Valuation of Power Plants and Abatement Costs in Carbon Markets

Transcription

1 Valuation of Power Plants and Abatement Costs in Carbon Markets d-fine GmbH Kellogg College University of Oxford A thesis submitted in partial fulfillment of the MSc in Mathematical Finance April 19, 2011

2 Abstract Carbon markets are relatively new markets, which have been created by the introduction of tradable allowance certificates. A company must own a sufficient number of them in order to avoid being charged a penalty for its carbon dioxide emissions. This is meant to encourage companies to reduce carbon emissions, for example by investing in new technologies or by switching to different fuel sorts. The emission certificates will obviously also have an influence on the value of power plants, as electricity producers will not only have to buy the fuel but also emission certificates to cover the produced emissions. The aim of the thesis will be to give a short introduction into the field of carbon markets and to model the allowance price by considering it as a derivative on the demand and on the total emissions to date. This will lead to a nonlinear PDE for the allowance price, the properties of which will be investigated. The gained knowledge will be used for a real options approach for the valuation of a power plant which takes into account the costs for the allowance certificates. The difference in value to the case, when no emission certificates are involved can be interpreted as the abatement costs for the emissions.

3 Acknowledgements I would like to thank my employer d-fine for letting me participate in this master programme. I also would like to thank Professor Sam Howison for supervising this thesis and for suggesting this interesting topic to me.

4 Contents 1 Introduction Real Options and Electricity Markets Carbon Markets and Abatement Costs Structure of the Thesis Valuation of a Power Plant using Real Options Real Options Supply and Demand Based Approach The Bid Stack Stochastic Processes Valuation of a Power Plant Carbon Markets Equilibrium Approach Reduced Form Approach Influence on the Value of a Power Plant Hybrid Approach The Model Value of the Emission Certificates Special Cases Numerical Results Comparison of MC and FD for the linear case Dependence on the parameters Valuation of a Power Plant in Carbon Markets General Approach Value of a Power Plant and Abatement Costs i

5 5 Conclusion and Outlook 48 6 Appendix 50 A Finite Difference Algorithm B Some Results for an Alternative Bid Stack Function ii

6 Nomenclature α α B Emission rate parameter (dependence on price of emission certificates) Parameter in Barlow s bid stack function α Z β γ κ κ G Offset parameter in model for C t Emission rate parameter (dependence on demand) Emission rate parameter (prefactor determining the overall scale) Mean reversion speed of demand process Mean reversion speed of gas price process κ Z Mean reversion speed in model for C t λ i Parameters for exponential distribution in model for C t (i = 1,..., 4) µ Mean reversion level of demand process µ (i) G Mean reversion levels (i = 1) and drifts (i = 0) of gas price processes µ Z Mean reversion level in model for C t π σ Penalty for each missing allowance Volatility of demand process σ (i) G Volatilities of gas price processes (i = 1, 2) σ Z Ŵ t, W t B t, W t C e Volatility in model for C t Standard Brownian motion, independent of W t, B t and of each other. Standard Brownian motions with correlation ρ Emission rate parameter (emission rate for maximal allowance price) iii

7 C P C t D t D max G t K M D M E N P N S N T Capacity per time of power plant (constant) Capacity (stochastic process) (Electricity) demand Maximal value of demand D t (FD algorithm) Gas price Emission cap Number of steps in D-direction (FD algorithm) Number of steps in E-direction (FD algorithm) Number of time steps (for lifetime of power plant) Number of simulations Number of time steps (for compliance period) p i Switching probabilities in model for C t (i = 1,..., 4) P t r T T P V P V SS RMSE Coal price Risk-free rate End of compliance period End of operating time of power plant Value of power plant Value of spark spread option Root mean square error iv

8 Chapter 1 Introduction 1.1 Real Options and Electricity Markets The valuation of physical assets like factories or power plants 1 has always been important, as the value attributed to a factory or a power plant is a crucial piece of information which can for example influence investment decisions: Depending on the result of the valuation, the investor can decide in favour of one technology and against another for example she could favour a gas-powered plant over a coal-powered plant. The more traditional approach to this valuation problem used to be the net present value approach [1] which proposes to look at the present value of all expected future profits of the project and the present value of all expected costs and substract these two from each other. If the net present value is positive one should invest in this project. This approach has the disadvantage that it makes several implicit assumptions. In particular it assumes that either the investment is reversible or if it is irreversible it has to be realized now without the chance of making the investment at a later point in time. It neglects the value which is contained in the possibility of postponing the project to a later stage when market conditions are better or new information is available. As an example let us consider a very simple model with discrete time steps (following [1]) in which a power plant can produce electricity from coal. The costs for coal are constant at 50 Euro, whereas the (sell) price for electricity produced from this is 80 Euro at time t = 0 and may go up to 120 Euro with probabiltiy q or go down to 40 Euro with probability 1 q at time t = 1 and stay constant afterwards. The net present value approach would tell us to invest in the power plant already at time t = 0 as the net present value is positive (80 Euro 50 Euro = 30 Euro). However, if we wait until time t = 1 we will only invest if the price of electricity has gone up. The net present value approach clearly fails take into account the value of the optionality to delay the investment to a later time. 1 The main interest of this thesis will be the case of a power plant. Therefore I often simply talk about a power plant although many statements about real options also hold true for other kind of assets. 1

9 Figure 1.1: Illustration of the example. The electricity price can go up to 120 Euro with probability q or down to 40 Euro with probability 1 q. The net present value approach fails to take into account the value of the optionality to delay the investment. One has here an analogy to financial options and similar techniques can be used to value the opportunity costs this approach is by now familiar under the name Real Options approach [1]. From the example it can also seen that the problem involves the price processes of the quantities determining the costs and the payoff. In the example above this would be the price of electricity (which the power plant will sell) and the price of the fuel used to produce it (which will be part of the costs). Hence it is important to model these price drivers mathematically. The most important quantities are the following: ˆ Fuel: On the one hand, one has the fuel used to produce electrity. Among the many possibilities I will restrict myself in this thesis to gas (denoted by G t ) and sometimes coal (denoted by P t ). For gas different stochastic models exist, which will be introduced in detail in chapter 2, whereas for coal it is often sufficient to use the forward curve, which also reduces the dimensionality of the problem [2]. ˆ Capacity: In more sophisticated models like the one introduced in [2] one has to try to capture effects caused for example by outages which affect the production process. This can be achieved by modelling the capacity as described in chapter 2. ˆ Electricity: Electricity is the end product of the power plant which is meant to be sold on to the consumer. It has some unique features compared to other commodities which make it particularly difficult to model. It is essentially non-storable which enforces a immediate balancing of supply and demand. This can lead for example to large price spikes. 2

10 The common approaches to model electricity (see for instance [3, 4]) have the drawback that although some of them manage to capture fairly well the actual price process, a clear connection to market fundamentals like the demand is missing. This gap is filled by looking at the so-called bid stack function which connects the electricity price to the demand. This approach mimics the price formation mechanism on the bid stacks in real electricity markets. It has the advantage of still being sufficiently simple to obtain handable expressions (for example for forward prices or some derivative products) while offering an insight into market mechanisms as well. 1.2 Carbon Markets and Abatement Costs A new element has been added to this in recent years by the introduction of emission trading schemes. These so-called cap-and-trade schemes are political tools to reduce the total emissions of carbon dioxide and other harmful greenhouse gases at the lowest social costs as possible. They essentially work in such a way that at the end of each compliance period (for instance after a year) the producer of the emissions has to come up with a certain amount of emission allowances which offset the emissions she has produced during the last compliance period. If she is unable to do that she has to pay a penalty for each missing allowance. These allowances are freely tradable. It is obvious, that cap-and-trade schemes will significantly modify the market mechanisam sketched above. A producer will have to take into account the costs of the emissions when she makes her bid. This will change the price of electricity compared to the market without emission certificates. If a producer for example decides to change to a low-emission fuel she can offer a more competive bid which again changes the bid stack and so on. 2 The mechanics of carbon markets has been investigated already by several authors [5, 6, 7, 8, 9, 10]. The interesting questions for this thesis is how the introduction of emissions certificates will influence the market mechanics sketched above and in particular how the value of a power plant is going to change in comparison with the valuation depending on the demand only. One would expect the difference in value to be the abatement costs. Basically two kinds of them can arise: i.) temporary costs to which the firm can react by (temporarily) switching to a low - emission fuel, and ii.) permanent costs, which can justify the investment into a new plant. 2 An illustration of these dependencies can be found at the end of chapter 3 in figure 3.6 3

11 1.3 Structure of the Thesis The remainder of this thesis is structured in the following way: In chapter 2, I will show how a power plant can be valuated using a real options. I will also describe the bid stack model introduced in [11] and explain the stochastic processes used in the model. This is the setup, I will later refer to when valuating the power plant in presence of emission certificates. In chapter 3, I will introduce carbon markets and review some existing models before I present an alternative model in chapter 4, which is motivated by the bid stack model introduced in chapter 2. Then the price of the emission certificate will be determined numerically and the dependence on the various parameters of the model will be looked at. Also the effect of the existence of emission certificates (or rather the obligation to set off emission with them) on the price of the power plant will be computed, using the consideration from chapter 2 and the comparison will be made to the value of the power plant without emission certificates. Finally, I present my conclusions in chapter 5. The appendix contains the description of the finite difference scheme which has been used to solve the partial differential equation introduced in chapter 4, and some additional results for an alternative bid stack function. 4

12 Chapter 2 Valuation of a Power Plant using Real Options 2.1 Real Options A popular approach to value power plants and to deal with connected issues like investment decisions etc. are so-called Real Options as has already been pointed out in the previous chapter. As opposed to the net-market-value approach which simple discounts future cash flows it takes into account irreversiblities and can also be used to study more complicated situations like for example the influence of a temporary suspension of the project. A good introduction into this topic is for instance [1]. As the main emphasis of this thesis is not on the Real Options approach as such but rather on the new effects introduced by carbon markets, I will not go into all details of this approach and all the various questions which can be treated with it. I will consider the following simplified version of a power plant: ˆ The plant uses one type of fuel only (for example gas) which is transformed into electricity with heat rate K H, i.e. in order to produce one unit of electricity the plant needs K H units of fuel. The electricity price is denoted by S t, the price of the fuel by P t (for coal) or by G t (for gas). ˆ The power plant has a certain life span, denoted by T P, after which it is abandoned. For gas-power plants T P typically equals about 20 years, for coal-powered plants it can be much longer. The effeciency of the plant is assumed to be constant over the whole lifetime, there are no deteriorations or new investments which could change for example K H. 5

