Empirical performance of reduced-form models for emission permit prices

Abstract

The value of emission permits in environmental markets derives from the particular design features of the underlying cap-and-trade system. In this paper, we evaluate a model framework for the price dynamics of emission permits which accounts for these features in a reduced-form way. Based on permit futures and option data from the European Union Emissions Trading System, the world’s largest environmental market, we show that model variants which represent the design of the system most accurately provide the best fit to historical futures prices and achieve the best option pricing performance. Our results suggest that the specific design of cap-and-trade systems directly translates to the dynamic properties of emission permit prices, and that models tailored to environmental markets are the best choice for related pricing and risk management decisions.

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Notes

  1. 1.

    The G7 member states have agreed on this goal at the 41st G7 summit in June 2015. Subsequently, 195 countries worldwide have signed the Paris climate agreement, of which only the United States has announced the intention to withdraw.

  2. 2.

    By now, a double-digit number of cap-and-trade policies are in force all over the world, including systems in the European Union, in several states of Canada and the US, in South Korea, and in several provinces and cities in China.

  3. 3.

    Theory predicts that within a compliance period, the standard cost-of-carry relationship holds because permit spot and futures contracts can be traded without any frictions, see Hitzemann and Uhrig-Homburg (2018). Uhrig-Homburg and Wagner (2009) and Rittler (2012) show that this is also true empirically.

  4. 4.

    For \(R_t=0\), simply drop the last term \(e^{-r(T_m-{\overline{t}})^+}e^{\lambda _{m+1,t}}\) in (9) and all subsequent equations.

  5. 5.

    Unlike Grüll and Taschini (2009) and others, we exclude more sophisticated standard models, e.g., normal-inverse Gaussian (NIG) processes, from our analysis, since our focus is on the performance of reduced-form models within a pure Itô framework. We leave it open for future research to consider a reduced-form model framework for emission permit prices based on general Lévy processes.

  6. 6.

    The choice of this parametrization, which we label by CSCH due to its affinity to the hyperbolic cosecant, is mainly motivated by its simplicity and its favorable empirical behavior.

  7. 7.

    The ECX is the world’s largest carbon futures exchange and is operated by Intercontinental Exchange (ICE), http://www.theice.com.

  8. 8.

    In Phase I from 2005 to 2007, it was forbidden to bank emission permits into the next compliance period, which led to a collapse of permit prices once the market realized that the economy was long of permits. In the ongoing Phase III from 2013 to 2020, the EU ETS undergoes a structural reform that addresses the large surplus of emission permits in the system.

  9. 9.

    The only non-generic point is the initialization procedure. Since the transition equation is non-stationary in all of its components, we initialize the unscented Kalman filter by using diffuse priors (see Harvey 1989, pp. 121–122). In particular, we employ the approach of Rosenberg (1973) to treat the initial state as fixed and unknown, and infer it by maximum likelihood estimation (see Durbin and Koopman 2001, pp. 117–188).

  10. 10.

    For the model parametrization \(z^{CH}_k\), the parameters \(\alpha _1,\ldots ,\alpha _m\) are set to 1 and not included into the numerical optimization.

  11. 11.

    We abbreviate by writing (CS)CH when a statement holds for both the CH and the CSCH parametrization.

  12. 12.

    Similar two-stage procedures are frequently applied for other model classes, see for example Broadie et al. (2007).

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Correspondence to Steffen Hitzemann.

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We thank René Carmona, Martin Hain, Juri Hinz, Rüdiger Kiesel, Brenda López Cabrera, Marcel Prokopczuk, Claus Schmitt, Philipp Schuster, Nils Unger, as well as participants of the 2012 Financial Management Association Annual Meeting, 2013 EnergyFinance Conference, 2014 Institute of Mathematical Statistics Annual Meeting and Australian Statistical Conference, and seminar participants at Humboldt University Berlin for valuable discussions and helpful comments and suggestions. Financial support by the Graduate School 895 “Information Management and Market Engineering” at the Karlsruhe Institute of Technology (KIT), funded by Deutsche Forschungsgemeinschaft (DFG), is gratefully acknowledged.

Appendices

Appendix A: Dynamics of risk-neutral shortage probabilities

We derive the dynamics of the risk-neutral shortage probabilities \(A_k\) in general and show that all emissions processes (3) satisfying a narrow-sense linear SDE after transformation by a strictly increasing function lead to the same class of dynamics given by (6). For notational ease we omit the k indicating the compliance period in this section.

