Risk neutral dynamics of spot and forward
electricity prices
- Joint work with J.-M. Marin & N. Touzi -
Risk neutral
dynamics of
spot and
forward
electricity
prices
Luciano
Campi
Introduction
The Model
Luciano Campi
Electricity
forward
prices
CEREMADE, Laboratoire FiME (Paris-Dauphine)
Constant
coefficients
What’s next
Luciano Campi
Risk neutral dynamics of spot and forward electricity prices
CEREMADE, Laboratoire FiME (Paris-Dauphine)
Contents
1 Introduction
2 The Model
Risk neutral
dynamics of
spot and
forward
electricity
prices
Luciano
Campi
Introduction
3 Electricity forward prices
4 Constant coefficients
The Model
Electricity
forward
prices
Constant
coefficients
5 What’s next
Luciano Campi
Risk neutral dynamics of spot and forward electricity prices
What’s next
CEREMADE, Laboratoire FiME (Paris-Dauphine)
Introduction I : Motivations
In standard financial markets : Ft (T ) = St e r (T −t) . This
equality relies heavily on costless storability of financial
assets, it breaks down when St is spot price of electricity
A priori, no relations between spot and forward at least in
a market composed of electricity and bank account (see,
e.g., Geman-Vasicek (2001))
Risk neutral
dynamics of
spot and
forward
electricity
prices
Luciano
Campi
Introduction
The Model
Geman-Vasicek (2001) and Bessembinder-Lemon (2002)
show that short-term forward contract are (upward- or
downward-) biased estimator of spot prices, so ...
Electricity
forward
prices
... in mathematical terms, when t ↑ T , Ft (T ) does not
necessarily tend to ST
What’s next
Luciano Campi
Risk neutral dynamics of spot and forward electricity prices
Constant
coefficients
CEREMADE, Laboratoire FiME (Paris-Dauphine)
Introduction I : Motivations
In standard financial markets : Ft (T ) = St e r (T −t) . This
equality relies heavily on costless storability of financial
assets, it breaks down when St is spot price of electricity
A priori, no relations between spot and forward at least in
a market composed of electricity and bank account (see,
e.g., Geman-Vasicek (2001))
Risk neutral
dynamics of
spot and
forward
electricity
prices
Luciano
Campi
Introduction
The Model
Geman-Vasicek (2001) and Bessembinder-Lemon (2002)
show that short-term forward contract are (upward- or
downward-) biased estimator of spot prices, so ...
Electricity
forward
prices
... in mathematical terms, when t ↑ T , Ft (T ) does not
necessarily tend to ST
What’s next
Luciano Campi
Risk neutral dynamics of spot and forward electricity prices
Constant
coefficients
CEREMADE, Laboratoire FiME (Paris-Dauphine)
Introduction I : Motivations
In standard financial markets : Ft (T ) = St e r (T −t) . This
equality relies heavily on costless storability of financial
assets, it breaks down when St is spot price of electricity
A priori, no relations between spot and forward at least in
a market composed of electricity and bank account (see,
e.g., Geman-Vasicek (2001))
Risk neutral
dynamics of
spot and
forward
electricity
prices
Luciano
Campi
Introduction
The Model
Geman-Vasicek (2001) and Bessembinder-Lemon (2002)
show that short-term forward contract are (upward- or
downward-) biased estimator of spot prices, so ...
Electricity
forward
prices
... in mathematical terms, when t ↑ T , Ft (T ) does not
necessarily tend to ST
What’s next
Luciano Campi
Risk neutral dynamics of spot and forward electricity prices
Constant
coefficients
CEREMADE, Laboratoire FiME (Paris-Dauphine)
Introduction I : Motivations
In standard financial markets : Ft (T ) = St e r (T −t) . This
equality relies heavily on costless storability of financial
assets, it breaks down when St is spot price of electricity
A priori, no relations between spot and forward at least in
a market composed of electricity and bank account (see,
e.g., Geman-Vasicek (2001))
Risk neutral
dynamics of
spot and
forward
electricity
prices
Luciano
Campi
Introduction
The Model
Geman-Vasicek (2001) and Bessembinder-Lemon (2002)
show that short-term forward contract are (upward- or
downward-) biased estimator of spot prices, so ...
