In this paragraph we invent an appropriate method involving multi-dimensional Fourier transforms in order to derive explicit pricing formulas for carbon emission allowances as commonly traded in the EU ETS but newly under our improved multi-state market zone net position approach. We initially do this under the ordinary assumption that the market participants’ knowledge is such as modeled in (6.2.10). In other words, at time we suppose these uninitiated traders solely to have an idea about the histories of and during a time range 0 ≤ ≤ . On the contrary, in section 6.5 we will show how these pricing formulas alter if one presumes some additional forward-looking insider information about the market zone net position at a future time available to so-called informed traders, respectively market insiders.
As described at the end of Chapter 2 in [25], we firstly should notice that if the EU ETS market ends up long > 0 at the expiry date (meaning that there are still some firms holding carbon emission allowances that they do not need any more), then – under the assumption of no banking – EUA0 contracts will become worthless what drives their prices towards zero immediately. Theoretically, we assume the same to be valid for the rather unlikely but of course possible case wherein the market ends up in a precise equilibrium = 0 . Contrarily, if the market ends up short < 0 at the final time (meaning that there is a shortage situation throughout carbon emission allowances in the market), then – in this much more delicate market scenario – a EUA0 contract will not become worthless, since the latter easily can be turned into a EUA1 contract by paying an imposed penalty (cf. p.6 in [25]). Hence, slightly deviating from “(2.5) in [25]”, we yet define the price of a EUA0 contract via
(6.3.1)
≔ + = 0, ≥ 0+ , < 0
within a constant penalty value > 0 (given in EURO). Thus, the EUA0 price in (6.3.1) can be interpreted as a generalized contingent claim, respectively as a non-standard option on [25]. In other words, a EUA0 contract may be considered as some kind of exotic option on a EUA1 contract.
As explained before, in [25] the market zone net position is modeled as a two-state Markov-chain merely taking values in the set ≔ −1,1 , whereby the case = 1 ( = −1) in [25] corresponds to ≥ 0 ( < 0) in our setup. For this reason, in the Cetin-Verschuere-approach (see the top of page 8 in [25]) the contingent claim easily can be written in the extremely convenient form
(6.3.2)
= + =1 −2 + .
At this step, we notice a (preliminary) disadvantage of our innovative CPP approach (although the latter possibly appears more realistic from a modeling point of view), as the very tractable representation (6.3.2) unfortunately is no longer valid if we model the market zone net position by a linear combination of compound Poisson processes taking arbitrary many values in the compact set , ⊂ ℝ, instead of a net position with values in ≔ −1,1 , solely. In fact, regarding the structure of our contingent claim (6.3.1), it seems hardly possible to express the latter within a similar notational form as in (6.3.2). However, the claim (6.3.1) not at all reveals a European-type structure (as commonly known from popular plain-vanilla options), though on a superficial sight rather similar, since inside the indicator function not the process itself appears, but another process, namely .
141
Our key idea to overcome the just formulated problem reads as follows: We newly associate the contingent claim as given in (6.3.1) within a customized real function mapping
: 0, ∞ × , ⟶ 0, ∞
whereas we concretely define
(6.3.3) , ≔ +
so that , = proves true instantly. Unfortunately, the function is not integrable on the set ℳ ≔ 0, ∞ × , ⊂ ℝ with respect to the two-dimensional Lebesgue measure , in symbols , ∉ ℒ ℳ, . However, for later purposes we introduce the exponentially damped function (6.3.4)
, ≔ ,
defined on ℳ as well, within a real dampening parameter > 0, yet obeying , ∈ ℒ ℳ, on the opposite.
6.3.1 Pricing EUA0 contracts with Fourier transform methods
In this subsection we derive risk-neutral prices for EUA0 contracts such as traded in the EU ETS market by applying a customized Fourier transform procedure (also compare paragraph 3.2.4 above). Starting off, with respect to (6.3.1), (6.3.3) and (6.3.4), we immediately obtain
(6.3.5)
= , = , .
Moreover, in accordance to (2.4.3) [but with = 2 therein], we receive (6.3.6)
, = 21 ,
ℳ
, .
Next, for 0 ≤ ≤ the risk-neutral pricing formula [cf. (3.2.30)] can be written as (6.3.7)
= ℚ |ℱ = ℚ , ℱ .
Appealing to (6.3.5), (6.3.6) and the Fubini-Tonelli theorem, the conditional expectation on the right hand side of (6.3.7) becomes
(6.3.8) ℚ , ℱ = ℚ , ℱ
= 21 ,
ℳ
ℚ ℱ , .
Further on, the ℱ -measurability of and together with the independent increment property of the two latter ℚ-independent processes implies
142
(6.3.9) ℚ ℱ = ℚ ℱ
= ℚ ℚ =: × ℑ × ℑ .
Meanwhile, we proceed with the computation of the Fourier transform such as appearing inside (6.3.8): In accordance to (2.4.2), (6.3.3) and (6.3.4), we get
(6.3.10)
, = + ,
ℳ
= +
= 1 + ++ =1 + ++ × − 1.
Anyway, applying (6.2.6), (6.2.7), (6.2.15) and the Lévy-Khinchin formula [see Theorem 2.1.3], by common independency (and stationarity) arguments we next derive
(6.3.11)
ℑ ≔ ℚ = ℚ =
within characteristic exponents (6.3.12)
= − 1 .
What remains is the computation of the first multiplier ℑ : Using (6.2.19) while exploiting standard conditioning methods (cf. e.g. the last equality of the proof of Prop. 10.4 in [13]), we deduce
(6.3.13)
ℑ ≔ ℚ = ℚ
= ℚ + −12 − 1
≔
.
Furthermore, for 0 ≤ ≤ the stochastic process (6.3.14)
≔ = −12
143
ℚ = 1
and variance
ℚ = − 1.
As usual, we denote the latter fact by writing
ℚ = − 2 , , ≔
in shorthand notation. Hence, adhering to similar measure-transformation/conditioning arguments as applied in e.g. the proof of Prop. 10.4 in [13], with respect to (6.3.13) and (6.3.14) we next obtain (6.3.15) ℑ = ℚ ≔ = ℚ ≔ , = √2 − 1 2 + ≔ =: ,
which can be calculated further by standard numerical integration methods for Riemann integrals (see paragraph 19.3 in [19], for example). Merging (6.3.8), (6.3.9), (6.3.11) and (6.3.15) into (6.3.7), we finally end up with the expression
(6.3.16)
= 2 , , ,
ℳ
yielding the -penalized EUA0 price at time of a contingent claim paying as given in (6.3.1) at the expiry date , whereby , , and , are such as defined in (6.3.10), (6.3.12) and (6.3.15), respectively. In practical applications, the random processes and appearing inside (6.3.16) naturally have to be simulated, whereas the two-dimensional -integral over ℳ must be evaluated numerically.