Introduction
The Model
Equilibrium Prices without Banking
The Effect of Banking
Discussions
Conclusions
Equilibrium Pricing of Contingent Claims in
Tradable Permit Markets
Masaaki Kijima(a) , Akira Maeda(b) , Katsumasa Nishide()
(a)
(b)
(c)
Graduate School of Social Science, Tokyo Metropolitan University
Graduate School of Energy Science, Kyoto University
Interdisciplinary Research Center, Yokohama National University
The 39th ASTIN Colloquium – Helsinki, Finland
The full paper is available at http://ssrn.com/abstract=1344103.
Kijima, Maeda and Nishide
Equilibrium Pricing of Contingent Claims in Tradable Permit Markets
Introduction
The Model
Equilibrium Prices without Banking
The Effect of Banking
Discussions
Conclusions
Plan of Talk
1
Introduction
2
The Model
3
Equilibrium Prices without Banking
4
The Effect of Banking
5
Discussions
6
Conclusions
Kijima, Maeda and Nishide
Equilibrium Pricing of Contingent Claims in Tradable Permit Markets
Introduction
The Model
Equilibrium Prices without Banking
The Effect of Banking
Discussions
Conclusions
Introduction
Kijima, Maeda and Nishide
Equilibrium Pricing of Contingent Claims in Tradable Permit Markets
Introduction
The Model
Equilibrium Prices without Banking
The Effect of Banking
Discussions
Conclusions
Kyoto Protocol
The first five-year commitment period started on January
1, 2008.
Ratifying countries are required to meet the cap of
pollution emission amount.
Each country also imposes the emission limitation to its
industries.
How to meet the regulation:
Reduce the emission amount by self-efforts.
Buy pollution rights (pollution permit).
Kijima, Maeda and Nishide
Equilibrium Pricing of Contingent Claims in Tradable Permit Markets
Introduction
The Model
Equilibrium Prices without Banking
The Effect of Banking
Discussions
Conclusions
Statistics of carbon market (1)
Source: Carbon 2007.
Kijima, Maeda and Nishide
Equilibrium Pricing of Contingent Claims in Tradable Permit Markets
Introduction
The Model
Equilibrium Prices without Banking
The Effect of Banking
Discussions
Conclusions
Statistics of carbon market (2)
ECX CFI Options Volume
Puts
Calls
30,000,000
Volume (tonnes CO2)
25,000,000
20,000,000
15,000,000
10,000,000
5,000,000
0
7 7 7
7 7 7
7 7
6 6 6
8
7
7 7
7
t-0 v-0 -0 -0 -0 r-0 r-0 y-0 -0 l-0 g-0 -0 t-0 v-0 -0 -0
Oc No Dec Jan Feb Ma Ap Ma Jun Ju Au Sep Oc No Dec Jan
Date
Source: ECXpres-March2008.
Kijima, Maeda and Nishide
Equilibrium Pricing of Contingent Claims in Tradable Permit Markets
Introduction
The Model
Equilibrium Prices without Banking
The Effect of Banking
Discussions
Conclusions
Summary of the paper
The purpose of the paper is to propose a pricing formula
that evaluates any contingent claim in the pollution permit
market.
Literature review:
Cronshaw and Kruse (1996), Rubin (1996), Seifert et al.
(2008) only focus on the spot prices.
Maeda (2004): only the forward price is examined.
Daskalakis et al. (2007), Chesney and Taschini (2008): the
spot price (underlying) is exogenously given.
To our best knowledge, this paper is the first to study the
pricing of any contingent claim of permit markets in a
general equilibrium framework.
Kijima, Maeda and Nishide
Equilibrium Pricing of Contingent Claims in Tradable Permit Markets
Introduction
The Model
Equilibrium Prices without Banking
The Effect of Banking
Discussions
Conclusions
Pricing Functional
~
Let Y be the payoff of any contingent claim.
It is a well-known result in the finance literature that the
price of Y , denoted by Y , can be expressed with some
random variable as
~
~
(~)
(Y~ ) = E [~Y~ ℄:
The r.v.
~ is a so-called state price density.
Kijima, Maeda and Nishide
Equilibrium Pricing of Contingent Claims in Tradable Permit Markets
Introduction
The Model
Equilibrium Prices without Banking
The Effect of Banking
Discussions
Conclusions
The Model
Kijima, Maeda and Nishide
Equilibrium Pricing of Contingent Claims in Tradable Permit Markets
Introduction
The Model
Equilibrium Prices without Banking
The Effect of Banking
Discussions
Conclusions
The market
2-period economy: time 0 (present) and time 1 (future).
Market participants (all price takers)
regulated emitters who need to meet the cap of emission.
unregulated financial traders who are not obliged to follow
the regulation.