Electricity
forward
prices
... in mathematical terms, when t ↑ T , Ft (T ) does not
necessarily tend to ST
What’s next
Luciano Campi
Risk neutral dynamics of spot and forward electricity prices
Constant
coefficients
CEREMADE, Laboratoire FiME (Paris-Dauphine)
Introduction II : Main idea
Nevertheless, imagine an fictitious economy where
electricity is produced only out of coal, so that electricity
spot price Pt = cc Stc , and agents can trade coal, buy
electricity and have a bank account
Assume no-arbitrage in the market of coal and bank
account, i.e. there exists a risk-neutral measure Q for
S̃tc = e −rt Stc
A forward contract on spot electricity PT can be viewed as
a contract on coal necessary to produce 1 MWh of
electricity, with price cc Stc , so that
Risk neutral
dynamics of
spot and
forward
electricity
prices
Luciano
Campi
Introduction
The Model
Electricity
forward
prices
Constant
coefficients
What’s next
F0e (T ) = EQ [PT ] = EQ [cc STc ] = cc F0c (T )
Luciano Campi
Risk neutral dynamics of spot and forward electricity prices
CEREMADE, Laboratoire FiME (Paris-Dauphine)
Introduction II : Main idea
Nevertheless, imagine an fictitious economy where
electricity is produced only out of coal, so that electricity
spot price Pt = cc Stc , and agents can trade coal, buy
electricity and have a bank account
Assume no-arbitrage in the market of coal and bank
account, i.e. there exists a risk-neutral measure Q for
S̃tc = e −rt Stc
A forward contract on spot electricity PT can be viewed as
a contract on coal necessary to produce 1 MWh of
electricity, with price cc Stc , so that
Risk neutral
dynamics of
spot and
forward
electricity
prices
Luciano
Campi
Introduction
The Model
Electricity
forward
prices
Constant
coefficients
What’s next
F0e (T ) = EQ [PT ] = EQ [cc STc ] = cc F0c (T )
Luciano Campi
Risk neutral dynamics of spot and forward electricity prices
CEREMADE, Laboratoire FiME (Paris-Dauphine)
Introduction II : Main idea
Nevertheless, imagine an fictitious economy where
electricity is produced only out of coal, so that electricity
spot price Pt = cc Stc , and agents can trade coal, buy
electricity and have a bank account
Assume no-arbitrage in the market of coal and bank
account, i.e. there exists a risk-neutral measure Q for
S̃tc = e −rt Stc
A forward contract on spot electricity PT can be viewed as
a contract on coal necessary to produce 1 MWh of
electricity, with price cc Stc , so that
Risk neutral
dynamics of
spot and
forward
electricity
prices
Luciano
Campi
Introduction
The Model
Electricity
forward
prices
Constant
coefficients
What’s next
F0e (T ) = EQ [PT ] = EQ [cc STc ] = cc F0c (T )
Luciano Campi
Risk neutral dynamics of spot and forward electricity prices
CEREMADE, Laboratoire FiME (Paris-Dauphine)
The Model
Randomness : (W 0 , W ) = (W 0 , W 1 , . . . , W n ) Wiener
process defined on a given (Ω, F, P) and Ft = Ft0 ∨ FtW
models the market information flow.
Rt
Riskless asset St0 = exp 0 ru du, t ≥ 0, r is Ft0 -adapted
and ≥ 0.
Commodities market: n ≥ 1 commodities (coal, gas, ...)
whose prices S i to produce 1 MWh of electricity follows
X ij
dSti = Sti (µit dt +
σt dWtj ), t ≥ 0.
j
yi
For simplicity, assume that convenience yields = 0 for all
i = 1, . . . , n.
Electricity demand: D = (Dt )t≥0 Ft0 -adapted, positive
process; notice that D is independent of each S i .
Luciano Campi
Risk neutral dynamics of spot and forward electricity prices
Risk neutral
dynamics of
spot and
forward
electricity
prices
Luciano
Campi
Introduction
The Model
Electricity
forward
prices
Constant
coefficients
What’s next
CEREMADE, Laboratoire FiME (Paris-Dauphine)
The Model
Randomness : (W 0 , W ) = (W 0 , W 1 , . . . , W n ) Wiener
process defined on a given (Ω, F, P) and Ft = Ft0 ∨ FtW
models the market information flow.
Rt
Riskless asset St0 = exp 0 ru du, t ≥ 0, r is Ft0 -adapted
and ≥ 0.