13 ˆ The power plant can be ramped up or shut down quickly when this is economically necessary. This assumption is justified for gas powered plants but is for instance wrong in the case of a nuclear power plant. In order to value the power plant, I look at so-called spark spread options (see for example [3]). This is an exchange option which gives at maturity the right to exchange the electricity price S T for the price of the fuel 1 P T needed for its generation. In other words, the payoff at maturity T is P SS T (S T, K T ) = max(s T K H P T, 0) (2.1) This instrument can be valued using the usual option-pricing techniques. The price is given by the expectation V SS (t, S, P ; T ) = e r(t t) E Q ( P SS T (S T, P T ) S t = S, P t = P ) (2.2) where Q is a risk-neutral measure. Expression (2.2) can of course also be turned into a partial differential equation with the help of the Feynman-Kac theorem [12, 13]. The power plant offers at every time during its lifetime the same opportunity as an exchange option so that its value is the sum of spark spread options over its lifetime. Suppose we have devided the timeline [t, T P ] into N P equally spaced buckets [t i, t i+1 ] with t i = t + i t (for i = 0,..., N P and t = (T t)/n P ). Then the value V P (t, S, P ) of the plant ist given by Here C P N P V P (t, S, P ) = C P t V SS (t, S, P ; t i ) (2.3) i=0 is the capacity per unit time of the power plant, which will be assumed to be constant for simplicity. In this way C P (T t) is the capacity for the remaining lifetime and C P (T t)/n P = C P t the capacity per time bucket. This expression can (for example) be evaluated for daily buckets using daily average prices for both S t and P t. For theoretical purpuses one could also take the limit N P, which would result into TP V P (t, S, P ) = C P V SS (t, S, P ; ξ)dξ (2.4) t This formula can be useful when analytical expressions for V SS (t, S, P ; T ) are available, as for example described in [14]. With this general approach one can now proceed and introduce stochastic processes both for the electricity price S t and the fuel price P t. This will then allow to compute 1 In a slight abuse of notation P t will always denote a general fuel (not only coal as introduced above) in the general formulas of this section 6

14 the spark spread option and then integrate the values over the lifetime of the powerplant. This approach is for instance followed in [14]. Depending on which stochastic processes are chosen for S t and P t (2.2) either has to be evaluated numerically or in certain simple case analytically. One can deduce a closed-form expression for (2.2) if one uses for both S t and P t geometric Brownian motions or mean-reverting processes. Under the real-world measure P, the stochastic evolution would be given by or for mean-reverting processes by ds t = µ S S t dt + σ S S t dw t dp t = µ P P t dt + σ P P t d W t (2.5) ds t = κ S (µ S log(s t )) dt + σ S S t dw t dp t = κ P (µ P log(p t )) dt + σ P P t d W t (2.6) where the Brownian motions W t and W have correlation ρ. Recall that in the risk-neutral measure Q, the discounted quantities S t and P t do not have a drift term any more. Under the simple assumption (2.5) or (2.6) the price of the spark spread option can either be found from (2.2) by employing the change-of-numeraire technique [15] or by resorting to the equivalent partial differential equation and using a scaling approach [16]. Either way relies on the fact the under these simplifying assumptions the problem has a scaling symmetry under the transformation S t λs t, P t λp t and V SS λv SS for a constant positive parameter λ. This makes the problem solvable in this case with the result [16, 14] V SS (t, S, P ; T ) = S N (d + ) K H P N (d ) (2.7) with and d ± = d ± (t, S, P ) = ( ) log S K H P ± 1 2 Σ2 (T t) Σ T t (2.8) Σ 2 = σ 2 S + σ 2 P 2ρσ S σ P (2.9) The expression (2.7) then has to be introduced into the integral expression (2.4) or an corresponding sum, which can then be evaluated numerically. Below we show some figures of the resulting behaviour. As was to be expected, the value decreases when the remaining operating time diminishes. If the electricity price is fixed, a rising gas price will decrease the value of the power plant and vice versa a rising electricity price will lead to a higher value if the gas price remains constant. The basic expected behaviour is therefore captured by the simple models (2.5) and (2.6). However, the suggested processes (2.5) and (2.6) are actually too simple, especially for electricity. As mentioned earlier electricity usually shows 7

15 Figure 2.1: Value of the power plant for different remaining operating times. The volatility σ S of the electricity is assumed to be 0.5, the volatility σ P of the fuel price 0.2 and the correlation ρ between these two 0.5. The heat rate K H is taken to be 1 and the timeline was devided into 1000 equally spaced time slices (N P = 1000). extreme spikes which makes it necessary to consider more complicated processes involving jumps or regime-switching processes. In addition to this, the approach of modelling S t and P t directly has general disadvantages: A connection to fundamental price drivers like the demand is lacking and the models may be difficult to calibrate. Therefore I follow an approach proposed in [2] which connects the electricity price to demand (and other price drivers like capacity) via a so-called bid stack function. 2.2 Supply and Demand Based Approach The Bid Stack Following [11], I introduce the notion of a bid stack function. In a bid stack, generators make bids b (i) t (x i ) for amount the x i of electricity they want to sell, which are then arranged by price and form in this way the bid stack. The spot price S t for electricity is then the highest bid needed to match the demand D t. For example, if one had n generators, the electricity price would be given by { } S t = max b (i) t (x i ) 1 i n (2.10) 8

16 where the variables x i are such that they meet the demand, i.e. { } n {x 1,..., x n } = argmin x1,...,x n max 1 i n b(i) t (x i ) x i = D t i=1 (2.11) Assuming a very large number of generators, the bid stack function becomes a continous increasing function which will be denoted by B( ). A very simple form has been suggested by Barlow [17], which connects the electricity price and the demand in the following way: S t = B(D t ) = (1 + α B D t ) 1/α B (2.12) where α B is a parameter which controls the steepness of the bid stack function. α B = 1 would entail a linear dependence, α B 0 means exponential growth and α B < 0 leads to an even steeper behaviour with extreme spikes. We will refer to this model as Barlow s model or shorter model B in what follows. A more sophisticated model has been developed in [2]. It also takes into account outages or similar effects by introducing a new process, the capacity C t, in addition to the demand D t and the underlying price process P t. The connection between electricity price and the price drivers is then given by S t = B ( Dt C t ) (2.13) where the function B( ) depends on the process P t (and possible other price processes in the case of more than one fuel). For the one-fuel case (gas) the following form of the bid stack function is found to be reasonable: S t = α 0 + α 1 P t + (β 0 + β 1 P t ) (log(d t ) log(c t D t )) (2.14) which was originally derived for gas. This model we will call model CH in what follows. In [2] the case of two fuels (gas and coal) has also been considered with the result S t = x where B 1 (x) = D t C t (2.15) and B 1 (x) = ω 1 2 tanh ( ) x (α0 + α 1 P t ) + 1 ω 1 tanh 2(β 0 + β 1 P t ) 2 ( ) x (α0 + α 1 G t ) 2(β 0 + β 1 G t ) (2.16) but this will not be considered further in this thesis. 9

17 2.2.2 Stochastic Processes The specification of the actual stochastic processes for the quantities D t, G t, P t and C t is still missing. Some possibilities which have been discussed in the literature will now be presented: 1. The demand D t : Throughout this thesis I will always suppose that the demand D t is inelastic and is governed by a stochastic differential equation of the form dd t = µ(t, D t )dt + σ(t, D t )dw t. (2.17) Demand has to be positive and it usually also has a tendency to be mean-reverting. Also demand can have seasonal fluctations. I will only consider the following cases ˆ Geometric Brownian motion (GBM): This is a rather pathological case where µ(t, D t ) = ˆµD t and σ(t, D t ) = ˆσD t for two constants ˆµ, ˆσ. As this model does not capture any of the demanded features it is not relevant for practical purposes. However, it will be occasionally looked at as it is simple to solve. The stochastic differential equation is dd t = ˆµD t dt + ˆσD t dw t. (2.18) Seasonality can be included by adding a deterministic function f(t) to D t which has the form f(t) = a 1 + a 2 t + a 3 cos(2πt + a 4 ) + a 5 cos(4πt + a 6 ) (2.19) if not mentioned explicitely the parameters a i will take the values which have been determined in [2]: a 1 = 0.746, a 2 = 0, a 3 = 0.114, a 4 = 0.862, a 5 = 0.186, a 6 = ˆ Mean reversion (MR): A model of this type was suggested by Barlow [17] and includes mean reversion and seasonality. It assumes that the demand is given by D t = f(t) + Y t (2.20) where f(t) is the deterministic function (2.19) and the random variable Y t is determined by the stochastic differential equation dy t = κ( µ Y t )dt + σdw t (2.21) for constants κ, µ and σ. A drawback of this model is that the demand can still become negative. This can be avoided by modelling log(d t ) instead of D t directly. This leads to the third model considered. 10

18 ˆ Exponential mean reversion (EMR): The demand is given by (see for example [2]): D t = exp (f(t) + Y t ) (2.22) where f(t) incorporates the seasonality as in (2.19) and Y t is given by dy t = κ(µ Y t )dt + σdw t (2.23) for constants κ, µ and σ. guaranteed to be positive. This model has the advantage that the demand is 2. The gas price G t : For the gas price we use the following two models ˆ Geometric Brownian motion (as a toy model) dg t = µ G G t dt + σ G G t dŵt (2.24) ˆ The model used in [2] ( ) G t = exp h(t) + X (1) t + X (2) t (2.25) where the two stochastic processes X (1) t and X (2) t are driven by two independent Brownian motions Ŵt and W t 3. The coal price P t : dx (1) t = κ G (µ (1) G X(1) t )dt + σ (1) G dŵt (2.26) dx (2) t = µ (2) G dt + σ(2) G d W t We will assume the coal price to be deterministic and use the forward curve for the price P t, as discussed in [2], although there are by now indications that this approach may no longer by reasonable in the future. This means that [3] with the convenience yield q. 4. The capacity C t : P T = P t exp((r q)(t t)) (2.27) Finally in order to describe the capacity in model CH, one looks at the quantity M t = C t D t and uses a regime-switching process to describe it [2] { Z OU t 1 p log(m t ) = i Zt SP (2.28) p i 11

19 where the two processes Z OU t and Z SP t are given by dz OU t = κ Z (µ Z Z OU t )dt + σ Z d B t (2.29) Z SP t = α Z J, J = exp(λ i ) (2.30) The Brownian motion B t is correlated with the Brownian motion W t driving the demand and the correlation will be denoted by ρ. α Z is an offset parameter, J an exponential random variable parametrised by λ i and p i is the switching probability. Both λ i and p i can take four different values (i = 1,..., 4) depending on the season. In table 2.1 we give an overview of the models and stochastic processes used in this thesis. For the sake of completeness the table also contains the emission rate e t, which will be introduced and explained later (chapter 4). models by simply adding the keys given in the table. Sometimes we will use shorthands for the For example the model B-D2-G1 would be the model using Barlow s bid stack function, the mean reverting process (2.20) for the demand D t and the process (2.26) for the gas price. Quantity Symbol Equation Key Ref. dd t = ˆµD t dt + ˆσD t dw t D1 (2.18) Demand D t D t = f(t) + Y t, dy t = κ( µ Y t )dt + σdw t D2 (2.20) D t = exp (f(t) + Y t ), dy t = κ(µ Y t )dt + σdw t D3 (2.22) dg t = µ G G t dt + σ G d t dŵt G1 (2.24) Gas G t G t = exp(h(t) + X (1) t + X (2) t ), dx (1) t = κ G (µ (1) G X(1) t )dt +σ (1) G dŵt, dx (2) t = µ (2) G dt + σ(2) G d W t G2 (2.26) Coal P t P T = P t exp((r q) (T t)) Co1 (2.27) Bid Stack B(D) = (1 + α B D) 1/α B B (2.12) Function B( ) B(D/C) = α 0 + α 1 G t (β 0 + β 1 G t ) log(c/d 1) CH (2.14) Capacity C t M t = C t D t and log(m t ) = Zt OU (prob 1 p i ) or log(m t ) = Zt SP (prob. p i ) C1 (2.28) Emission Rate e t e t (A, D) = γd β (C e A α ) E1 (4.13) Table 2.1: Models used in this text. The stochastic drivers are W t, W t, Ŵt (and B t, see (2.29)). Generally I indicate the kind of model by connecting by simply adding the keys together. The emission rate has not been introduced yet, but will be explained in chapter 4 A conceptual advantage of the approach via bid stack functions is that this approach mimics the actual price formation process better than a simple prescription of a stochastic process for S t. The interplay between prices, price drivers and bid stack is illustrated in figure (2.2). 12