At time 0, there are two markets:
Spot market: spot permits of time 0 are traded.
Contingent claims market: contingent claims written on the
permits of time 1 are traded.
The contingent claims market is complete in the sense that
any contingent claim is tradable.
Kijima, Maeda and Nishide
Equilibrium Pricing of Contingent Claims in Tradable Permit Markets
Introduction
The Model
Equilibrium Prices without Banking
The Effect of Banking
Discussions
Conclusions
Glossary
ME
MS
Eit
it (Xit )
St
Rkt
r
set of emitters
set of financial traders
the net abatement target of emitter i 2 ME
at time t
the cost function to reduce the amount of
emission Xit by emitter i at time t,
spot permit price at time t,
exogenous income of agent k 2 ME [ MS ,
risk-free rate determined in the financial market.
Kijima, Maeda and Nishide
Equilibrium Pricing of Contingent Claims in Tradable Permit Markets
Introduction
The Model
Equilibrium Prices without Banking
The Effect of Banking
Discussions
Conclusions
Emitters
To match the regulation, emitter i can reduce emission by
her own effort and by buying the spot permits in the market.
The final wealth of emitter i is described as
Wi (!)
= (1 + r)[Ri i (Xi )
+ Ri (!) i (Xi (!))
0
~
1
0
1
S0 (Ei0
0
Xi0 ) (H~ i )℄
S1 (!)fEi1 (!) Xi1 (!)g + Hi(!);
1
where Hi is the contingent claim that emitter i purchases.
The preference of emitter i is represented by an CARA
utility with absolute risk-aversion coefficient i ;
E
[ exp f
Kijima, Maeda and Nishide
~ ig℄:
i W
Equilibrium Pricing of Contingent Claims in Tradable Permit Markets
Introduction
The Model
Equilibrium Prices without Banking
The Effect of Banking
Discussions
Conclusions
Financial traders
Financial traders have no incentive to enter the spot
market, and only trade in the contingent claims market to
hedge the risk of their exogenous income.
The final wealth of financial trader j is written as
Wj (!) =(1 + r)[Rj 0 (H~ j )℄ + Rj 1(!) + Hj (!); j 2 MS :
The preference of financial trader j is also represented by
an CARA utility;
E
[ exp f
Kijima, Maeda and Nishide
~ j g℄:
j W
Equilibrium Pricing of Contingent Claims in Tradable Permit Markets
Introduction
The Model
Equilibrium Prices without Banking
The Effect of Banking
Discussions
Conclusions
Equilibrium Prices without
Banking
Kijima, Maeda and Nishide
Equilibrium Pricing of Contingent Claims in Tradable Permit Markets
Introduction
The Model
Equilibrium Prices without Banking
The Effect of Banking
Discussions
Conclusions
Market clearing
Spot markets:
X
i2ME
X
i2ME
(Ei
0
(E i (! )
1
Xi0 ) = 0;
Xi1 (!)) = 0
Contingent claims market:
X
k2ME [MS
Hk (!) = 0
for almost all ! .
Kijima, Maeda and Nishide
Equilibrium Pricing of Contingent Claims in Tradable Permit Markets
Introduction
The Model
Equilibrium Prices without Banking
The Effect of Banking
Discussions
Conclusions
Cost function
Assumption
()
The cost function it is strictly increasing, continuously
differentiable, strictly convex, and satisfies
0it (0) = 0; 0it (1) = 1:
The assumption guarantees the existence and uniqueness
of the equilibrium.
Kijima, Maeda and Nishide
Equilibrium Pricing of Contingent Claims in Tradable Permit Markets
Introduction
The Model
Equilibrium Prices without Banking
The Effect of Banking
Discussions
Conclusions
Some notations
:= P
Let t
verify that t is a function of
i2ME it . Then, we P
Et in equilibrium, where Et
i2ME Eit , i.e., we have an
expression
:=
t = t (Et ):
Define
Rt :=
X
k2ME [MS
Rkt ; :=
Kijima, Maeda and Nishide
1
X
k2ME [MS
1:
k
Equilibrium Pricing of Contingent Claims in Tradable Permit Markets
Introduction
The Model
Equilibrium Prices without Banking
The Effect of Banking
Discussions
Conclusions
Pricing formula in the case of no banking
Proposition
The state price density is given by
(!) =
1 efh E !
1 + r E ef E
1 ( 1 ( ))
1 ( ~1 )
R1 (!)g
~1 g
R
i
:
Proof.
Apply the result of Bühlmann (1980).