Commodities market: n ≥ 1 commodities (coal, gas, ...)
whose prices S i to produce 1 MWh of electricity follows
X ij
dSti = Sti (µit dt +
σt dWtj ), t ≥ 0.
j
yi
For simplicity, assume that convenience yields = 0 for all
i = 1, . . . , n.
Electricity demand: D = (Dt )t≥0 Ft0 -adapted, positive
process; notice that D is independent of each S i .
Luciano Campi
Risk neutral dynamics of spot and forward electricity prices
Risk neutral
dynamics of
spot and
forward
electricity
prices
Luciano
Campi
Introduction
The Model
Electricity
forward
prices
Constant
coefficients
What’s next
CEREMADE, Laboratoire FiME (Paris-Dauphine)
The Model
Randomness : (W 0 , W ) = (W 0 , W 1 , . . . , W n ) Wiener
process defined on a given (Ω, F, P) and Ft = Ft0 ∨ FtW
models the market information flow.
Rt
Riskless asset St0 = exp 0 ru du, t ≥ 0, r is Ft0 -adapted
and ≥ 0.
Commodities market: n ≥ 1 commodities (coal, gas, ...)
whose prices S i to produce 1 MWh of electricity follows
X ij
dSti = Sti (µit dt +
σt dWtj ), t ≥ 0.
j
yi
For simplicity, assume that convenience yields = 0 for all
i = 1, . . . , n.
Electricity demand: D = (Dt )t≥0 Ft0 -adapted, positive
process; notice that D is independent of each S i .
Luciano Campi
Risk neutral dynamics of spot and forward electricity prices
Risk neutral
dynamics of
spot and
forward
electricity
prices
Luciano
Campi
Introduction
The Model
Electricity
forward
prices
Constant
coefficients
What’s next
CEREMADE, Laboratoire FiME (Paris-Dauphine)
The Model
Randomness : (W 0 , W ) = (W 0 , W 1 , . . . , W n ) Wiener
process defined on a given (Ω, F, P) and Ft = Ft0 ∨ FtW
models the market information flow.
Rt
Riskless asset St0 = exp 0 ru du, t ≥ 0, r is Ft0 -adapted
and ≥ 0.
Commodities market: n ≥ 1 commodities (coal, gas, ...)
whose prices S i to produce 1 MWh of electricity follows
X ij
dSti = Sti (µit dt +
σt dWtj ), t ≥ 0.
j
yi
For simplicity, assume that convenience yields = 0 for all
i = 1, . . . , n.
Electricity demand: D = (Dt )t≥0 Ft0 -adapted, positive
process; notice that D is independent of each S i .
Luciano Campi
Risk neutral dynamics of spot and forward electricity prices
Risk neutral
dynamics of
spot and
forward
electricity
prices
Luciano
Campi
Introduction
The Model
Electricity
forward
prices
Constant
coefficients
What’s next
CEREMADE, Laboratoire FiME (Paris-Dauphine)
Electricity spot price Pt when all technologies are
available :
∆i > 0 denotes the capacity of i-th commodity for
electricity at every instant, a constant known to the
producer
(1)
(n)
order commodities prices St (ω) ≤ . . . ≤ St (ω) from the
cheapest to the most expensive, giving an FtW -adapted
random permutation πt (ω) of {1, . . . , n}
look at the demand Dt :
"k−1
!
k
X
X
(k)
∆πt (i) ,
∆πt (i) ⇒ Pt = St
Dt ∈ Ikπt :=
i=1
... so that Pt =
P
k
i=1
Risk neutral
dynamics of
spot and
forward
electricity
prices
Luciano
Campi
Introduction
The Model
Electricity
forward
prices
Constant
coefficients
What’s next
(k)
St 1I πt (Dt ) for t ≥ 0.
k
Luciano Campi
Risk neutral dynamics of spot and forward electricity prices
CEREMADE, Laboratoire FiME (Paris-Dauphine)
Electricity spot price Pt when all technologies are
available :
∆i > 0 denotes the capacity of i-th commodity for
electricity at every instant, a constant known to the
producer
(1)
(n)
order commodities prices St (ω) ≤ . . . ≤ St (ω) from the
cheapest to the most expensive, giving an FtW -adapted
random permutation πt (ω) of {1, . . . , n}
look at the demand Dt :
"k−1
!