20 Figure 2.2: Interplay between electricity price, price drivers like demand and the bid stack. Illustration adapted from [11] 2.3 Valuation of a Power Plant In section 2.1 it was discussed how the power plant can be evaluated. Using the bid stack model introduced in the last section the value of the spark spread option is described by V SS (t, D, P ) = e r(t t) E Q( max ( B(D T ) K H P T, 0 ) ) Dt = D, P t = P (2.31) where the function B( ) is given by the models described above. This can be computed by Monte-Carlo simulation and then be introduced into formula (2.3). Here and for all other numerical investigations, I only consider model B for the bid stack function. For the gas price I used model G2 and for the demand model D3. In figure 2.3 it is shown for different remaining life times how the value of the power plant changes as a function of the demand and the gas price. The parameters used are given in the figure caption, see also the nomenclature for the meaning of the symbols. For longer operating times the surface seems to be more bumpy. This happens because in this case the Monte- Carlo noise has more time to add up. Increasing significantly the number of simulations (here N S = was used) will smoothen the surface also for longer operating times. In the shown simulation, the order of magnitude of the RMSE of the surface for T P = 10 was 0.06, whereas for T P = 1, it was only As comparison, for N S = 1000 simulations only, the RMSE was in the order of magnitude of 0.2 and 0.02 for the T P = 10 and T P = 1 respectively. In section 4.2.2, I will investigate how this behaviour is modified by the introduction of emission certificates. The difference to the present results can then be interpreted as abatement costs. 13

21 Figure 2.3: Price of a power plant in dependence on the demand and the fuel price for different remaining operating times. Parameters are as follows: N P = 100, N S = 10000, C P = 1, µ (1) G = 1.33, µ(2) G = 0.053, κ G = , σ (1) G = 5.836, σ(2) G = 0.53,µ = 1, σ = 1.393, κ = 4.24, α B = 1.4, T = 2.1, K H = 1.1. Compare also figure 2.1. For a discussion of the Monte-Carlo noise seen in the upper left figure see main text. 14

22 Chapter 3 Carbon Markets In the last decades, it has become scientific consensus that global CO 2 emissions contribute significantly to global warming with all its threatening consequences. 1 Hence, the necessity to reduce carbon emissions is by now a major political issue, which has been at the center of several political initiatives aiming to come up with effective means to enforce emission reductions. Key decision were taken during the Kyoto Conference in One of the consequences of this conference was to impose mandatory limits on CO 2 emissions to signatory nations [18]. In order to implement these reductions, the countries of the European Union have decided on the introduction of a so-called Cap-and-Trade Scheme. Such schemes have been used successfully in the past to reduce SO X and NO X emissions in the United States [19]. The basic idea of such schemes is that in order to be allowed to emit a certain quantity of carbon dioxide a company has to come up in certain period time intervals (for example once the year) with emission allowances, which cover all emissions of the preceding year or else has to pay a penalty. The allowances are allocated at the beginning of (or even during) each compliance period and can be freely traded. In this way, firms can invest in low-emission technologies and then make a profit by selling on the allowances they do not need to firms which have not made this investment. Vice versa, firms which are using energy-intense technologies can avoid paying the penalty by buying emission certificates on the market. The regulator issues only enough allowances for total emissions up to a certain cap and can achieve a reduction of emissions in this way. 2 1 CO 2 (carbon dioxide) is not the only gas responsible for the green house effect (another example would be CH 4 (methane)) As the difference does not really matter here, I will use the words CO 2 and green house gases synonymously. 2 One could of course also choose other means to achieve the reduction - for example taxation - but cap-and-trade schemes offer the advantage the same goal at the lowest possible costs for society, as will be discussed in section

23 There are two important aspects in which the regulation can be varied. The first one consists of mechanisms which connect the current year with the preceding or the following one, for example ˆ Banking: This allows for the (possibly limited) transfer of unused allowances from the present to the next period. ˆ Withdrawal: This is an additional penalty for firms which fail to produce a sufficient number of allowances at the end of a year. It obliges such a firm to provide the missing allowance during the next year (in addtion to paying the penalty) or in other words the missing allowances are substracted from the initial allocation of allowances in the next year. ˆ Borrowing: This allows for the (possibly limited) transfer of allowances from the next period to the present one. It seems that regulators tend to allow the first two of these mechanisms, but not to include borrowing [20]. In this thesis, I will not consider these extensions but restrict myself mostly to a simple model, in which the different compliance periods are not connected. This allows to consider a single compliance period at a time, as the certificates will behave similarly in each compliance period. Note however that these extensions have been discussed in the mathematical literature, for instance in [8] the authors discuss the effects of banking and withdrawal in the context of their model. The second important point are the initial allocations of allowances at the beginning of the compliance period. The possibilities to allocate the emission certificates include ˆ Free allocation: The allowances are distributed for free among market participants. This method has been criticized as it led to huge windfall profits for some firms [21]. ˆ Auctioning: The allowances can be auctioned among market participants at the beginning of the compliance year. ˆ Relative Allocation: This is an allocation scheme introduced in [20] in order to reduce windfall profits, which involves subsidies paid in form of allowances. It is of course possible to conceive other allocation mechanisms and the topic is actually quite intensively discussed [22]. However, I will only briefly discuss some facts about the influence of the initial allocations in section 3.1 of this chapter. Unless explicitly stated to the contrary all mathematical models in particular the model presented in chapter 4 16

24 Figure 3.1: Prices of EU-Allowances during the first and the second phase of the European union emission trading scheme. In the first phase, most certificates were allocated for free (no auctioning). Furthermore, too many certificates were given away: Altogether about 2150 million certificates per year, where only 2012 million tonnes (2005), 2034 million tonnes (2006) and 2050 million tonnes (2007) of CO 2 were produced [23]. Finally, the certificates could not be carried into the next period. All this led to a price drop to zero. In the second phase the allocations were more restrictive. Data source: EXAA (see [24]), downloaded the 26th march 2011 will not deal with auctioning or how the initial distribution of certificates is achieved. Cap-and-trade schemes are not completely new. They have been used in the United States in the 90 s following the establishment of the Clean Air Act in The aim was to reduce emissions of SO X and NO X and the regulations have generally been considered a success. The example I will mostly refer to in this thesis is the emission trading scheme implemented by the European Union. In this case, the exact regulation looks as follows. There are two compliance periods. The first one has already expired and took place from 2005 to 2007, the second period runs from 2008 to At the 30th April of each year each market participant 4 has to produce a sufficient number of allowances for the preceding year or, if unable to do so has to pay a penalty for each tonne of emissions which is not covered by an allowance ( for example in original design of EU-ETS the penalty would amount to 40 Euro per metric ton of carbon dioxide equivalent). In addition, the market participant has to produce the missing allowances at the end of the next year. I will denote by A t the price of an EUA or emission certificate at time t. The price may 3 The exact regulation after 2012 is currently still unclear 4 By market participant I mean all companies obliged by the regulators to acquire (by allocation or purchase) allowances to cover their emissions 17

25 depend on several different variables like demand or fuel price, depending on the model which is being used. There are different mathematical approaches which I will present in what follows. I will start with an equilibrium approach as it offers the greatest insight into the market mechanisms. 3.1 Equilibrium Approach A fairly sophisticated equilibrium approach has be developed in [9, 20, 5, 6, 7] which we shall briefly look at. Most of this section is based on [20] where more detailed information can be found. The authors consider a set I of firms which produce and sell K different goods at times 0,..., T 1. Each firm i I has access to a set J i,k of different technologies to produce good k K which generates emissions. To each technology j J i,k belong the following parameters: ˆ Production level ξ i,j,k t : This denotes the quantity of goods k the firm i produces at time t using technology j. It is capped by the production capacity κ i,j,k, that is one has the inequality for all t = 0,..., T 1, all i I, all j J i,k and all k K. 0 ξ i,j,k t κ i,j,k (3.1) i,j,k ˆ Marginal production costs C t : This quantity denotes the marginal cost (for company i) of producing one unit of good k at time t using technology j. ˆ Emission factor e i,j,k : This quantity measures the volume of pollutants emitted during the production of good k by firm i using technology j. ˆ Daily demand D k t : The daily demand for each good k is denoted by D k t for t = 0, 1,..., T. It is reasonable to assume that the demand is always smaller than the total production capacity for this good 0 D k t i I j J i,k κ i,j,k (3.2) ˆ Production strategy ξ i = {ξ i t} t=0,...,t 1 : Each firm i follows a production strategy. Using the expression ξ i,j,k t introduced above, we write ξ i = {(ξ i,j,k t ) k K,j J i,k} t=0,...,t 1 for the production strategy of firm i. An admissible production strategy has to respect the restriction (3.1). ˆ Trading strategy θ i = {θ i t} t=0,...,t : This describes the trading strategy in emission certificates, that is the firm holds θ i t emissions certificates at time t where θ i t < 0 denotes a short position and θ i t > 0 a long position. 18

26 Using the introduced notation, the final position at time T in emission certificates is given by T 1 θt(a i t+1 A t ) θt i A T (3.3) t=0 Denoting by Γ i the amount of emissions which can be offset by the emission certificates firm i owns, the total penalty paid at time T is given by where π(γ i + Π i (ξ i ) θ i T ) + = π max(γ i + Π i (ξ i ) θ i T, 0) (3.4) Π i (ξ i ) := T 1 t=0 e i,j,k ξ i,j,k t (3.5) (j,k) M i is the total amount of emissions the firm i has produced and M i = {(j, k) : j J i,k, k K}. Γ i can for example contain emissions on which the firm i has no control or emission credits gained from the Clean Development Mechanism. 5 S k t The spot price of good k is denoted by and the T -forward price by S k t = e r(t t) Sk t. In the same way, the forward price for the marginal production costs is written as C i,j,k t net gain from producing goods is given by T 1 t=0 (j,k) M i ( = e S k t C i,j,k t r(t t) Ci,j,k t. With this notation the total ) ξ i,j,k t (3.6) At maturity each firm i needs to offset its emissions or pay a penalty. All these considerations lead to the following expression for the terminal wealth, which combines the expressions (3.3),(3.4) and (3.6) L A,S,i (θ i, ξ i ) := + (T 1) t=0 (j,k) M i ( S k t C i,j,k t ) ξ i,j,k t T 1 θt i (A t+1 A t ) θt i A T (3.7) t=0 π ( Γ i + Π i (ξ i ) θ i T ) + Each market participant will try to maximize this expression. In addition to this assumption, one still needs the concept of market equilibrium. The authors of [20] define equilibrium in the following way: The pair (S, A ) form an equilbrium, if there is a pair (ξ i, θ i ) such that 5 The Clean Development Mechanism has been defined in Article 12 of the Kyoto protocol and allows a country to implement emission-reduction projects in developing countries, which will earn tradable certified emission reduction (CER) credits. The country can use the CER towards meeting its own emission target 19