Kijima, Maeda and Nishide
Equilibrium Pricing of Contingent Claims in Tradable Permit Markets
Introduction
The Model
Equilibrium Prices without Banking
The Effect of Banking
Discussions
Conclusions
Forward price
Corollary
The price of forward contract is derived as
F
Proof.
Verify that S1
=
E
h
01 (E~1 )e f1 (E~1 )
E
h
ef
~1 )
1 (E
~1 g
R
~1 g
R
i
i
:
= 0 (E ) in equilibrium.
1
1
Kijima, Maeda and Nishide
Equilibrium Pricing of Contingent Claims in Tradable Permit Markets
Introduction
The Model
Equilibrium Prices without Banking
The Effect of Banking
Discussions
Conclusions
Normal risks (1)
Suppose
E~ R~ N
1
1
2
R ; E E2R
E R
R
E
The price whose payoff at time 1 is g
(g(S~1 )) =
E
h
(S ) is given by
1
h(Z~ )e1 (Z~ )
h
i
i;
(1 + r)E e Z where h() = g (0 ()), and Z~ is a normal with mean
E
1( ~ )
1
E R and variance E2 .
Kijima, Maeda and Nishide
Equilibrium Pricing of Contingent Claims in Tradable Permit Markets
Introduction
The Model
Equilibrium Prices without Banking
The Effect of Banking
Discussions
Conclusions
Normal risks (2): comparative statics
Corollary
Let
=E [h0 (Z~ )℄
1
C [h(Z~ ); 01 (Z~ )℄;
where E and C are the expectation and covariance operators
under the risk-neutral measure, respectively. Then,
(g)
E
= 1 +1 r ;
1
Kijima, Maeda and Nishide
(g)
= 1 +E rR :
1
Equilibrium Pricing of Contingent Claims in Tradable Permit Markets
Introduction
The Model
Equilibrium Prices without Banking
The Effect of Banking
Discussions
Conclusions
Quadratic cost function
Suppose furthermore that the cost function of each emitter
^it 2
is quadratic as it x
x . Then, the social cost function
2
is given by
( )=
1
2
t (Et ) = ^t Et2 ;
^ = Pi2M
where t
1
1 .
E ^it
The forward price is derived as
^ Z
E R )
=
= ^ (1E 1 ;
^ E
where Z := E E R and := ^ E .
F
1
1
1
2
1
2
This is the case that Maeda (2004) considered.
Kijima, Maeda and Nishide
Equilibrium Pricing of Contingent Claims in Tradable Permit Markets
Introduction
The Model
Equilibrium Prices without Banking
The Effect of Banking
Discussions
Conclusions
The Effect of Banking
Kijima, Maeda and Nishide
Equilibrium Pricing of Contingent Claims in Tradable Permit Markets
Introduction
The Model
Equilibrium Prices without Banking
The Effect of Banking
Discussions
Conclusions
Banking (and borrowing)
Banking: additional reduction amount in the current period
can be used in later periods.
When banking and borrowing are allowed, the current and
future spot markets are connected.
The market clearing condition of the permit:
1
X
X
t=0 i2ME
(Eit
or equivalently
X
i2ME
( Ei + B i
0
0
Xi0 ) = 0 and
X
i2ME
Xit ) = 0
(Ei (!)
1
Bi0
Xi1 (!)) = 0:
where Bi0 denotes the banking of emitter i at time 0.
Kijima, Maeda and Nishide
Equilibrium Pricing of Contingent Claims in Tradable Permit Markets
Introduction
The Model
Equilibrium Prices without Banking
The Effect of Banking
Discussions
Conclusions
No arbitrage condition
No arbitrage condition implies
(1 + r) |0 (E {z+ B }) =
0
=
0
E
0
Spot price
|
h
01 (E~1 B0 )e f1 (E~1
E
h
ef
=
~1
1 (E
~1 g
B0 )+R
~1 g
B0 )+R
i
{z
i
;
}
Forward price
(NAC)
=P
where B0
i2ME Bi0 .
The aggregated amount of banking in equilibrium is
determined by (NAC).
Kijima, Maeda and Nishide
Equilibrium Pricing of Contingent Claims in Tradable Permit Markets
Introduction
The Model
Equilibrium Prices without Banking
The Effect of Banking
Discussions
Conclusions
Normal risks (1)
Consider the case of Example 1 and suppose that the
system of banking and borrowing is introduced.
The aggregated amount of banking is derived as
B0 =
^1 (E E R )
2
1 ^1 E
(1 + r)^ +
0
(1 + r)^ E
0
1
0
^1
2
^1 E
:
The forward price is given by
FWB =
^1 (E0 + E E R )
1 ^1 E2 + (1+^r1)^0
Kijima, Maeda and Nishide
= 1 ^ (E++ Z )
1
0
^1
(1+r )^
0
:
Equilibrium Pricing of Contingent Claims in Tradable Permit Markets
Introduction
The Model
Equilibrium Prices without Banking
The Effect of Banking
Discussions
Conclusions
Normal risks (2)
Comparing F and FWB , we have
F
E
F F
< 0
F
> 0;
E
WB
WB
The introduction of banking and borrowing lowers the
sensitivity of exogenous parameters on the forward price
(smoothing effect).