k
X
X
(k)
∆πt (i) ,
∆πt (i) ⇒ Pt = St
Dt ∈ Ikπt :=
i=1
... so that Pt =
P
k
i=1
Risk neutral
dynamics of
spot and
forward
electricity
prices
Luciano
Campi
Introduction
The Model
Electricity
forward
prices
Constant
coefficients
What’s next
(k)
St 1I πt (Dt ) for t ≥ 0.
k
Luciano Campi
Risk neutral dynamics of spot and forward electricity prices
CEREMADE, Laboratoire FiME (Paris-Dauphine)
Electricity spot price Pt when all technologies are
available :
∆i > 0 denotes the capacity of i-th commodity for
electricity at every instant, a constant known to the
producer
(1)
(n)
order commodities prices St (ω) ≤ . . . ≤ St (ω) from the
cheapest to the most expensive, giving an FtW -adapted
random permutation πt (ω) of {1, . . . , n}
look at the demand Dt :
"k−1
!
k
X
X
(k)
∆πt (i) ,
∆πt (i) ⇒ Pt = St
Dt ∈ Ikπt :=
i=1
... so that Pt =
P
k
i=1
Risk neutral
dynamics of
spot and
forward
electricity
prices
Luciano
Campi
Introduction
The Model
Electricity
forward
prices
Constant
coefficients
What’s next
(k)
St 1I πt (Dt ) for t ≥ 0.
k
Luciano Campi
Risk neutral dynamics of spot and forward electricity prices
CEREMADE, Laboratoire FiME (Paris-Dauphine)
Electricity spot price Pt when all technologies are
available :
∆i > 0 denotes the capacity of i-th commodity for
electricity at every instant, a constant known to the
producer
(1)
(n)
order commodities prices St (ω) ≤ . . . ≤ St (ω) from the
cheapest to the most expensive, giving an FtW -adapted
random permutation πt (ω) of {1, . . . , n}
look at the demand Dt :
"k−1
!
k
X
X
(k)
∆πt (i) ,
∆πt (i) ⇒ Pt = St
Dt ∈ Ikπt :=
i=1
... so that Pt =
P
k
i=1
Risk neutral
dynamics of
spot and
forward
electricity
prices
Luciano
Campi
Introduction
The Model
Electricity
forward
prices
Constant
coefficients
What’s next
(k)
St 1I πt (Dt ) for t ≥ 0.
k
Luciano Campi
Risk neutral dynamics of spot and forward electricity prices
CEREMADE, Laboratoire FiME (Paris-Dauphine)
Technologies failures I : The case of two
commodities
If n = 2 we have St1 ≤ St2 or St2 ≤ St1 , let’s consider the
first case πt = {1, 2}
Introduce two r.v.’s it , i = 1, 2 such that
it = 1 when technology i is available, otherwise it = 0
it = 0 implies that jt = 1 for i 6= j
Only three cases may happen at each time t
1t = 2t = 1 then Pt = St1 1[0,∆1 ) (Dt ) + St2 1[∆1 ,∆1 +∆2 ) (Dt )
2 1t = 1, 2t = 0 then Pt = St1 1[0,∆1 ) (Dt )
3 1t = 0, 2t = 1 then Pt = St2 1[0,∆1 +∆2 ) (Dt )
1
To sum up:
Pt = St1 1[0,∆1 1t ) (Dt ) + St2 1[∆1 1t ,∆1 1t +∆2 2t ) (Dt )
Luciano Campi
Risk neutral dynamics of spot and forward electricity prices
Risk neutral
dynamics of
spot and
forward
electricity
prices
Luciano
Campi
Introduction
The Model
Electricity
forward
prices
Constant
coefficients
What’s next
CEREMADE, Laboratoire FiME (Paris-Dauphine)
Technologies failures I : The case of two
commodities
If n = 2 we have St1 ≤ St2 or St2 ≤ St1 , let’s consider the
first case πt = {1, 2}
Introduce two r.v.’s it , i = 1, 2 such that
it = 1 when technology i is available, otherwise it = 0
it = 0 implies that jt = 1 for i 6= j
Only three cases may happen at each time t
1t = 2t = 1 then Pt = St1 1[0,∆1 ) (Dt ) + St2 1[∆1 ,∆1 +∆2 ) (Dt )
2 1t = 1, 2t = 0 then Pt = St1 1[0,∆1 ) (Dt )