Let \(D_{t}(x_{t,T};.)\) be the cumulative density function of \(x_{T}\) given \(x_{t,T}\) at time t. By definition, it follows

$$\begin{aligned} A_{t}=1-D_{t}(x_{t,T};q)=:G_{t}(x_{t,T}). \end{aligned}$$
(27)

Applying Itô’s Lemma directly yields

$$\begin{aligned} { dA}_{t}=g_{t}(x_{t,T})\sigma (t,x_{t,T}) dW_t =g_{t}(G^{-1}_{t}(A_{t}))\sigma (t,G^{-1}_{t}(A_{t})) dW_t, \end{aligned}$$
(28)

where \(g_{t}\) is the first derivative of \(G_{t}\). Equation (28) describes the general dynamics of the risk-neutral shortage probabilities.

Aiming at simple dynamics of \(A_t\), we assume that \(x_{t,T}\) can be transformed by a strictly increasing function \(\upsilon \) in such a way that \(\upsilon (x_{t,T})\) follows a linear SDE in the narrow sense, i.e.,

$$\begin{aligned} d\upsilon (x_{t,T})=(a_1(t)\upsilon (x_{t,T})+a_2(t))dt+b(t)dW_t, \end{aligned}$$
(29)

where \(a_1:(0,T)\rightarrow \mathbb {R}\) and \(a_2:(0,T)\rightarrow \mathbb {R}\) are continuous, bounded functions and \(b:(0,T)\rightarrow \mathbb {R}^+\) is continuous and square-integrable. This especially covers a geometric Brownian motion as proposed by Carmona and Hinz (2011) or the case that \(\ln {x_{t,T}}\) follows an Ornstein-Uhlenbeck process \(d\ln {x_{t,T}}=\kappa (\mu (t)-\ln {x_{t,T}}) dt+\sigma (t) dW_t\). Given \(x_{t,T}\), we can write \(\upsilon (x_{T})\) explicitly as

$$\begin{aligned} \upsilon (x_{T})=e^{\int _t^T a_1(s)ds}\upsilon (x_{t,T})+\int _t^T e^{\int _s^T a_1(u)du}a_2(s)ds +\int _t^T e^{\int _s^T a_1(u)du}b(s)dW_s,\nonumber \\ \end{aligned}$$
(30)

see Karatzas and Shreve (1991), pp. 360–361. Particularly, \(\upsilon (x_{T})\) is normally distributed with mean

$$\begin{aligned} \mu _{\upsilon t,T}(x_{t,T})=e^{\int _t^T a_1(s)ds}\upsilon (x_{t,T})+\int _t^T e^{\int _s^T a_1(u)du}a_2(s)ds \end{aligned}$$
(31)

and standard deviation

$$\begin{aligned} \sigma _{\upsilon t,T}=\sqrt{\int _t^T e^{2\int _s^T a_1(u)du}b^2(s)ds}. \end{aligned}$$
(32)

It follows that the cumulative density function is given by

$$\begin{aligned} D_{t,T}(x_{t,T};y)=\Phi \left( \frac{\upsilon (y)-\mu _{\upsilon t,T}(x_{t,T})}{\sigma _{\upsilon t,T}}\right) , \end{aligned}$$
(33)

and we obtain

$$\begin{aligned} g_{t,T}(x_{t,T})=\Phi '\left( \frac{\upsilon (q)-\mu _{\upsilon t,T}(x_{t,T})}{\sigma _{\upsilon t,T}}\right) \frac{\mu _{\upsilon t,T}'(x_{t,T})}{\sigma _{\upsilon t,T}}. \end{aligned}$$
(34)

Inserting into (28) yields

$$\begin{aligned} \begin{aligned} { dA}_{t}&= \Phi '\left( \Phi ^{-1}(A_{t})\right) \frac{\mu _{\upsilon t,T}'(x_{t,T})\sigma (t,x_{t,T})}{\sigma _{\upsilon t,T}} dW_t \\&=\Phi '\left( \Phi ^{-1}(A_{t})\right) \frac{e^{\int _t^T a_1(s)ds}\upsilon '(x_{t,T})\sigma (t,x_{t,T})}{\sqrt{\int _t^T e^{2\int _s^T a_1(u)du}b^2(s)ds}} dW_t \end{aligned} \end{aligned}$$
(35)

for the risk-neutral shortage probability. Now observe that the dynamics of \(\upsilon (x_{t,T})\) is also given by

