A quantitative approach to carbon price risk modeling - CiteSeerX

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next, utilize (24) in to obtain T∑ −1 π(Γ i − ξt i − θT i ) + = π(Γ i − ϑ i T ) + (30) t=0 and subtract (30) from (29) to verify (26). Remark Before entering the proof of the Proposition 1, let us give some feeling about the mechanism responsible for the equilibrium carbon price formation. The objective (25) shows two effects working on opposite directions i) the impact of carbon price on emission reduction strategies ii) the overall effect of emission reduction on the carbon price Namely, the last term in (25) is responsible for i), illustrating that the fuel switching ξt i > 0 is optimally performed if and only if the own switching price falls below the market carbon price: A t − Et i ≥ 0. This emission reduction, in turn, is connected to ii) through the expected final carbon price A T . The higher is the current price A t , the more agents apply fuel switching decreasing so the probability that the market will be short of credits at the end of the compliance period. However decreasing this probability also decreases the expected final need for carbon allowances and consequently lowers the expected final carbon price A T . Being a financial asset, carbon allowances can not exhibit a significant price decay from A t to A T , that is, the fuel switching at time t also diminishes A t . Thus, there is an equilibrium level to be reached by carbon price at any time. At time t, the equilibrium carbon price depends on the expected overall compliance gap E(Γ | F t ) and on the expected future prices (E(E s | F t )) T s=t −1 of fuel switching. We are now prepared to prove the main Theorem 1. Proof. Due to the Proposition 2, the equilibrium property of (A ∗ t ) T t=0 can be shown by an explicit construction of (ϑ i∗ , ξ i∗ ) ∈ L 1 × L 1 × U i , i = 1, . . . , N which fulfill (27) and (28). To proceed so, let ξ ∗ ∈ U be from the Proposition 1 and define by (ϑ i∗ ) N i=1 ∈ (L 1 × L 1 ) N ϑ ∗i t = t∑ s=0 ξ ∗i s for all i = 1, . . . , N, t = 0, . . . , T − 1, ϑ i∗ T = Γ i − (Γ − Π(ξ ∗ ))/N. The restriction (28) is obviously fulfilled, so we focus on (27). 18
For ϑ ∈ (L 1 × L 1 ) N , ξ ∈ U , and carbon price processes (16), express (25) as T∑ −1 T∑ −1 I A∗ ,i T (ϑ i , ξ i ) = ϑ i t(A ∗ t+1 − A ∗ t ) − ϑ i T A ∗ T − π(Γ i − ϑ i T ) + + ξt(A i ∗ t − Et) i (31) t=0 which gives the expected value T∑ −1 E(I A∗ ,i T (ϑ i , ξ i )) = E(−ϑ i T A ∗ T − π(Γ i − ϑ i T ) + ) + E( ξt(A i ∗ t − Et)) i since the process A ∗ ∈ L ∞ follows a martingale by definition (16) and (ϑ i t) T −1 t=0 is an element from L 1 . Thus, to show (27), it suffices to prove that t=0 t=0 ϑ i T ↦→ E(−ϑ i T A ∗ T − π(Γ i − ϑ i T ) + ) is maximized on L 1 at ϑ i∗ T , (32) T∑ −1 whereas ξ i ↦→ E( ξt(A i ∗ t − Et)) i is maximized on U i at ξ i∗ . (33) t=0 First, we turn to (32) showing that the maximum is attained pointwise. By (16), ω ∈ {Γ − Π(ξ ∗ ) < 0} implies that A ∗ T (ω) = 0 and ϑi∗ T (ω) > Γi (ω). Moreover, the maximum of z ↦→ −zA T (ω) − π(Γ i (ω) − z) + (34) is attained on each point from [Γ i (ω), ∞[, thus ϑ i∗ T (ω) is a maximizer. In the other case, ω ∈ {Γ − Π(ξ ∗ ) ≥ 0}, we have A ∗ T (ω) = π and ϑi∗ T (ω) ≤ Γi (ω). Here the maximum of (34) is attained on [0, Γ i (ω)], thus ϑ ∗ T (ω) is a maximizer. Now we turn to (33). It suffices to show that for each i = 1, . . . , N and t ∈ {0, . . . , T − 1} the following inclusions hold almost surely {A ∗ t − Et i > 0} ⊆ {ξt ∗i = λ i }, (35) {A ∗ t − Et i < 0} ⊆ {ξt ∗i = 0}. (36) First, we emphasize that the dynamical principle of optimal control implies that for any ξ ∈ U with ξ s = ξs ∗ for s = 0, . . . , t − 1. E(G(ξ)|F t ) ≤ E(G(ξ ∗ )|F t ) holds almost surely (37) This assertion is seen by the following argumentation: On the contrary, one uses the F t –measurable set M := {E(G(ξ)|F t ) > E(G(ξ ∗ )|F t )} of positive measure P (M) > 0, to outperform ξ ∗ by ξ ′ as ξ ′ s = 1 M ξ s + 1 Ω\M ξ ∗ s for all s = 0, . . . , T − 1. (38) 19
Page 1 and 2: A quantitative approach to carbon p
Page 3 and 4: • for the IET mechanism, these cr
Page 5 and 6: and JI- projects. On the contrary,
Page 7 and 8: are default values provided by the
Page 9 and 10: With these notations, the equilibri
Page 11 and 12: period 2008-2012 and, more importan
Page 13 and 14: in Mega tons of carbon dioxide. Num
Page 15 and 16: Let us summarize our findings. The
Page 17: Given the fuel switching policy (ξ
Page 21 and 22: which with (40) implies that the fo
Page 23 and 24: 40 30 20 10 ∆X 0 -10 -20 -30 -40
Page 25: [15] J. Seifert, M. Uhrig-Homburg,

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