3 1t = 0, 2t = 1 then Pt = St2 1[0,∆1 +∆2 ) (Dt )
1
To sum up:
Pt = St1 1[0,∆1 1t ) (Dt ) + St2 1[∆1 1t ,∆1 1t +∆2 2t ) (Dt )
Luciano Campi
Risk neutral dynamics of spot and forward electricity prices
Risk neutral
dynamics of
spot and
forward
electricity
prices
Luciano
Campi
Introduction
The Model
Electricity
forward
prices
Constant
coefficients
What’s next
CEREMADE, Laboratoire FiME (Paris-Dauphine)
Technologies failures I : The case of two
commodities
If n = 2 we have St1 ≤ St2 or St2 ≤ St1 , let’s consider the
first case πt = {1, 2}
Introduce two r.v.’s it , i = 1, 2 such that
it = 1 when technology i is available, otherwise it = 0
it = 0 implies that jt = 1 for i 6= j
Only three cases may happen at each time t
1t = 2t = 1 then Pt = St1 1[0,∆1 ) (Dt ) + St2 1[∆1 ,∆1 +∆2 ) (Dt )
2 1t = 1, 2t = 0 then Pt = St1 1[0,∆1 ) (Dt )
3 1t = 0, 2t = 1 then Pt = St2 1[0,∆1 +∆2 ) (Dt )
1
To sum up:
Pt = St1 1[0,∆1 1t ) (Dt ) + St2 1[∆1 1t ,∆1 1t +∆2 2t ) (Dt )
Luciano Campi
Risk neutral dynamics of spot and forward electricity prices
Risk neutral
dynamics of
spot and
forward
electricity
prices
Luciano
Campi
Introduction
The Model
Electricity
forward
prices
Constant
coefficients
What’s next
CEREMADE, Laboratoire FiME (Paris-Dauphine)
Technologies failures I : The case of two
commodities
If n = 2 we have St1 ≤ St2 or St2 ≤ St1 , let’s consider the
first case πt = {1, 2}
Introduce two r.v.’s it , i = 1, 2 such that
it = 1 when technology i is available, otherwise it = 0
it = 0 implies that jt = 1 for i 6= j
Only three cases may happen at each time t
1t = 2t = 1 then Pt = St1 1[0,∆1 ) (Dt ) + St2 1[∆1 ,∆1 +∆2 ) (Dt )
2 1t = 1, 2t = 0 then Pt = St1 1[0,∆1 ) (Dt )
3 1t = 0, 2t = 1 then Pt = St2 1[0,∆1 +∆2 ) (Dt )
1
To sum up:
Pt = St1 1[0,∆1 1t ) (Dt ) + St2 1[∆1 1t ,∆1 1t +∆2 2t ) (Dt )
Luciano Campi
Risk neutral dynamics of spot and forward electricity prices
Risk neutral
dynamics of
spot and
forward
electricity
prices
Luciano
Campi
Introduction
The Model
Electricity
forward
prices
Constant
coefficients
What’s next
CEREMADE, Laboratoire FiME (Paris-Dauphine)
Technologies failures II : The general case
Let η be a new process, values in {0, 1, . . . , n}, with
interpretation
event {ηt = i} means “i-th technology not available”, for
1≤i ≤n
event {ηt = 0} means that all technologies are available
Hidden assumption: only one failure at the time is allowed.
Define it := 1{ηt 6=i} , 1 ≤ i ≤ n, so that
it = 1 means “i-th technology available at time t
it = 0 means “i-th technology not available at time t
Set
Ikπt (t)
:=
hP
... so that Pt =
πt (i) Pk
π (i)
k−1
, i=1 ∆πt (i) t t
i=1 ∆πt (i) t
(k)
π
k St 1Ik t (t)
P
Risk neutral
dynamics of
spot and
forward
electricity
prices
Luciano
Campi
Introduction
The Model
Electricity
forward
prices
Constant
coefficients
What’s next
for t ≥ 0.
Luciano Campi
Risk neutral dynamics of spot and forward electricity prices
CEREMADE, Laboratoire FiME (Paris-Dauphine)
Technologies failures II : The general case
Let η be a new process, values in {0, 1, . . . , n}, with
interpretation
event {ηt = i} means “i-th technology not available”, for
1≤i ≤n
event {ηt = 0} means that all technologies are available
Hidden assumption: only one failure at the time is allowed.