$$\begin{aligned} d\upsilon (x_{t,T})=(\upsilon '(x_{t,T})\mu (t,x_{t,T})+\frac{1}{2}\upsilon ''(x_{t,T})\sigma ^2(t,x_{t,T}))dt +\upsilon '(x_{t,T})\sigma (t,x_{t,T})dW_t,\nonumber \\ \end{aligned}$$
(36)

applying Itô’s Lemma to (3). Comparing this to (29) yields

$$\begin{aligned} \upsilon '(x_{t,T})\sigma (t,x_{t,T})=b(t), \end{aligned}$$
(37)

and we can write (35) as

$$\begin{aligned} \begin{aligned} { dA}_{t}&=\Phi '\left( \Phi ^{-1}(A_{t})\right) \frac{e^{\int _t^T a_1(s)ds}b(t)}{\sqrt{\int _t^T e^{2\int _s^T a_1(u)du}b^2(s)ds}} dW_t \\&=\Phi '\left( \Phi ^{-1}(A_{t})\right) \frac{c(t)}{\sqrt{\int _t^T c^2(s)ds}} dW_t \end{aligned} \end{aligned}$$
(38)

with \(c(t)=e^{\int _t^T a_1(s)ds}b(t)\). Carmona and Hinz (2011) show that for every continuous function \(z:(0,T)\rightarrow \mathbb {R}^+\) satisfying

$$\begin{aligned} \lim _{t\rightarrow T} \int _0^t z(s) ds=\infty , \end{aligned}$$
(39)

there exists a square-integrable continuous function \(c:(0,T)\rightarrow \mathbb {R}^+\) fulfilling \(\frac{c^2(t)}{\int _t^T c^2(s)ds}=z(t)\). Therefore we can construct every continuous function z satisfying (39) by the choice \(a_1(t)=0\) and \(b(t)=c(t)\). The other way round, \(c(t)=e^{\int _t^T a_1(s)ds}b(t)\) is a square-integrable continuous and positive function by the properties of \(a_1\) and b, and it follows that \(z(t)=\frac{c^2(t)}{\int _t^T c^2(s)ds}\) is positive and continuous and satisfies (39), see Carmona and Hinz (2011).

Altogether, for all dynamics of \(x_{t,T}\) for which a strictly increasing function \(\upsilon \) exists such that \(\upsilon (x_{t,T})\) follows a narrow-sense linear SDE (29), the class of possible dynamics for \(A_t\) is completely characterized by (6), with a continuous function \(z:(0,T)\rightarrow \mathbb {R}^+\) satisfying (39). Particularly, all possible dynamics for \(A_t\) in this class can be obtained by choosing an arithmetic Brownian motion (4) for the expected cumulative emissions process \(x_{t,T}\).

Appendix B: Evaluation of option pricing formulas

To calculate theoretical option prices within our reduced-form model framework, Eq. (14) requires the numerical evaluation of an \(m+1\)-dimensional integral. While this is straightforward in the one-dimensional case (\(m=0\)), the following transformations may help to improve computational efficiency for models with more price components. We demonstrate these transformations for the case of \(m=1\) and an additional component \(R_t\) as well as for \(m=2\) and \(R_t=0\).

In both cases, we decompose the bivariate normal distribution as proposed by Carmona and Hinz (2011) by defining

$$\begin{aligned} \mu ^{2,c}(x_1)=\mu ^2+\frac{\nu ^{1,2}}{\nu ^{1,1}}\left( x_1-\mu ^1\right) \quad \text {and}\quad \nu ^{2,2,c}=\nu ^{2,2}-\frac{(\nu ^{1,2})^2}{\nu ^{1,1}}, \end{aligned}$$
(40)

and we can write the integral in (14) as

$$\begin{aligned}&\int _{\mathbb {R}}\int _{\mathbb {R}} \left( e^{-r(T_1-{\overline{t}})}\Phi (x_1)p_1+e^{-r(T_1-{\overline{t}})}e^{x_{2}}-K\right) ^+\nonumber \\&\quad \cdot \,\varphi \left( \mu ^{2,c}(x_1),\nu ^{2,2,c};x_2\right) dx_2 \varphi \left( \mu ^1,\nu ^{1,1};x_1\right) dx_1 \end{aligned}$$
(41)

for the case \(m=1\) with an additional component \(R_t\) or as

$$\begin{aligned}&\int _{\mathbb {R}}\int _{\mathbb {R}} \left( e^{-r(T_1-{\overline{t}})}\Phi (x_1)p_1+e^{-r(T_2-{\overline{t}})}\Phi (x_2)p_2-K\right) ^+ \nonumber \\&\quad \cdot \,\varphi \left( \mu ^{2,c}(x_1),\nu ^{2,2,c};x_2\right) dx_2 \varphi \left( \mu ^1,\nu ^{1,1};x_1\right) dx_1 \end{aligned}$$
(42)

for \(m=2\) and \(R_t=0\).