27 ˆ Supply meets demand for all goods k K and all times t = 0,..., T i I j J i,k ξ i,j,k t = D k t (3.8) ˆ All financial positions are in zero net supply for all times t = 0,..., T, i.e. i I θ i t = 0 (3.9) ˆ Each firm i I is satisified by its own strategy in the sense that E[L A,S,i (θ i, ξ i )] E[L A,S,i (θ i, ξ i )] (3.10) for all other trading and production strategies ξ i and θ i. For the case of vanishing penalty (π = 0) one has what the authors of [20] call the Business-As-Usual scenario. In this case it turns out that the equilibrium price is given by ( St k = max i I,j J i,k C i,j,k t ) 1 {ξ i,j,k t >0} (3.11) We observe that this is in line with the price determined the classical merit order (see expression (2.10)) if one assumes that the bids are identical to the production costs (the firm would have no profit margin in this case). Furthermore the authors provide necessary and sufficient conditions for the existence of such an equilibrium. In particular, they show that if (A, S ) is an equilibrium in the described sense with associated optimal strategies (θ, ξ ) then, under certain additional technical assumptions, the allowance price A t is a described by A t = πe [ 1 {Γ+Π(ξ) 0} F t ] (3.12) and the spot prices S k and the optimal production strategy ξ i correspond to a merit-type equilibrium with adjusted costs C i,j,k t + e i,j,k A t (3.13) We will encounter a similar expression in chapter again when we modify expression (2.1) in order to evaluate the power plant in the presence of a carbon market (compare expressions (3.30) and (4.19)). 20

28 Figure 3.2: Consumer and producer costs for the standard trading scheme. Consumer costs are significantly higher than social costs. Figure taken from [5, 6, 7] Figure 3.3: Comparison of emissions for different trading schemes. The standard and the relative allocation scheme are compared to a scenario without penalty (Business-As-Usual) and to a tax scheme, in which emission are simply taxed. Figure taken from [5, 6, 7] 21

29 Figure 3.4: Comparison of windfall profits for different trading schemes. Standard and relative allocation scheme are compared to a tax scheme (emissions are simply taxed) and auctioning of certificates. The relative allocation of certificates produces significantly less windfall profits than the other schemes. Figure taken from [5, 6, 7] The authors of [20] prove further results about this model and perform numerical simulations, considering one good in the economy, namely electricity. They define the social costs as difference in the production costs in a market with non-vanishing penalty and the Business-As-Usual scenario, in which there is no penalty. Windfall profits are the excess profits of a company over what the profit would have been, had the same dispatching schedule been used, and the target price been charged to the end consumer without the cost of pollution. One of their results is illustrated in figure 3.2. The costs for the end consumer are much higher than the overall social costs. In other words, the standard trading scheme leads to huge windfall profits for the producers, who fully pass on the abatement costs to the consumer [21]. This has indeed been a major criticisms of cap-and-trade scheme oponents. The authors then demonstrate that a different allocation mechanism for initial allowances can remedy this deficiency. Apart from the standard allocation scheme which has been considered up until here, in which initial allowances are given away for free, the authors introduce what they call a relative allocation scheme. In this scheme the marginal penalty for the production of one good increases only by (e i,j,k y k )A t (instead of e i,j,k A t ) where y k can be seen as a kind of subsidy granted in form of allowances. The authors also consider a simple tax scheme for comparison. One of the main results of their paper is that the efficiency of the cap-and-trade scheme crucially depends on the allocation scheme used, as is illustrated in figures 3.3 and

30 These results demonstrate how important many details in the setup of cap-and-trade schemes are if one wants them to work properly. In what follows however the focus will be less on the issue of a working carbon market but rather on the description of the certificates themselves and of derivatives on them. This will be needed later for the valuation of the power plant in carbon markets. 3.2 Reduced Form Approach While the approach described above allows to gain insight into markets mechanisms and the general driving processes of cap-and-trade schemes it is less suitable for derivative pricing. For this one would rather like to prescribe a suitable stochastic process for the price A t, calibrate it to market data and then write derivatives on this and possibly additional processes. In this subsection I closely follow the paper [8], in which the authors use the following setup: All processes take place on a filtered probability space (Ω, F, P, (F t ) t [0,T ] ). equivalent probability measure Q is fixed (the spot martingale measure). Furthermore the event of non-compliance is denoted by N F T. The general expression for the price of the emission certificate A t is given by An A t = πe Q (1 N F t ) (3.14) compare for example (3.12). The set N can more formally be defined as N = {E T /K 1} (3.15) where E t denote the total emissions at time t and K is the total number of credits (i.e. the emission cap). Furthermore it is assumed that at time T the emissions follow a lognormal distribution ( T E T = E 0 exp σ s dw s 1 T ) σ sds 0 (3.16) for some square-integrable deterministic function σ t. The authors of [8] prove that the price of an emission certificate under these assumptions is given by T A t = πφ φ 1 (a 0 ) 0 σ2 s + t 0 σ sdw s (3.17) T t σsds 2 where φ is the cumulative density function of the standard normal distribution. The normalised certificate a t = π 1 A t solves the stochastic differential equation da t = φ (φ 1 (a t )) z t dw t (3.18) 23

31 where z t = T t σ 2 t σ 2 udu The proof of these statements is by direct calculation [8]: We have (3.19) a t = E Q ( 1 {ET /K 1} F t ) = Q (ET /K 1) (3.20) Using expression (3.16) this is easily seen to be equal to φ ln(e t/k) 1 T 2 t σsds 2 = φ ln(e 0/K) 1 T 2 T T t σ s ds 0 σ sds 0 σ2 sds which can be transformed into (3.17) using the initial condition a 0 = φ ln(e 0/K) 1 2 T 0 σ2 sds T 0 σ2 sds T 0 σ2 sds + T t σsds 2 t 0 σ sdw s T t σsds 2 (3.21) (3.22) Formula (3.17) describes the stochastic evoluation of A t (under the quoted assumptions) with initial values at t = 0. Similarly, one can compute the value of A t, given that we know the emissions at time t already. In this case one obtains A t = πφ ln(e t/k) 1 T 2 t σsds 2 (3.23) T t σsds 2 The dependence of the price on the total emissions is illustrated in figure 3.5. Calibration: One possibility to calibrate the model introduced above is to parametrise z t as z t = z t (ˆα, ˆβ) = ˆβ(T t) ˆα (3.24) for t (0, T ]. It can then be shown that the parameters ˆα and ˆβ must satisfy ˆβ > 0 and ˆα 1 (3.25) The volatility corresponding to this choice is given by ( σ t (ˆα, ˆβ) 2 = z t (ˆα, ˆβ) exp = t 0 ) z u (ˆα, ˆβ)du { ( ) ˆβ(T t) ˆα exp ˆβ T 1 ˆα (T t) 1 ˆα 1 ˆα for ˆβ > 0, ˆα > 1 ˆβ(T t) ˆβ 1 T ˆβ (3.26) for ˆβ > 0, ˆα = 1 24

32 Figure 3.5: Dependence of the price of the emission certificate on the emissions in as described by formula (3.23). Compare this also to figure 4.3 in the next chapter. Using this result, one can for example consider options written on A t with maturity τ [0, T ]. With the parametrisation just described one has for a plain-vanilla call on A t at time t τ with strike price ˆK C t = e τ t rsds R ( πφ(x) ˆK) + p (µ t,τ,ν 2 t,τ ) (x)dx (3.27) where p (µt,τ,ν2 t,τ ) (x) is a normal distribution density with mean µ t,τ and variance ν 2 t,τ, which are given by ( ) ( ) µ t,τ = φ 1 At T t ˆβ (3.28) π T τ ( ) T t ˆβ νt,τ 2 = 1 (3.29) T τ Other models have been discussed in the literature, but the two I have presented here are representative for the two general modelling approaches which have been used so far for the description of emission certificates. The interested reader may for example refer to ([10]), where a comparison of some of these models is presented (including the models presented here) and issues like calibration are discussed. 3.3 Influence on the Value of a Power Plant The considerations above have shown that by the introduction of freely traded emission certificates the market mechansisms are modified considerably. The price of the emission 25

33 certificates has an influence on the bid stack which is central for the price formation. The bid stack itself influences the demand for emission certificates and couples back in this way to the carbon price. Figure 2.2 shown in chapter 2 thus has to be modified as shown in figure 3.6. Figure 3.6: Illustration of market mechanism in carbon markets. Picture adapted from [11] As concerns the valuation of the power plant, the payoff of the spark spread option as introduced in (2.1) will have to be modified in the following way: P SS T (S T, P T, A T ) = max(s T K H (P T + L H A T ), 0) (3.30) where K H is as before the heat rate, i.e. the number of units of fuel need for the generation of one unit of electricity. L H is similar to the emission rate seen in expression (3.13) above. It tells us how many units of emissions are produced by the combustion of one unit of fuel and is hence a measure for how dirty the fuel is. As we want to connect the electricity price S T to the demand via the bid stack function as presented in section 2.3, we still have to develop a model for the price of A T, which does not only take into account the dependence of the certificate price on the emissions (like the model seen in the previous section 3.2) but also the dependence on the demand. Then one can use this to compute the price of the spark spread option and from this the value of the power plant in a similar way as described in chapter 2. Doing this is the plan for the rest of the thesis. 26

34 Chapter 4 Hybrid Approach In the last chapter, I have presented two modelling approaches for emission certificates and derivatives on them. While equilibrium models allow to gain insight into market mechanisms it is difficult to use them directly to price options or similar products. The reduced form approach on the other hand does not offer a deep insight into the driving mechanisms of the market but is more suitable for the pricing of derivatives. One drawback of the reducedform models presented so far is however that there is no real connection to fundamental quantities like the demand which actually drive the evolution of prices, and it can be difficult to calibrate them. Therefore in this chapter, I will describe a slightly different approach first presented in [25] which avoids these problems and rather follows the spirit of [11] in the way derivatives are connected to market fundamentals. 4.1 The Model I will make a simplifying assumption and consider a single compliance period model only. Under the condition that one excludes banking or any other of the mechanisms connecting different compliance periods described in chapter 3, this can easily be extended to several compliance periods, see for instance the discussion for the valuation of power plants before (4.22) below. The more involved situation involving banking of certificates will not be considered. The model will have the following two fundamental input parameters 1. The demand D t : This is the only stochastic input for the model and is supposed to be inelastic. Some possibilities to model this quantity have already been introduced in chapter 2. They are always of the form dd t = µ(t, D t )dt + σ(t, D t )dw t (4.1) 27