Define it := 1{ηt 6=i} , 1 ≤ i ≤ n, so that
it = 1 means “i-th technology available at time t
it = 0 means “i-th technology not available at time t
Set
Ikπt (t)
:=
hP
... so that Pt =
πt (i) Pk
π (i)
k−1
, i=1 ∆πt (i) t t
i=1 ∆πt (i) t
(k)
π
k St 1Ik t (t)
P
Risk neutral
dynamics of
spot and
forward
electricity
prices
Luciano
Campi
Introduction
The Model
Electricity
forward
prices
Constant
coefficients
What’s next
for t ≥ 0.
Luciano Campi
Risk neutral dynamics of spot and forward electricity prices
CEREMADE, Laboratoire FiME (Paris-Dauphine)
Technologies failures II : The general case
Let η be a new process, values in {0, 1, . . . , n}, with
interpretation
event {ηt = i} means “i-th technology not available”, for
1≤i ≤n
event {ηt = 0} means that all technologies are available
Hidden assumption: only one failure at the time is allowed.
Define it := 1{ηt 6=i} , 1 ≤ i ≤ n, so that
it = 1 means “i-th technology available at time t
it = 0 means “i-th technology not available at time t
Set
Ikπt (t)
:=
hP
... so that Pt =
πt (i) Pk
π (i)
k−1
, i=1 ∆πt (i) t t
i=1 ∆πt (i) t
(k)
π
k St 1Ik t (t)
P
Risk neutral
dynamics of
spot and
forward
electricity
prices
Luciano
Campi
Introduction
The Model
Electricity
forward
prices
Constant
coefficients
What’s next
for t ≥ 0.
Luciano Campi
Risk neutral dynamics of spot and forward electricity prices
CEREMADE, Laboratoire FiME (Paris-Dauphine)
Technologies failures II : The general case
Let η be a new process, values in {0, 1, . . . , n}, with
interpretation
event {ηt = i} means “i-th technology not available”, for
1≤i ≤n
event {ηt = 0} means that all technologies are available
Hidden assumption: only one failure at the time is allowed.
Define it := 1{ηt 6=i} , 1 ≤ i ≤ n, so that
it = 1 means “i-th technology available at time t
it = 0 means “i-th technology not available at time t
Set
Ikπt (t)
:=
hP
... so that Pt =
πt (i) Pk
π (i)
k−1
, i=1 ∆πt (i) t t
i=1 ∆πt (i) t
(k)
π
k St 1Ik t (t)
P
Risk neutral
dynamics of
spot and
forward
electricity
prices
Luciano
Campi
Introduction
The Model
Electricity
forward
prices
Constant
coefficients
What’s next
for t ≥ 0.
Luciano Campi
Risk neutral dynamics of spot and forward electricity prices
CEREMADE, Laboratoire FiME (Paris-Dauphine)
No-arbitrage assumption on commodities market.
Let T > 0. There exists Q ∼ P on FTW such that :
1
each S̃ i /S 0 is a Q-martingale w.r.t. F W
2
the laws of W 0 and η do not change
3
filtrations
(Ft0 ), (FtW ), (Ftη )
Risk neutral
dynamics of
spot and
forward
electricity
prices
Luciano
Campi
are Q-independent
Introduction
Remarks
The Model
W 0, W
1. Property 3 above is satisfied if
and η are constructed
on the canonical product space and the change of measure
affects only the factor where W is defined.
2. Being D not tradable, this market is not complete. We
choose Q as the pricing measure.