Defining \(K^*(x_1)=K-e^{-r(T_1-{\overline{t}})}\Phi (x_1)p_1\), we note that the inner integral in (41) can completely be settled analytically according to

$$\begin{aligned} \begin{aligned}&\int _{\mathbb {R}}\left( e^{-r(T_1-{\overline{t}})}e^{x_{2}}-K^*(x_1)\right) ^+ \cdot \varphi \left( \mu ^{2,c}(x_1),\nu ^{2,2,c};x_2\right) dx_2 \\&=\left\{ \begin{array}{ll} e^{-r(T_1-{\overline{t}})} \int _{\mathbb {R}} e^{x_{2}} \cdot \varphi \left( \mu ^{2,c}(x_1),\nu ^{2,2,c};x_2\right) dx_2 -K^*(x_1), &{} \hbox {if }\quad K^*(x_1)\le 0; \\ \int _{\mathbb {R}}\left( e^{-r(T_1-{\overline{t}})}e^{x_{2}}-K^*(x_1)\right) ^+ \cdot \varphi (\mu ^{2,c}(x_1),\nu ^{2,2,c};x_2) dx_2, &{}\quad \hbox {otherwise.} \end{array} \right. \\&=\left\{ \begin{array}{ll} e^{-r(T_1-{\overline{t}})}\cdot e^{\mu ^{2,c}(x_1)+\frac{\nu ^{2,2,c}}{2}}-K^*(x_1), &{} \hbox {if }\quad K^*(x_1)\le 0; \\ e^{-r(T_1-{\overline{t}})}\cdot e^{\mu ^{2,c}(x_1)+\frac{\nu ^{2,2,c}}{2}}\Phi (d_1)-K^*(x_1)\Phi (d_2), &{}\quad \hbox {otherwise.} \end{array} \right. \end{aligned} \end{aligned}$$
(43)

with

$$\begin{aligned} d_1=\frac{\mu ^{2,c}(x_1)-\ln (\frac{K^*(x_1)}{e^{-r(T_1-{\overline{t}})}})}{\sqrt{\nu ^{2,2,c}}}+\sqrt{\nu ^{2,2,c}} \quad \text {and} \quad d_2=d_1-\sqrt{\nu ^{2,2,c}}. \end{aligned}$$
(44)

For (42), Carmona and Hinz (2011) show that numerical integration of the inner integral is only necessary if \(0<K^*(x_1)<e^{-r(T_2-{\overline{t}})}p_2\) since outside this interval we have

$$\begin{aligned} \begin{aligned}&\int _{\mathbb {R}}\left( e^{-r(T_2-{\overline{t}})}\Phi (x_{2})p_2-K^*(x_1)\right) ^+ \cdot \varphi \left( \mu ^{2,c}(x_1),\nu ^{2,2,c};x_2\right) dx_2 \\&=\left\{ \begin{array}{ll} 0 &{}\quad \hbox {if }\quad K^*(x_1)\ge e^{-r(T_2-{\overline{t}})}p_2; \\ e^{-r(T_2-{\overline{t}})}p_2\Phi \left( \frac{\mu ^{2,c}(x_1)}{\sqrt{1+\nu ^{2,2,c}}}\right) -K^*(x_1) &{}\quad \hbox {if }\quad K^*(x_1)\le 0. \end{array} \right. \end{aligned} \end{aligned}$$
(45)

It is straightforward again to evaluate the outer integral of (41) and (42) once the inner integral is calculated.

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Hitzemann, S., Uhrig-Homburg, M. Empirical performance of reduced-form models for emission permit prices. Rev Deriv Res 22, 389–418 (2019). https://doi.org/10.1007/s11147-018-09152-7

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Keywords

  • Emission permits
  • Price dynamics
  • Option pricing
  • Carbon derivatives
  • Environmental finance

JEL Classification

  • G13
  • Q50