35 µ(d) σ(d) BM ˆµ (D f(t)) + f (t) ˆσ (D f(t)) MR κ( µ D + f(t)) + f (t) σ EMR D [κ(µ ln(d) + f(t)) + f (t) σ2 ] σ D Table 4.1: Overview over all models discussed in chapter 2. Recall that the function f describing seasonality is given by f(t) = a 1 + a 2 t + a 3 cos(2πt + a 4 ) + a 5 cos(4πt + a 6 ) where the a i are constants. and I simply recall in table 4.1 the three possibilities outlined in chapter 2. It should be noted that I often omit the arguments for the drift and volatility term in order to ease the notation. That is, I write µ = µ(t, D) and σ = σ(t, D). The tilde then distinguishes them from the constants µ, µ, ˆµ and σ, σ, ˆσ. 2. Total emissions E t : The total emissions from the beginning of the compliance period up to today are supposed to be determined by the emission rate e t = e t (A t, D t ) which depends itself on the price of the emission certificate and the demand. It is determined by an instantaneous merit order: The producers take into account the price of A t when they submit a bid to the market. The total emissions are then given by integration over time up to time t E t := t 0 e s (A s, D s )ds (4.2) Furthermore e t (A t, D t ) is assumed to be previsible. The stochastic differential equation describing E t is given by de t := e t (A t, D t )dt (4.3) so there is only a drift term and the stochasticity enteres only via D t and A t. Let us now consider a general derivative V t = V (t, D t, E t ) on D t and E t. Note that this includes A t which can also be considered as a derivative on demand and emissions. Using Itô s lemma and the forms (4.3) and (2.17) one obtains [25] dv t = [ t V t + µ D D V t + e t (A t, D t ) E V t + 12 σ2d 2DV ] t dt + σ D D V t dw t (4.4) because all the second order and cross terms except the term involving dd t vanish due to the previsibility (4.2) of E t. If we now equate the drift term to rv t (where r is the risk-free rate) we can deduce a partial differential equation for V t. For A t itself this gives t A t σ2 D 2 DA t + µ D D A t + e t (A t, D t ) E A t ra t = 0 (4.5) 28

36 which is non-linear due to the term e t (A t, E t ) through which A t influences its own evolution. For a single compliance period model a terminal condition at expiry t = T (i.e. at the end of the compliance period) has to be specified. As the emission certificate issued cover total emissions up to a cap K their value is zero if total emission remain under the cap. In this case, there are more emission certificates in the market than needed, which reduces their value to zero. When the cap is exceeded however, there are not enough emission certificates to cover all emissions so that some market participants will have to pay the penalty amount π, which has been fixed by the regulator. This amount is just the price one would be willing to pay in the market for an emission certificate in order not be forced to pay the penalty. Altogether this means that one has a digitial option with payoff { π if ET > K A T (D T, E T ) = π H(E T K) = 0 else (4.6) where K denotes the cap for the emissions at the end of the compliance period and H(x) := 1 {x>0} is the Heaviside function. Let us come back to equation (4.4) and the case of an arbitrary derivative V t = V (t, D t, E t ). As already described above, equating the drift term of (4.4) to rv t yields the following partial differential equation for V t : t V t σ2 D DV 2 t + µ D D V t + e t (A t, D t ) E V t rv t = 0 (4.7) This equation is linear but contains the prefactor e t (A t, D t ) in front of the derivative with respect to E, through which the value of A t (given by the solution of (4.5)) enters the problem. I will briefly come back to (4.7) later when I look at the valuation of the power plant. I first want to analyse equation (4.4). In order to specify this equation completely one still needs boundary conditions (in addition to the terminal condition (4.6)). We need two boundary conditions for D and one for E, as I now will explain: ˆ For the demand the situation is as follows at the lower boundary (D t = 0): Suppose the process for D t is absorbing (like for the exponential mean reverting process (EMR)) and that the emission rates drop to zero when there is no demand e t (A t, D t = 0) = 0 (4.8) In this case, when there is no demand then either the total emissions already exceed the cap or they do not. In the first case the emission certificate has already reached 29

37 its (discounted) maximal value, in the latter case its value is zero. Hence A t (D t = 0, E t ) = exp( r(t t)) π H(E t K) (4.9) If the process can leave D t = 0 again like for example for a mean-reverting process (MR) or if emissions also happen for D t = 0, the situation is more complicated. A special case occurs when the emission rate does not depend on the demand at all, for example for a constant emission rate, which will be considered in the next section. In this case condition (4.8) is not satisfied and (4.9) does not hold. ˆ Let us consider the situation at the upper boundary for D t. I will make here the following simplifying assumption: When the demand is maximal (including possibly infinity) the price of the emission certificate will not matter to the producers as they are making huge profits. Mathematically this can be expressed by the following boundary condition at D = D max A t D (D t = D max, E t ) = 0 (4.10) This says that an infinitesimal increase in demand does not change the value of A t as the producers do not modify their behaviour in this case. This pragmatic approach involves the described assumption but as D max is very large and the probability of D t reaching this point is small, the error in this approximation should not disturb the behaviour of the solution to much. ˆ For the total emissions we observe that there are only first derivatives involved. Furthermore, looking at the characteristics yields t E λ = 1 and λ = e t < 0 (where λ parametrises the characteristics). This shows that information flows from the upper boundary into the region we are interested in, and from the lower boundary out of this region. In other words we need to specify one boundary condition at the maximal value of E. When emission are very large, so that the cap K is already exceeded at time t the value of A t will certainly be π at time T as the emission do not decrease. Therefore the value of A t is given by and by the same value for all E t > K. A t (D t, E t = K) = exp( r(t t)) π (4.11) Equation (4.5), the terminal condition (4.6) and the boundary conditions (4.9)-(4.11) completely specify the price of A t. 30

38 Of course, we need to know the exact form of e t (A t, D t ) also. In principle it should be determined from empirical data of quoted prices on the bid stack which include the effect of the emission certificates. However this task is highly nontrivial and shall not be part of this thesis. Therefore, I will simplify the situation and assume a parameteric form for e t (A t, D t ). I discuss below a possible parametrisation, using the asymptotic behaviour of e t (A t, D t ) as a guide, which will be used for the numerical inverstigations to follow. ˆ When the price A t gets zero one has de t = e t (D t )dt and the model for A t becomes linear. The expression for e t (D t ) has to vanish for D t 0 (see below). A simple form for e t (D t ) in this case is to assume e t (D t ) = const D t. Likewise when A t reaches its maximal value e r(t t) π, the emission rate can possibly (but not necessarily) become zero and will be in any case smaller than for lower values of A t. Otherwise the effect of the emission certificates on the production behaviour would be the opposite of what is intended. A simple expression to capture the dependence on A t is the factor C e A α t (with α 0), where the constant C e has to be larger than or equal to π α. ˆ When considering the dependence of D t one can make two observations: First, when demand is zero, no one will produce and therefore there are no new emissions e t (A t, 0) = 0 (4.12) Second, when demand increases the emission rate will certainly not decrease as it is more advantageous than before for market participants to produce. e t (A t, D t ) should be a nondecreasing function of D t. Therefore These considerations still leave room for many possible parametrisations. Here I simply suggest a very simple form. Taking into account the considerations above, one way to parametrise e t would be the following expression e t (A t, D t ) = γ D β t (C e A α t ) (4.13) There are four parameters involved: α, β, γ and the constant C e which has to be larger than π α.the prefactor γ determines the overall strength of the emission rate. β describes how strongly emissions go up when the demand increases, whereas α measures the non-linearity of the model. The larger α gets the stronger is the feedback effect of A t on itself. As far as the constant C e is concerned, all that matters is the difference to π α as it determines how strong emissions are when A t reaches its maximal value. These parameters in principle have to be determined by calibrating the model to historical data. Next, I will consider solutions of the differential equation (4.5) with payoff (4.6), boundary conditions (4.9), (4.10) and (4.11) and the form (4.13) of the emission rate. 31

39 4.2 Value of the Emission Certificates Special Cases In order to gain more insight into equation (4.5), I will look at some simple special cases first. ˆ Vanishing emission rate : This very simple case with e t = 0 has the solution A t (E t, D t ) = e r(t t) π H(E t K) (4.14) As emissions do not change over time, the value is just the discounted terminal value. ˆ Constant emission rate : Also in this case with e t (A t, D t ) =: e = const the solution is simple, as the behaviour of the total emissions is deterministic. If the total emissions are E t at time t then at maturity the total emissions will be E t + e (T t), so that A t (E t, D t ) = e r(t t) π H(E t + e (T t) K) (4.15) Graphically this is just a simple step function (as function of E t ) for which the step moves gradually towards K. Note that the boundary condition (4.9) does not hold as (4.8) is not fulfilled. ˆ Emission rate independent of the price of the certificates: In this case, where e t (A t, D t ) = e t (D), there is no feedback effect of the price A t on itself the partial differential equation is linear and governed by the stochastic process of D t which drives the evolution of E t. Although this case in not analytically solvable in this generality, it can be easily treated by Monte-Carlo simulation. One simply simulates the stochastic process for D t and from this then deduces the total emissions E t. dd t = µ D (D t )dt + σ D (D t )dw t de t = e t (D t )dt (4.16) This offers a convenient way of checking the general finite difference approach to the solution of the general equation (4.5). The result will be presented in the next section. One should note however that for all the special cases just presented the allowance price does not affect the emissions as the emission rate does not depend on A t. This is obviously not what is intended by the introduction of an emission trading scheme. Therefore this simple cases should only considered as a mathematical aid to understand better the behaviour of the partial differential equation (4.5) rather than a statement about the functioning of emission trading schemes. 32

40 4.2.2 Numerical Results After some simple cases have been considered in the last section, I will now look at the numerical solution of the equation (4.5). It will be solved using an implicit finite difference scheme which is described in detail in the appendix A. I will always use the form (4.13) and the exponentially mean-reverting process (2.22) for D t. In table 4.2, I recall the meaning of the parameters involved. In addition to these there are the parameters a i of function f(t) introduced in (2.19) which describes the seasonality. They have always been chosen as a 1 = 0.746, a 2 = 0, a 3 = 0.114, a 4 = 0.862, a 5 = 0.186, a 6 = which corresponds to the values determined in [2]. Their influence has turned out to be rather small. Numerical Parameters Parameters emission rate Parameters Demand/Payoff N S Number of simulations γ overall strength µ long-term mean N T Number of time steps α strength of non-linearity κ mean-reversion speed M D Number of D steps β dependence on D σ volatility M E Number of E steps C e strength for maximal A t K emission cap D max Maximal value of D π penalty Table 4.2: Relevant parameters for the computations. See also the nomenclature on pages iii and iv. The numerical parameters have been introduced in the appendix A, the emission rate in this chapter and the demand process in chapter Comparison of MC and FD for the linear case When the parameter α is equal to zero, the model is linear as there is no feedback effect of the price of the emission certificate on itself. As a simple consistency check I solve the model in this case once with Monte-Carlo simulation and once with the finite difference scheme developed in the appendix. To start with, the case β = 0 is considered, which was solved analytically above. Figure 4.1 and 4.2 show how the price of the emission certificate develops when the time to maturity gets short. One observes a step function, which is in line with the analytical solution (4.15). Furthermore, the consistency is also checked for values of β which are different from zero. Figure 4.3 shows that deviations exist but are comparatively small. They are due to finite size effects and the approximations in the boundary conditions. Figure 4.3 also gives an insight into the dependence of the model on β. Whereas for β = 0 the price of the emission certificate is a step function, the step gets more and more rounded when β increases, and it also moves further towards lower emissions. 33

41 Figure 4.1: The plot shows the evolution in time for the simple case of a constant emission rate (β = 0 and α = 0), computed via the finite difference scheme described in the appendix. The remaining parameters are: K = 20, π = 40, r = 0.05, γ = 3.5, C e = π α + 1, σ = 1.393, κ = 4.24, µ = 1. Numerical parameters were as follows: N T = 100, M D = 32, M E = 64, D max = 40. Figure 4.2: The plot shows the evolution in time for the simple case of a constant emission rate (β = 0) and (α = 0), computed via Monte-Carlo simulations with 100 simulations. The remaining parameters are: K = 20, π = 40, r = 0.05, γ = 3.5, C e = π α + 1, σ = 1.393, κ = 4.24, µ = 1. 34