Luciano Campi
Risk neutral dynamics of spot and forward electricity prices
Electricity
forward
prices
Constant
coefficients
What’s next
CEREMADE, Laboratoire FiME (Paris-Dauphine)
Electricity forward prices I
The pay-off of a forward contract on spot electricity is
P (k)
PT = k ST 1I πT (T ) (DT ) so it can be viewed as an
k
option on commodities
Use no-arbitrage assumption on commodities to get
" n
#
X (k)
QT
QT
Ft (T ) = E [PT |Ft ] = E
ST 1I πT (T ) (DT )|Ft
k
k=1
where QT is the forward risk-neutral measure on FT :
RT
exp t ru du
d QT
=
RT
dQ
EQ [exp
ru du|Ft ]
Risk neutral
dynamics of
spot and
forward
electricity
prices
Luciano
Campi
Introduction
The Model
Electricity
forward
prices
Constant
coefficients
What’s next
t
(Notice that QT = Q if r is non-random)
Luciano Campi
Risk neutral dynamics of spot and forward electricity prices
CEREMADE, Laboratoire FiME (Paris-Dauphine)
Electricity forward prices I
The pay-off of a forward contract on spot electricity is
P (k)
PT = k ST 1I πT (T ) (DT ) so it can be viewed as an
k
option on commodities
Use no-arbitrage assumption on commodities to get
" n
#
X (k)
QT
QT
Ft (T ) = E [PT |Ft ] = E
ST 1I πT (T ) (DT )|Ft
k
k=1
where QT is the forward risk-neutral measure on FT :
RT
exp t ru du
d QT
=
RT
dQ
EQ [exp
ru du|Ft ]
Risk neutral
dynamics of
spot and
forward
electricity
prices
Luciano
Campi
Introduction
The Model
Electricity
forward
prices
Constant
coefficients
What’s next
t
(Notice that QT = Q if r is non-random)
Luciano Campi
Risk neutral dynamics of spot and forward electricity prices
CEREMADE, Laboratoire FiME (Paris-Dauphine)
Electricity forward prices II : The main formula
Proposition
Under previous assumptions and if STi ∈ L1 (QT ), 1 ≤ i ≤ n :
for all t ∈ [0, T ]
Ft (T ) =
n X
X
Risk neutral
dynamics of
spot and
forward
electricity
prices
Luciano
Campi
π(i)
cπ(i) Ft (T )QT [DT
∈ Iiπ (T )|Ft0 ]
i=1 π∈Πn
π(i)
×QT [πT
Introduction
The Model
=
π|FtW ]
Electricity
forward
prices
where :
Πn is the set of all permutations of {1, . . . , n}
Constant
coefficients
What’s next
Fti (T ) is forward price of i -th commodity, delivery date T
π(i)
π(i)
π(i)
d QT /d QT = ST /EQT [ST ] on FTW
Luciano Campi
Risk neutral dynamics of spot and forward electricity prices
CEREMADE, Laboratoire FiME (Paris-Dauphine)
Electricity forward prices II : The main formula
Proposition
Under previous assumptions and if STi ∈ L1 (QT ), 1 ≤ i ≤ n :
for all t ∈ [0, T ]
Ft (T ) =
n X
X
Risk neutral
dynamics of
spot and
forward
electricity
prices
Luciano
Campi
π(i)
cπ(i) Ft (T )QT [DT
∈ Iiπ (T )|Ft0 ]
i=1 π∈Πn
π(i)
×QT [πT
Introduction
The Model
=
π|FtW ]
Electricity
forward
prices
where :
Πn is the set of all permutations of {1, . . . , n}
Constant
coefficients
What’s next
Fti (T ) is forward price of i -th commodity, delivery date T
π(i)
π(i)
π(i)
d QT /d QT = ST /EQT [ST ] on FTW
Luciano Campi
Risk neutral dynamics of spot and forward electricity prices
CEREMADE, Laboratoire FiME (Paris-Dauphine)
Electricity forward prices II : The main formula
Proposition
Under previous assumptions and if STi ∈ L1 (QT ), 1 ≤ i ≤ n :
for all t ∈ [0, T ]
Ft (T ) =
n X
X
Risk neutral
dynamics of
spot and
forward
electricity
prices
Luciano
Campi
π(i)
cπ(i) Ft (T )QT [DT
∈ Iiπ (T )|Ft0 ]
i=1 π∈Πn
π(i)
×QT [πT
Introduction
The Model
=
π|FtW ]
Electricity
forward
prices
where :
Πn is the set of all permutations of {1, . . . , n}
Constant
coefficients
What’s next
Fti (T ) is forward price of i -th commodity, delivery date T
π(i)
π(i)
π(i)
d QT /d QT = ST /EQT [ST ] on FTW
Luciano Campi
Risk neutral dynamics of spot and forward electricity prices
CEREMADE, Laboratoire FiME (Paris-Dauphine)
Electricity forward prices III : some remarks
This model is able to explain three basic features of electricity
market as :
Observed spikes in electricity spot prices dynamics
Non-convergence of electricity forward prices towards spot
(day-ahead) prices as t ↑ T . Indeed,
Ft (T ) → FT (T ) =
n
X
Risk neutral
dynamics of
spot and
forward
electricity
prices
Luciano
Campi
Introduction
The Model
(i)
ST QT [x
∈
Iiπ (T )]|x=DT ,π=πT .
i=1
FT (T ) 6= PT whenever η is non-degenerate.
The paths of electricity forward prices are much smoother
than the corresponding spot prices.