The sensitivity of the forward price is partly absorbed by
the change of the current spot price.
Kijima, Maeda and Nishide
Equilibrium Pricing of Contingent Claims in Tradable Permit Markets
Introduction
The Model
Equilibrium Prices without Banking
The Effect of Banking
Discussions
Conclusions
Discussions
Kijima, Maeda and Nishide
Equilibrium Pricing of Contingent Claims in Tradable Permit Markets
Introduction
The Model
Equilibrium Prices without Banking
The Effect of Banking
Discussions
Conclusions
Price spike: cost function and spot price
~
~
Suppose the normality of E1 and R1 , and that the social
abatement cost8
function is given by
t (x) = ^L 1
>
<
0
x
x2
2
1
>2
2
:
( 1)XB x + XB
2
where .
The spot price of8time 1 is derived as
S1
0
= ^L >E~
>
<
1
2
E~1 0;
E~1 XB ;
E~1 > XB :
( 1)XB E~
B , and the parameter The spot price kinks at x = X
:
E~1
1
x 0;
x X B ;
x > X B ;
1
measures the magnitude of the kink.
Kijima, Maeda and Nishide
Equilibrium Pricing of Contingent Claims in Tradable Permit Markets
Introduction
The Model
Equilibrium Prices without Banking
The Effect of Banking
Discussions
Conclusions
Price spike: no banking case
Forward price with different :
10
8
XB
6
F
1.0
2.0
3.0
4
1.0
Parameter values:
^L
1.0
E
X B
0.3
0.0
3.0
2
0
1
1.5
2
2.5
3
3.5
4
muE
The spike level is lower in the forward price.
Kijima, Maeda and Nishide
Equilibrium Pricing of Contingent Claims in Tradable Permit Markets
Introduction
The Model
Equilibrium Prices without Banking
The Effect of Banking
Discussions
Conclusions
Price spike: introduction of banking
Forward price with different :
10
8
6
F, S1
WithBank
NoBank
S1
4
1.0
Parameter values:
^L
1.0
E
X B
0.3
0.0
3.0
kappa = 3.0
2
0
1
1.5
2
2.5
3
3.5
4
muE, E1
The price spike is significantly mitigated by the introduction of
banking and borrowing.
Kijima, Maeda and Nishide
Equilibrium Pricing of Contingent Claims in Tradable Permit Markets
Introduction
The Model
Equilibrium Prices without Banking
The Effect of Banking
Discussions
Conclusions
Comparative statics
By simply differentiating the forward price formula, we have
E % ) indeterminate
& ) indeterminate
&
R %
)
)
F
F
%
&
The impact on the risk-premium and the correlation with
the exogenous income should be considered.
Kijima, Maeda and Nishide
Equilibrium Pricing of Contingent Claims in Tradable Permit Markets
Introduction
The Model
Equilibrium Prices without Banking
The Effect of Banking
Discussions
Conclusions
Conclusions
Kijima, Maeda and Nishide
Equilibrium Pricing of Contingent Claims in Tradable Permit Markets
Introduction
The Model
Equilibrium Prices without Banking
The Effect of Banking
Discussions
Conclusions
Conclusions
In this paper,
we have proposed a formula that prices any contingent
claim in the tradable permit market.
we have analyzed the equilibrium price in the permit
market with the case of normal distributions.
we have shown that the prices in permit market have some
specific properties that are not observed in ordinary
financial markets.
Kijima, Maeda and Nishide
Equilibrium Pricing of Contingent Claims in Tradable Permit Markets
Introduction
The Model
Equilibrium Prices without Banking
The Effect of Banking
Discussions
Conclusions
Summary
Key determinants of the pricing:
(i)
(ii)
(iii)
(iv)
Social cost function to meet the abatement target
Uncertainty of the required abatement level
Risk-aversion of market participants
Correlation of the abatement level with
exogenous income.
System design:
Banking and borrowing,
Limitation of market participants.
Kijima, Maeda and Nishide
Equilibrium Pricing of Contingent Claims in Tradable Permit Markets
Introduction
The Model
Equilibrium Prices without Banking
The Effect of Banking
Discussions
Conclusions
Thank you for your attention
Kijima, Maeda and Nishide
Equilibrium Pricing of Contingent Claims in Tradable Permit Markets