42 Figure 4.3: Comparison of the Monte Carlo simulation and the finite differenc scheme for the linear case (α = 0). Different values of β are considered and the time to maturity is equal to 2.1.The remaining parameters are: K = 30, π = 40, r = 0.05, γ = 3.5, const = π α + 1, σ = 1.393, κ = 4.24, µ = 1, T = 2.1. The demand is D = 50. Numerical parameters are as follows: N T = 100, N S = 1000 (Monte-Carlo part), M D = 64, M E = 64, D max = 100. This figure shows also that the dependency on the emissions is different than in the model presented in section 3.2, compare figure 3.5. Error bars (RMSE) for the Monte Carlo simulation are smaller than the symbol size. Figure 4.4: Dependence of the price of the emission certificate on the demand and the emissions for different values of α, β and γ. The remaining parameters are:k = 20, π = 40, r = 0.05, C e = π α + 1, σ = 1.393, κ = 4.24, T = 2.1. Numerical parameters were as follows: N T = 100, M D = 64, M E = 64, D max =

43 Dependence on the parameters Next, I will consider the dependence of the price of the emission certificate as described by this model on some of the parameters involved. First, there are the parameters of the emission rate e t (A t, D t ) as already described above. ˆ γ: This parameter is a simple prefactor and determines the overall scale of the emission rate. ˆ β: This parameter determines how strongly the emission rate reacts to a change in demand. If β is very large, the emission rate will increase more rapidely when demand rises, whereas small β indicates a weak dependency. ˆ α: This parameter determines the nonlinearity of the model. For α = 0 the model is linear: There is no feedback effect of the price of the emission certificate on itself. When α gets larger the feedback effect becomes larger as well. Figure 4.5: Dependence of the price of the emission certificate on the parameter γ for different values of β. The remaining parameters were: K = 30, π = 40, r = 0.05, α = 0.7, C e = π α + 1, σ = 1.393, κ = 4.24, µ = 1, T = 2.1. The position on the grid is D = 1 2 D max, E = 2 3 K. Numerical parameters are as follows: N T = 100, M D = 64, M E = 64, D max = 100. Second, there are the parameters of the stochastic driver D t. Recall that we are using an exponential mean-reverting process. The mean-reverting process which is exponentiated has the following parameters: ˆ σ: The volatility parameter, which determines how strongly the process fluctuates. 36

44 ˆ µ: The mean reversion level towards which the process tends in the long term. ˆ κ: The mean reversion speed of the demand process, that is the strength with which the mean-reverting process is pulled back towards the mean reversion level. First, I consider the parameters of the emission rate. The results are shown in figures 4.4, 4.5, 4.6 and 4.7. In figure 4.4 it is first demonstrated how the shape of the surface presented for the simple linear case in figure 4.1 changes when the parameters of the emission rate are varied. As one can see changing the value of β distorts the step function in the region of large values for both D and E, whereas increasing α has the effect of making the transition region of the step function larger so that the change in value for increasing E does not happen so suddenly any more. In figure 4.5 the dependence on γ for several value of β is shown. All curves increase with increasing γ, which was to be expected as the overall emissions increase when γ goes up. This in turn means that the emission certificate is more likely to be in the money when γ is larger. One can also see that for larger values of β the increase happens earlier and more suddenly. This is consequence of the fact that for larger β, the emission rate e t increases much more rapidly for increasing D than for smaller β. Figure 4.6: Dependence of the price of the emission certificate on the parameter α for different values of β and for the cases when for all α the term C e π α (left panel) or the term C e (right panel) is constant. The remaining parameters are:k = 20, π = 40, r = 0.05, σ = 1.393, κ = 4.24, γ = 0.1 µ = 1, T = 2.1. The position on the grid is D = 1 2 D max, E = 7 10 K. Numerical parameters are as follows: N T = 20, M D = 64, M E = 32, D max =

45 In figure 4.6 the dependence on α is shown for several values of β. Here two different situations can be considered. The first one is when the constant C e is the same for all values of α considered, that is in this case C e = π The disadvantage of this representation is that for smaller values of α, the term C e in the emission rate dominates the term A α t too strongly so that the dependence on α becomes hardly noticable as is shown in the inset of figure 4.6. Therefore the situation is considered when for each value of α the constant C e is chosen such that C e = π α + 1 (main panel of figure 4.6). One then observes a pronounced increase in value for increasing α. This was to be expected as larger α means higher emissions (for A t and D t fixed) which makes the certificate more likely to be valuable at the end of the compliance period. For the different values of β the shape of the curves does not change much, but the curve is shifted sideways. Finally in figure 4.7 the dependence on β for several values of γ is shown. One can see Figure 4.7: Dependence of the price of the emission certificate on the parameter β for different values of γ. The remaining parameters are:k = 30, π = 40, r = 0.05, α = 0.7 C e = π α + 1, σ = 1.393, κ = 4.24, µ = 1, T = 2.1. Position on the grid is D = 1 2 Dmax, E = 1 2 K. Numerical parameters are as follows: N T = 100, M D = 64, M E = 64, D max = 100. that for increasing β the value of the emission certificate rises as could already been seen on the previous figures. This happens because (for fixed D t ) a larger value of β leads to higher emissions which in turn makes the certificate more likely to be in the money at expiry. The value of α (which was chosen to be 0.7 in the figure) seems to have an influence on the curve especially for lower values of β. In fact it turned out that for vanishing α all curves 38

46 become almost step functions, i.e. the value of the emission certificate rises quite rapidely from 0 to its maximal value (see inset of figure 4.7). Outside this transition region the price of the emission certificate is then almost independent of β and γ influences the curves only insofar as an increasing γ pushes the transition region towards smaller values β. Figure 4.8: Dependence of the price of the emission certificate on the parameter κ for different values of β. The remaining parameters are:k = 20, π = 40, r = 0.05, α = 1.1, γ = 0.05, C e = π α + 1, σ = 1.393, κ = 4.24, µ = 1, T = 2.1. The position on the grid is D = 1 2 D max, E = 5 6 K. Numerical parameters are as follows: N T = 100, M D = 32, M E = 32, D max = 100. The dependence on the parameters of the stochastic driver D is shown in figures 4.8, 4.9 and In figure 4.8 we see the dependence on κ for different values of β. All curves are decreasing, for smaller values of β the decrease happens more rapidly than for larger values. In figure 4.9 the dependence on σ is shown for different values of β. They all increase for increasing σ, as a rising volatility increases the price just as for other derivatives. In figure 4.10 the dependence on µ is shown for different values of β. Larger values of µ mean that the long-term mean of the demand process is higher and therefore the value of the certificate also rises as it gets more valuable for a higher demand. 39

47 Figure 4.9: Dependence of the price of the emission certificate on the parameter σ for different values of β. The remaining parameters were:k = 30, π = 40, r = 0.05, α = 0.3, γ = 0.5, C e = π α + 1, σ = 1.393, κ = 4.24, µ = 1, T = 2.1. The position on the grid is D = 1 2 D max, E = 1 2 K. Numerical parameters were as follows: N T = 100, M D = 32, M E = 32, D max = Valuation of a Power Plant in Carbon Markets General Approach We recall from chapter 2 (equation (2.3)) that when one wants to compute the value of the power plant one sums (or integrates) over a spark spread option with different times to maturity N P V P (t, D, E, A, P,...) = C P t V SS (t, D, E, A, P,... ; t i ) (4.17) where T P is the time when the plant will stop operating and the time span from t to T P has been divided into N P equally spaced segements. The capacity C P will always be assumed to be equal to one in what follows. As opposed to chapter 2, I now want to take the emission certificates into account. This means that the expression for the exchange option gets modified, which has already been indicated in formula (4.17) by the inclusion of A into the list of variables in V SS (t, D, E, A, P,... ; t i ) on which the value of the power plant depends. The dots indicate that there can be a further dependence on additional variables (like the capacity introduced in (2.28)), depending on the model. i=0 The producer now has to buy not only the fuel but also emission certificates to offset 40

48 Figure 4.10: Dependence of the price of the emission certificate on the parameter µ for different values of β. The remaining parameters are: K = 30, π = 40, r = 0.05, α = 1.2, γ = 0.01, C e = π α + 1, σ = 1.393, κ = 4.24, T = 2.1. The position on the grid is E = 5 6 K, D = 1 2 D max. Numerical parameters were as follows: N T = 100, M D = 64, M E = 32, D max = 40. the emission generated by the combustion of the fuel. Therefore the value of the exchange option V SS (t, D, E, A, P,... ; t i ) is now given by V SS (t, D, E, A, P,... ; t i ) = e r(ti t) E Q( ) P ti D t = D, E t = E, A t = A,... (4.18) where the payoff profile is given by the following expression P ti = max (S ti K H (P ti + L H A ti ), 0) (4.19) As before, K H is the number of units of fuel needed for generation of one unit of electricity. L H is a new parameter which describes the number of certificates needed to set off the emissions generated by the combustion of one unit of fuel. For L H = 0 we are back in the case without emission certificates. For a comparatively clean kind of fuel L H will be small, whereas a very dirty kind of fuel will mean that L H is larger. The electricity price S ti is connected to the demand D t via the bid stack function as described in chapter 2. We will only consider the bid stack function introduced by Barlow in detail, i.e. the simple form S t = B(D t ) = (1 + α B D t ) 1/α B (4.20) where α B is a parameter. In more sophisticated models as was also described in chapter 2 the concrete form of the bid stack function may involve other stochastic variables like 41

49 the gas price or the capacity. Some results for a more complicated bid stack function are shown in appendix B. In general, the live time of a power plant will span several compliance periods. Suppose that there are n compliance periods and the end of each period is denoted by T (i) for i = 1,..., n. All compliance periods except the last one end within the remaining live time T P of the power plant and when one compliance period has ended, the next one starts immediately. So one formally has t T (1), T P T (n), T (i 1) < T (i) (4.21) for i = 2,..., n. I exclude banking or any other mechanism connecting the different compliance periods, which were described at the beginning of chapter 3. Then one can formally define A P t (D, E) = A T (1) t (E, D) 1 {t T (1) } + n i=2 A T (i) t (E, D) 1 {T (i 1) <t T (i) } (4.22) where A T (i) t (E, D) denotes the emission certificate as treated before, expiring at time T (i). In what follows, I will only consider the situation when there is only one compliance period which ends at T P for which the formulas (4.17), (4.18) and (4.19) hold. However one could extend the following investigations by replacing A t (D, E) by A P t (D, E) in the formulas. 1 In principle there are two approaches to determine the value of the exchange option. 1. Finite Differences: As shown in (4.7), a derivative on A t of the form V SS t = V SS (t, D, A) obeys the partial differential equation t V SS t + µ D D V SS t + e t (A t, D t ) E V SS t with the terminal condition (4.19) σ2 D 2 DV SS t rv SS t = 0 (4.23) This holds if D is the only stochastic driver. This equation can be discretised in the same way as described for the corresponding equation for A t. Then it can be solved with a finite difference scheme. However there are the following problems with this approach ˆ In addition to the terminal condition (4.19) boundary conditions have to be specified. This can become rather involved in this case. 1 Another approach would be to argue that a power plant with a long remaining lifespan should have a value nearly independent of the demand, i.e. today s demand would only affect the closest spark spread options. In this case on would expect the spark spread options to be the same for each compliance period except for the first one. However, this line of thought will not be persued any further in this thesis. 42

