Quantifying the CO₂ Permit Price Sensitivity

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.

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

(16)

(17)

where

(18)

(19)

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) Log-normal (moment matching)

  

  (21)

Proof

(a) Follows from (3) and (6).

(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)

By (2) and (6) we have

Let Z∼N(0,1). Using (3) yields

1.4 6.4 Proof of Lemma 2b (Log-normal moment matching approach)

By (2) we have

Let Z∼N(0,1). Using (21) yields

Lemma 5

(Moments of log-normal 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 log-normal 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

$$\hat{S}_t = S_t + B_t := \gamma_t \hat{S}_t + (1-\gamma_t) \hat{S}_t $$

where γ _t ∈(0,1). Especially, we have that

$$\frac{B_t}{\hat{S}_t} = 1 - \gamma_t $$

and

$$\hat{S}_t - B_t = \gamma_t \hat{S}_t $$

As Φ⁻¹(⋅) is an increasing function, the implied change in permit availability h is positive if

$$\hat{S}_t - B_t < \hat{S}_{t-\Delta} - B_{t-\Delta} $$

which is equivalent to

(22)

By assumption a permit price drop coincides with an increase in the share of the banking value, i.e.

$$\frac{B_t}{\hat{S}_t} > \frac{B_{t-\Delta}}{\hat{S}_{t-\Delta}} $$

which is equivalent to

(23)

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)):