions. Umut Çetin (LSE) Pricing carbon 24 / 30

To be more concrete, suppose at some t 0 < T the true position of the zone is revealed to the market. To ease the calculations we further assume that there will be no further announcements before time T. Now, we redefine θ so that θ t = E[θ t F S t, θ t0 ] for t t 0. This implies that the dynamics of θ changes to for t t 0. t t α θ t = θ t0 2 λθ s ds + t 0 t 0 σ (1 θ 2 s )dw s, (8) Therefore, typically, there will be a jump in θ at time t 0 since θ t0 will be different than θ t0 as long as 0 < P(θ t0 = 1) < 1. Umut Çetin (LSE) Pricing carbon 25 / 30

Pricing under intermediate announcements Theorem 7 Suppose that the true state of θ is revealed at time t 0 < T. The fair price of EUA0 contracts is given by { h(t, S St, θ P t = t ) Z t +K t 2, for t t 0, Zt h + h(t, S t, θ t ), for t < t 0, where h is a continuous solution of a certain PDE and Z h t = E [h(t 0, S t0, θ t0 ) h(t 0, S t0, θ t0 ) F S t ], for t t 0. P has a jump at t 0 and the jump size equals P t0 = h(t 0, S t0, θ t0 ) h(t 0, S t0, θ t0 ) Z h t 0. Umut Çetin (LSE) Pricing carbon 26 / 30

Introducing banking Since the beginning of Phase II all unused carbon permits can be banked for future use. Therefore, the spot contracts are not necessarily worthless even if the zone is net long at their maturities. This necessitates a modification of the spot-forward relationship that we introduced earlier. Since the allowance for the current year can be used next year, in case the market is short at time T, EUA0 contract and EUA1 contract should have the same value. Therefore, { ST + K, if θ P T = T = 1, S T if θ T = 1. This implies that P t = S t + K P [θ T = 1 G t ] for every t [0, T ]. Umut Çetin (LSE) Pricing carbon 27 / 30

Consider a digital option which pays e1 if the market is short at time T. The fair price process, D, of this option is given by where h is the solution to D t = h(t, θ t ) Z t 2, and h t (t, y) + Lh(t, y) = 0, h(t, y) = 1 y 2, Lh(t, y) := 1 α 2 2 σ 2 (1 y 2 ) 2 h yy (t, x, y) ( h y (t, x, y) 2λy + α ) σ 2 (1 y 2 )(µ + αy). Umut Çetin (LSE) Pricing carbon 28 / 30

Moreover, the optimal strategy under incomplete information associated to the digital option is given by ξ D = (ξ D t ) 0 t T where ξ D t := α σ 2 1 θ 2 t S t h y (t, θ t ), for each t [0, T ]. Consequently, for every t < T, P T = S t + Kh(t, θ t ), and the optimal hedging strategy for the spot is given by 1 + K α σ 2 1 θ 2 t S t h y (t, θ t ). Umut Çetin (LSE) Pricing carbon 29 / 30

We expect that under banking the effect of intermediate announcements will be much less pronunced and will only depend on the markets s expectation of non-compliance. This is in line with what has been observed so far after the introduction of banking. The argument in the previous slides can be applied to get a relationship between different forward prices. More precisely, if P n denotes the price of the forward contract with maturity nt, then P n 1 t = P n t + P [θ (n 1)T = 1 G t ]. In view of the Markov property of θ, one has E [P(T, θ (n 1)T, 1) G t ], where P is the transition function of θ. Umut Çetin (LSE) Pricing carbon 30 / 30