50 ˆ As mentioned, equation (4.23) only holds if D t is the only stochatic driver. When we look at a plant which uses coal as fuel, it can be a reasonable approximation to simply use the forward curve for the coal price [2], so that D t is indeed the only stochastic variable. However, this will not be correct any more in the case of an gas-powered plant. Also, other stochastic drivers can enter the problem if a more complicated model for the bid stack function is used, for instance the form (2.14) involving the capacity. Although one could in principle extend equation (4.23) to include also terms analogous to the terms containing the derivatives in D, this becomes a very high-dimensional problem for which a simple finite difference approach tends to become very slow. 2. Monte-Carlo Simulation: This is more suitable for higher-dimensional models with more variables and will be used for the investigations which follow. One proceeds in the following way: (a) The starting values at time 0 are given by E, D etc. The starting value of A t at time t 0 = 0 is computed by the algorithm described in appendix A. (b) Given the values at time t j, the value of the emissions at time t j+1 is computed by E tj+1 = E tj + e tj (A tj, D tj ) ( t) (4.24) where t = t j+1 t j is the grid length on has chosen for the discretisation of the time axis from t to the expiry date t i of the spark spread option. 2 Note that this need not necessarily be the same length as in the finite difference algorithm used for the computation of A tj. (c) For the stochastic drivers the values D tj+1, P tj+1 (possibly C tj+1 ) at time t j+1 are given by their respective stochastic processes. As long as they do not involve A tj or E tj they can also be computed in a separate loop at the beginning of the whole computation. (d) The value of the emission certificate A tj+1 (D tj+1, E tj+1 ) at time t j+1 is now computed with the algorithm described in the appendix. (e) Steps (b) to (d) are repeated until the terminal time t i is reached. At t i the payoff profile is evaluated and discounted to time t. (f) This is repeated N S times and the mean is computed. 2 If the expiry dates are not equally spaced, one would have to adapt the formuly accordingly 43

51 The above procedure yields the value of the spark spread option with expiry time t i at time t. To obtain the value of the power plant one has to this for all expiry times t i and then compute the sum in (4.17) Value of a Power Plant and Abatement Costs I will now follow the Monte Carlo approach described in the last section to compute the value of the power plant. I will discuss the following case: 1. The simple bid stack function (2.12) as introduced by Barlow is used, so that we have the addtional parameter α B in the model. For comparison, in appendix B some results for the bid stack function (2.14) are also presented. 2. The plant is gas-powered and the gas price is described by the processes (2.25) and (2.26). 3. As it was already the case for the discussion of the emission certificates, I will only consider the parameteric form (4.13) for the emission certificates. I look first at the dependence on the parameter L H. In figure 4.11 one can see the expected behaviour: For L H = 0 the value of the power plant is the same as without emission certificates. The more L H increases, the more the value of the power plant decreases. This also shows in particular that a cleaner fuel can bring the value of the power plant much closer to the initial value without emission certificates, whereas a fuel which produces many emissions can sigificantly reduce its value if abatement costs are taken into account. In appendix B the same plot is shown for the bid stack function (2.14). As one can see in figure A.4, the decay looks similar although one needs to look at larger values for L H. In figure 4.12 we show the dependency on the demand D for different values of L H. We see that L H has a decisive influence on the behaviour. For a cleaner fuel (which means smaller L H ) the value of the plant is a concave function of D, i.e. the value increases relatively rapidely with rising demand. This changes for larger value of L H, that is for dirtier fuels. The value of the power plant becomes a convex function of the demand, i.e. the value first increases only slowly with increasing demand. The reason for this comparatively complex behaviour is that there are two competing mechanisms in the expression for the spark spread option. On the one hand the electricity price will go up when the demand rises. On the other hand the price of the emission certificate also increases with increasing demand, as we have seen in the last section. As this price is added to the costs of the fuel, it counteracts the increase of the electricity price. As the price of the emission certificate is 44

52 Figure 4.11: Value of the power plant as function of L H with and without emission certificates. The remaining parameters are K = 20, π = 40, r = 0.05, α = 1.2, β = 0.1, γ = 1, C e = π α + 1, σ = 1.393, κ = 4.24, µ = 1, T = 2.1, α B = 1.4, µ (1) G = 1.33, µ(2) G = 0.053, κ G = , σ (1) G = 5.836, σ(2) G = Position on the grid is D = 5 and P = 5, initial emission are E = 10. Numerical parameters were as follows: N T = 100, N P = 20, N S = 1000, M D = 16, M E = 16, D max = 100. Error bars indicate the RMSE bounded by π, but the electricity price is not, the value of the power plant will eventually always increase for large enough values of D. We remark also, that the bid stack function seems to have a strong influence on this, as can be seen by comparing with figure A.1 which shows a similar computation for the bid stack function (2.14). In this bid stack function the dependence on the demand is logarithmic. Although the value of the power plant does descrease for larger L H also in this case, the shape of the line looks more stable (i.e. no sign change of the second derivative). In figure 4.13 we consider the dependence on the fuel price G t for different values of L H. Here the bevaviour is not changed too much by the introduction of the emission certificate, as it simply increases the costs. Hence in this case, the inclusion of the emission certificates essentially causes a parallel shift of the line in figure This is also true when the bid stack function (2.14) is used (compare figure A.2). However, in the latter case the shape of the curves is different from the case when Barlow s bid stack functions was used due to the fact that the gas price G t turns up as a linear factor in the bid stack function now. In figure 4.14 a three-dimensional plot of this shown, which illustrates the dependence on demand and gas price for different values of L H. Finally in figure 4.15 the dependence on demand and gas price is shown again, this time for different remaining operating times and one fixed value of L H = This can be compared to figure

53 Figure 4.12: Value of the power plant as a function of the demand. The remaining parameters are:k = 20, π = 40, r = 0.05, α = 1.2, β = 0.1, γ = 1, C e = π α + 1, σ = 1.393, κ = 4.24, T = 2.1, α B = 1.4, K H = 1.1, L H = 0.025, µ (1) G = 1.33, µ(2) G = 0.053, κ G = , σ (1) G = 5.836, σ(2) G = The fuel price is P = 5, total emission are E = 10. Numerical parameters are as follows: N T = 100, N P = 20, N S = 10000, M D = 16, M E = 16, D max = 100. Error bars (RMSE) are smaller than symbol size Figure 4.13: Value of the power plant as function of the fuel price with and without emission certificates. The remaining parameters are:k = 20, π = 40, r = 0.05, α = 1.2, β = 0.1, γ = 1, C e = π α + 1, σ = 1.393, κ = 64.24, µ = 0.001, T = 2.1, α B = 1.4, K H = 1.1, L H = 0.025, µ (1) G = 1.33, µ(2) G = 0.053, κ G = , σ (1) G = 5.836, σ(2) G = Demand is D = 5, total emissions are E = 10. Numerical parameters are as follows: N T = 100, N P = 20, N S = M D = 16, M E = 16, D max = 100. Error bars indicate the RMSE. 46

54 Figure 4.14: Value of the power plant for different values of L H. The remaining parameters are:k = 20, π = 40, r = 0.05, α = 1.2, β = 0.1, γ = 1, C e = π α + 1, σ = 1.393, κ = 4.24,µ = 1, T = 2.1, α B = 1.4, K H = 1.1, L H = 0.025, µ (1) G = 1.33, µ(2) G = 0.053, κ G = , σ (1) G = 5.836, σ(2) G = Total emissions are E = 10. Numerical parameters are as follows: N T = 100, N P = 20, N S = M D = 16, M E = 16, D max = 100. The RMSE is in the order of magnitude of 0.02 (top left panel), (top right panel), (bottom left panel) and (bottom right panel). As the relative RMSE gets higher for larger values of L H (stochasticity of A t enters more strongly in this case) the bottom figures look slightly more bumpy. Figure 4.15: Value of the power plant for different remaining operating times. The remaining parameters are:k = 20, π = 40, r = 0.05, α = 1.2, β = 0.1, γ = 1, C e = π α + 1, σ = 1.393, κ = 4.24,µ = 1, α B = 1.4, K H = 1.1, L H = 0.025, µ (1) G = 1.33, µ(2) G = 0.053, κ G = , σ (1) G = 5.836, σ(2) G = Total emissions are E = 10. Numerical parameters are as follows: N T = 100, N P = 20, N S = 1000 M D = 16, M E = 16, D max = 100. The RMSE is in the order of magnitude of (top left panel), (top right panel), (bottom left panel) and (bottom right panel). For longer remaining operating times the noise has more time to add up, so the surfaces look more bumpy in this case. 47

55 Chapter 5 Conclusion and Outlook The subject of this thesis has been on the one hand the valuation of a power plant with real option techniques, and on the other hand and more importantly the influence the introduction of emission certificates and the development of carbon markets has on the result. This question matters as emission certificates are an integral part of the measures which are taken against global warming caused by greenhouse gases. Ideally the inclusion of the emission certificates into the valuaion of the power plant will encourage investors to invest in cleaner technologies. I have first considered an economy without emission certificates and briefly recalled how the power plant can be evaluated in this case. I have mainly concentrated on an approach suggested by Coulon and Howison [2] which connects the demand to the electrictiy price via the bid stack function. After having introduced the main aspects of carbon markets I have then discussed a model which generalises this concept and includes also emission certicates. At the center of this model is a nonlinear partial differential equation which describes the value of the emission certificate as a function of demand and total emissions. After having considered some special cases, I have then developed a finited difference scheme to solve this equation numerically. I have investigated into the behaviour of this equation and its dependence of the various parameters of the model. Finally, I have considered the influence of the emission certificates on the value of the power plant within the context of this specific model. The investigation of the model for the emission certificate showed that the dependence on the different model parameters is reasonable. In particular the dependence on the parameters of the emission rate was what was generally to be expected: For an increasing emission rate the price of the emission certificate increases as well as the emission certificate becomes more likely to be in the money. At the same time the investigation demonstrated the importance of the function e t (D t, A t ) describing the emission rate. The main conclusion for 48

56 the emission price is that the model shows a reasonable behaviour but crucially depends on the concrete form of e t (D t, A t ), in particular on the interplay between D t and A t in its functional form. When the connection was made between the model for the emission certificate and the real option model for the value of the power plant, it turned out that the value of the power plant depends crucially on the parameter L H. This parameter describes how much emissions are generated when one unit of fuel is burned. For large values of L H the value of the power plant decreases significantly, which shows in particular that one statement of the model is, that emission certificates will indeed provide an incentive for producers of electricity to use cleaner fuels. For example gas roughly only produces half as much carbon dioxide as coal 1 [26, 27]. This means that L H is only half as big and this can significantly enhance the value of the plant (compare for example figure (4.13)). This is the main conclusion for the valuation of the power plant: A working carbon market can provide incentives to switch to cleaner fuels and can influence investment decisions to build new power plants towards cleaner technologies, as the abatement costs induced by the emission certificates can be significant during the lifetime of the power plant. Of course there is still plenty of room for further investigations. A decisive question will be how the actual form of the emission rate e t (D, A) looks like. This is a highly nontrivial question for which one would need to have actual bid stack data available. But also for the simple form of the emission rate which was used here, it is important to find out the actual value of the parameters in order to make a statement about the order of magnitude the observed effects will have. These calibration issues have not yet been touched. Finally one can look at the obvious extensions and consider more complicated bid stack function (like the one presented in chapter 2, equation (2.14)) involving also factors like the capacity which were not considered in detail here. Also it would be worth looking at alternative processes for the stochastic drivers or try to incorporate banking and similar mechanisms into the model. 1 the exact number depend on many factors including the conditions under which the gas and the coal resources are exploited 49

