Abstract
Equilibrium models have been widely used in the literature with the aim of showing theoretical properties of emissions trading schemes. This paper applies equilibrium models to empirically study permit prices and to quantify the permit price sensitivity. In particular, we demonstrate that emission trading schemes both with and without banking are inherently prone to price jumps.
Zusammenfassung
Zur Untersuchung der theoretischen Eigenschaften von Emissionshandelssystemen wurden in der Literatur meist Gleichgewichtsmodelle genutzt. In diesem Artikel nutzen wir Gleichgewichtsmodelle zur empirischen Untersuchung von Zertifikatepreisen und ihren Sensitivitäten. Wir zeigen insbesondere, daß Emissionshandelssysteme (auch wenn Banking zugelassen ist) Preisprozesse generieren, die Sprünge enthalten.
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Notes
The corresponding Europeanstyle call option pays off the maximum of zero and the difference between the asset price at maturity and the strike price, i.e. max(S _{ T }−K,0). Asianstyle options are popular because they are cheaper compared to Europeanstyle options as the averaging reduces the volatility.
For a more detailed explanation see Chap. 3 in Grüll (2010).
The permit price trajectories are obtained by plugging the trajectory of x _{ t } (see Fig. 1) into the permit price formulae of Lemma 1.
Lower and upper bounds are obtained by considering the permit prices of \(\mathbb {E}(q_{[0,t]} ) \pm c \sqrt{Var (q_{[0,t]} )}\) which is the confidence band for outstanding emissions.
In phase I of the EU ETS the banking of permits was not allowed, i.e. permits that were not used in phase I (2005–2007) could not be handed in for compliance purposes in phase II (2008–2012). From phase II on the banking of permits is allowed.
The derivation of this result is to be found in the Appendix.
The volatility of the outstanding emissions of all regulated companies in the EU ETS is assumed to be smaller than 5 percent. This is motivated by the verified emissions data of the years 2005–2010. The annual percentage change remained in the confidence band [−10 %,+10 %] that is implied by our assumption.
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Appendix: Derivation of Permit Price Formulae and Sensitivities
Appendix: Derivation of Permit Price Formulae and Sensitivities
The derivation of the permit prices for the different approximation approaches is split up into three parts. We start by presenting the first two moments of the outstanding emissions in the time period [t _{1},t _{2}] denoted by \(q_{[t_{1},t_{2}]} = \int_{t_{1}}^{t_{2}} Q_{s} ds\) (see Sect. 6.1). Then, we derive the parameters of the random variables that are used to approximate the outstanding emissions (cf. Lemma 4 in Sect. 6.2). Finally, the permit price formulae are derived in Sect. 6.3.
1.1 6.1 Moments of Outstanding Emissions
Milevsky and Posner (1998) prove the following lemma for the integral over a geometric Brownian motion.
Lemma 3
(Moments of \(q_{[t_{1},t_{2}]}\))
Let \(Q_{s} = Q_{0} \exp \{ ( \mu \frac{\sigma^{2}}{2} ) s + \sigma W_{s} \}\) be a geometric Brownian motion. Then the first two moments of \(q_{[t_{1},t_{2}]} = \int_{t_{1}}^{t_{2}} Q_{s} ds\) are given by
where
Lemma 3 shows that \(\alpha_{t_{2}t_{1}}\) is approximately linear and that \(\beta_{t_{2}t_{1}}\) is approximately a quadratic function for small μ and σ. Therefore, expected outstanding emissions are approximately linear.
1.2 6.2 Moment Matching/Determining Parameters of Approximating Random Variables
Lemma 4
(Overview on approximations)
Let τ=T−t and Z∼N(0,1). The outstanding emissions q _{[t,T]} are approximated by the following random variables

(a)
Linear (interval length times GBM)
(20) 
(b)
Lognormal (moment matching)
(21)
Proof
(b) The parameters μ _{ L }(t,T) and σ _{ L }(t,T) are chosen such that the first two moments of \(q^{Log}_{[t,T]} = \log N(\mu_{L}(t,T), \sigma_{L}^{2}(t,T))\) match those of q _{[t,T]}.
Hence by Lemmas 3 and 5
Note that \(\sigma_{L}^{2}(t,T)\) is independent of Q _{ t }, μ _{ L }(t,T) not. □
1.3 6.3 Derivation of Permit Price Formulae
1.3.1 6.3.1 Proof of Lemma 2a (Linear Approximation Approach)
Let Z∼N(0,1). Using (3) yields
1.4 6.4 Proof of Lemma 2b (Lognormal moment matching approach)
By (2) we have
Let Z∼N(0,1). Using (21) yields
Lemma 5
(Moments of lognormal random variable)
Let X∼logN(μ,σ ^{2}). Then

(a)
The kth moment of X is given by
$$\mu_k^{(X)} = e^{k\mu+ k^2 \frac{\sigma^2}{2}} $$ 
(b)
Let X∼logN(μ,σ ^{2}) be a lognormal random variable with \(\mathbb {E}(X)=m_{1}\) and \(\mathbb {E}(X^{2})=m_{2}\). Then
Proof
(b) The first two moments of X∼logN(μ,σ ^{2}) are given by
By assumption \(m_{2} = m^{2}_{1} e^{\sigma^{2}}\) and \(m_{1} = e^{\mu}\sqrt {e^{\sigma^{2}}}\). □
1.5 6.5 Proof of Lemma 3
As the proof for both approximation approaches is the same we only present the proof of part (a), i.e. (14). We assume that the observed permit prices \(\hat{S}_{t\Delta}\) and \(\hat{S}_{t}\) can be described by the equilibrium price formula of the linear approximation approach given by (5) and Lemma 1, i.e.
Assuming that \(S_{t\Delta}^{Lin} ( x_{t\Delta}  \mu, \sigma ) \approx S_{t}^{Lin} ( x_{t\Delta}  \mu, \sigma )\) implies
This yields
1.6 6.6 Proof that Price Drop Implies a Positive h (Implied Change in Permit Availability) Assuming that the Permit Price and the Share of Banking Value are Negatively Correlated
Without loss of generality the observed permit price at time t can be written as
where γ _{ t }∈(0,1). Especially, we have that
and
As Φ^{−1}(⋅) is an increasing function, the implied change in permit availability h is positive if
which is equivalent to
By assumption a permit price drop coincides with an increase in the share of the banking value, i.e.
which is equivalent to
A decrease in the permit price and an increase in the share of the banking value imply that h is positive (cf. (22) and (23)):
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Grüll, G., Kiesel, R. Quantifying the CO_{2} Permit Price Sensitivity. Z Energiewirtsch 36, 101–111 (2012). https://doi.org/10.1007/s1239801200824
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DOI: https://doi.org/10.1007/s1239801200824