57 Chapter 6 Appendix A Finite Difference Algorithm In this section of the appendix, I describe briefly the finite difference algorithm used for the computation of the price of the emission certificates, i.e. for the numerical solution of the partial differential equation (4.5): t A t + µ(t, D) D A t + e t (A, D) E A t σ2 (t, D) 2 DA t ra t = 0 (A.1) The finite difference algorithm is developed by generalising the technique described in [28] to two dimensions. Another useful reference is [29]. In addition to equation (A.1), one has to take into account the terminal condition (4.6) and the boundary conditions (4.9),(4.10) and (4.11). Condition (4.10) is a Neumann boundary condition (imposed on the derivative At D ), all other boundary conditions are of Dirichlet type. In order to solve this equation numerically it will be discretised. To this end, I introduce first some shorthand notation. I will use equal spacing and write D i = i D i = 0,..., M D + 1 (A.2) E j = j E j = 1,..., M E (A.3) t k = k t k = 0,..., M T (A.4) where the values i = 0 and i = M D belong to the boundaries in the direction of D and j = M E to the boundary of E at the value E = K. The point i = M D + 1 is only formally included here in order to be able to build in the Neumann boundary condition as will be explained below. T = t M T is the end of the compliance period. Furthermore I write A k i,j := A(t k, D i, E j ) e t i,j := e t (A i,j, D j ) σ i := σ D (D i ) µ i := µ(t i, D i ) (A.5) 50

58 and for the Dirichlet boundary conditions at time t k : 1 ḡi k := A k i,m E i = 0,..., M D (A.6) fj k := A k 0,j j = 1,..., M E 1 (A.7) For the discretisation of the derivatives in D-direction, I use central differences, for the discretisation of the first derivative in E-direction right-sided differences. D A k i,j = Ak i+1,j Ak i 1,j 2 D (A.8) 2 DA k i,j = Ak i+1,j 2Ak i,j + Ak i 1,j (2 D) 2 (A.9) E A k i,j = Ak i,j+1 Ak i,j E (A.10) With the help of (A.8) the Neumann boundary condition (4.10), which tells us that the derivative with respect to D should vanish at the upper boundary for D, can now be expressed by the following equality A k M D +1,j = A k M D 1,j j = 1,..., M E 1 (A.11) This allows to eliminate the terms on the artificial boundary i = M D + 1 so that the index i then only runs from 0 to M D. For the derivative in time-direction, I will use forward differences which corresponds to a fully implicit scheme. t A k i,j = Ak+1 i,j A k i,j t (A.12) The results for the price of emission ceritificates presented in this thesis were all obtained using such a scheme. In fact, explicit and Crank-Nicolson schemes were also briefly considered. However, the explicit scheme seemed to produced significantly inferior results, while the results from a Crank-Nicolson scheme did not seem to differ very much from the implicit scheme. Therefore only the expressions for the implicit scheme are been quoted here, but it is straigthforward to adapt them to find the other schemes too. Introducing (A.8)-(A.10) and (A.12) into (A.1) and rearrangig terms yields the following 1 The boundary conditions have been discussed in the main text. In the notation introduced here the Dirichlet boundary conditions (4.9) and (4.11) read ḡ k i = exp( r(t t k )) π and f k j = exp( r(t t k )) π H(E K) 51

59 relation for the values of A k+1 i,j between two time-slices ( A k+1 i,j = A k i,j 1 + t + σ2 i t ( D) 2 + e ) i,j t + E }{{} A k i+1,j A k i 1,j A k i,j+1 =:a i,j ( µ i 2 D σ2 i t ) 2( D) }{{ 2 + } b i,j ( µi 2 D σ2 i t ) 2( D) }{{ 2 + } c i,j ( e ) i,j t = 2 E }{{} d i,j a i,j A k i,j + b i,j A k i+1,j + c i,j A k i 1,j + d i,j A k i,j+1 (A.13) which is valid for i = 1,..., M D 1 and j = 1,..., M E 1. For the boundary i = M D one uses (A.11) to replace the terms A k M D +1,j by Ak M D 1,j for all j. Equation (A.13) remains formally valid and the matrices a i,j and d i,j are defined in the same way as before also for the value i = M D. The other two matrices have a slightly different appearance on this boundary, namely b MD,j = 0, and c MD,j = σ2 M D t ( D) 2 j = 1,..., M E 1 (A.14) Now I define a (M D + 1) M E -dimensional vector ξ k = (ξ 1,..., ξ (MD +1)M E ) t with ξ k M E i+j := A k i,j (A.15) for i = 0,..., M D and j = 1,..., M E. This vector contains also the boundary terms. The vector η k is defined in such a way that it only contains the interior points and the points belonging to the Neumann boundary condition at i = M D. η k (M E 1) i+j := Ak i,j (A.16) for i = 1,..., M D and j = 1,..., M E 1. This vector has dimension M D (M E 1). Then one can rewrite the equation (A.13) in the following way η k+1 = (a + b + c + d + e }{{} =:A ) ξ k (A.17) 52

60 where the matrices have dimension M D (M E 1) times (M D + 1) M E. They look as follows a = M E {}}{ a 1,1... a 1,ME 1 0 a 2,1... a MD,M E 1 0 (A.18) b = 2M E {}}{ b 1, c 1,1... c = d = b 1,ME 1 0 c 1,ME 1 0 M E +1 {}}{ d 1,1... b 2,1... c 2,1... d 1,ME 1 0 b MD 1,M E M E 1. 0 c MD,M E }{{} M E d 2,1... (A.19) (A.20) (A.21) d MD,M E 1 One can now define reduced matrices of dimension M D (M E 1) times M D (M E 1) by crossing out the first M E columns plus columns (i + 1) M E (for i = 1,..., M D ). The 53

61 reduced matrices (the sum of which will be denoted by A) have the following form: a 1,1... a = a 1,ME 1 (A.22)... amd,me 1 b = c = M E 1 {}}{ b 1, M E 1 0 c 2, d 1,1... d = b 1,ME 1 c 2,ME 1 d 1,ME 2 b 2,1... c 2, M E 1 b MD 1,M E c MD,M E 1 0 }.{{.. 0 } M E 1 d 2,1... d MD,M E 2 0 (A.23) (A.24) (A.25) We now split up the vector ξ k into a vector containing the unknowns η k (interior points plus points on the Neumann boundary as defined above) and one containing the known boundary parts ξb k. In this way, we obtain the following equation Aη k = η k+1 A ξ k b (A.26) 54

62 The form of the term A ξb k on the righthand side can be easily computed. It is given by the sum of the following terms: a ξ k = 0 (A.27) b ξ k = 0 c 1,1 f1 k. c ξ k =. 0 d ξ k = c 1,ME 1f k M E M E 2 0 d 1,ME 1ḡ k 1 0. M E 2 0 d 2,ME 1ḡ k 2. d MD 1,M E 1ḡ k M D 1 (A.28) (A.29) (A.30) With these expressions and the value of η k+1 known, the system (A.26) can be solved for η k. Starting at the terminal condition and looping over all time-slices this will yield the numerical solution of (A.1) at time t = 0. The implementation of this algorithm and of all other numerical routines used in this thesis was done in matlab. In order to avoid bloating the thesis too much I have not listed the codes here. They are available on request from the author. B Some Results for an Alternative Bid Stack Function In this part of the appendix some result for the bid stack function (2.14) are shown, which should be compared to figures 4.12, 4.13, 4.14 and For the capacity, a slightly simplified version of the model described by equations (2.28)-(2.30) is used, in which all p i (probabilities for being in the jump regime) and λ i (parameter of the exponential random variable) are equal. They are denoted by p and λ. 55

63 Figure A.1: Value of the power plant as function of the demand with and without emission certificates. The remaining parameters are: K = 20, π = 40, r = 0.05, α = 1.2, β = 0.1, γ = 1, C e = π α + 1, σ = 1.393, κ = 64.24,µ = 0.001, T = 2.1, K H = 1.1, µ (1) G = 1.33, µ (2) G = 0.053, κ G = , σ (1) G = 5.836, σ(2) G = 0.53, p = 0.1, λ = 1, A 0 = 15, A 1 = 6,B 0 = 7, B 1 = 1, κ Z = 80, µ Z = 1, σ Z = 8, α Z = 2, ρ = 0.3. Gas price is G = 5, total emissions are E = 10. Numerical parameters are as follows: N T = 100, N P = 20, N S = M D = 16, M E = 16, D max = 100. Error bars indicate the RMSE. Figure A.2: Value of the power plant as function of the fuel price with and without emission certificates. The remaining parameters are: K = 20, π = 40, r = 0.05, α = 1.2, β = 0.1, γ = 1, C e = π α + 1, σ = 1.393, κ = 64.24,µ = 0.001, T = 2.1, K H = 1.1, L H = 1.0, µ (1) G = 1.33, µ(2) G = 0.053, κ G = , σ (1) G = 5.836, σ(2) G = p = 0.1, λ = 1, A 0 = 15, A 1 = 6,B 0 = 7, B 1 = 1, κ Z = 80, µ Z = 1, σ Z = 8, α Z = 2, ρ = 0.3. Demand is D = 5, total emissions are E = 10. Numerical parameters are as follows: N T = 100, N P = 20, N S = M D = 16, M E = 16, D max = 100. Error bars indicate the RMSE. 56

64 Figure A.3: Value of the power plant as function of the fuel price and demand for different values of L H. The remaining parameters are: K = 20, π = 40, r = 0.05, α = 1.2, β = 0.1, γ = 1, C e = π α + 1, σ = 1.393, κ = 64.24,µ = 0.001, T = 2.1, K H = 1.1, µ (1) G = 1.33, µ (2) G = 0.053, κ G = , σ (1) G = 5.836, σ(2) G = p = 0.1, λ = 1, A 0 = 15, A 1 = 6,B 0 = 7, B 1 = 1, κ Z = 80, µ Z = 1, σ Z = 8, α Z = 2, ρ = 0.3. Total emissions are E = 10. Numerical parameters are as follows: N T = 100, N P = 20, N S = M D = 16, M E = 16, D max = 100. The RMSE is in the order of magnitude of 2.7 (top left), 2.8 (top right), 2.6 (bottom left), 2.6 (bottom right). Figure A.4: Value of the power plant as function of L H with and without emission certificates. The remaining parameters are: K = 20, π = 40, r = 0.05, α = 1.2, β = 0.1, γ = 1, C e = π α + 1, σ = 1.393, κ = 64.24,µ = 0.001, T = 2.1, K H = 1.1, µ (1) G = 1.33, µ (2) G = 0.053, κ G = , σ (1) G = 5.836, σ(2) G = p = 0.1, λ = 1, A 0 = 15, A 1 = 6,B 0 = 7, B 1 = 1, θ Z = 80, µ Z = 1, α Z = 2, ρ = 0.3. Demand is D = 5, total emissions are E = 10. Numerical parameters are as follows: N T = 100, N P = 20, N S = M D = 16, M E = 16, D max = 100. Error bars indicate the RMSE. 57