Luciano Campi
Risk neutral dynamics of spot and forward electricity prices
Electricity
forward
prices
Constant
coefficients
What’s next
CEREMADE, Laboratoire FiME (Paris-Dauphine)
The constant coefficients model : more explicit
formulae
Si
Commodities prices
follow n-dim Black-Scholes model :
volatilities σ ij > 0 and interest rate r > 0 constant so that,
in particular, QT = Q
Fti (T ) = e r (T −t) Sti for all commodities 1 ≤ i ≤ n
Demand of electricity : D follows a OU process
dDt = a(b − Dt )dt + δdWt0 ,
Ikπ (T )|Ft0 ]
Under these assumptions probabilities Q[DT ∈
π(i)
and QT [πT = π|FtW ] can be computed explicitly as
functions of the parameters.
Risk neutral dynamics of spot and forward electricity prices
Luciano
Campi
Introduction
The Model
D0 > 0
with a, b, δ > 0.
Luciano Campi
Risk neutral
dynamics of
spot and
forward
electricity
prices
Electricity
forward
prices
Constant
coefficients
What’s next
CEREMADE, Laboratoire FiME (Paris-Dauphine)
The constant coefficients model : more explicit
formulae
Si
Commodities prices
follow n-dim Black-Scholes model :
volatilities σ ij > 0 and interest rate r > 0 constant so that,
in particular, QT = Q
Fti (T ) = e r (T −t) Sti for all commodities 1 ≤ i ≤ n
Demand of electricity : D follows a OU process
dDt = a(b − Dt )dt + δdWt0 ,
Ikπ (T )|Ft0 ]
Under these assumptions probabilities Q[DT ∈
π(i)
and QT [πT = π|FtW ] can be computed explicitly as
functions of the parameters.
Risk neutral dynamics of spot and forward electricity prices
Luciano
Campi
Introduction
The Model
D0 > 0
with a, b, δ > 0.
Luciano Campi
Risk neutral
dynamics of
spot and
forward
electricity
prices
Electricity
forward
prices
Constant
coefficients
What’s next
CEREMADE, Laboratoire FiME (Paris-Dauphine)
The constant coefficients model : more explicit
formulae
Si
Commodities prices
follow n-dim Black-Scholes model :
volatilities σ ij > 0 and interest rate r > 0 constant so that,
in particular, QT = Q
Fti (T ) = e r (T −t) Sti for all commodities 1 ≤ i ≤ n
Demand of electricity : D follows a OU process
dDt = a(b − Dt )dt + δdWt0 ,
Ikπ (T )|Ft0 ]
Under these assumptions probabilities Q[DT ∈
π(i)
and QT [πT = π|FtW ] can be computed explicitly as
functions of the parameters.
Risk neutral dynamics of spot and forward electricity prices
Luciano
Campi
Introduction
The Model
D0 > 0
with a, b, δ > 0.
Luciano Campi
Risk neutral
dynamics of
spot and
forward
electricity
prices
Electricity
forward
prices
Constant
coefficients
What’s next
CEREMADE, Laboratoire FiME (Paris-Dauphine)
The constant coefficients model : more explicit
formulae
Si
Commodities prices
follow n-dim Black-Scholes model :
volatilities σ ij > 0 and interest rate r > 0 constant so that,
in particular, QT = Q
Fti (T ) = e r (T −t) Sti for all commodities 1 ≤ i ≤ n
Demand of electricity : D follows a OU process
dDt = a(b − Dt )dt + δdWt0 ,
Ikπ (T )|Ft0 ]
Under these assumptions probabilities Q[DT ∈
π(i)
and QT [πT = π|FtW ] can be computed explicitly as
functions of the parameters.
Risk neutral dynamics of spot and forward electricity prices
Luciano
Campi
Introduction
The Model
D0 > 0
with a, b, δ > 0.
Luciano Campi
Risk neutral
dynamics of
spot and
forward
electricity
prices
Electricity
forward
prices
Constant
coefficients
What’s next
CEREMADE, Laboratoire FiME (Paris-Dauphine)
What’s next ?
Pricing of options on forward electricity : it can be reduced
to pricing of basket options of commodities
Risk neutral
dynamics of
spot and
forward
electricity
prices
Luciano
Campi
Simulations and estimation of parameters in progress ...
Make the model more complex, e.g. add stochastic
convenience yields and interest rate, more than one failure
at the time ...
Study the risk premium π(t, T ) = Ft (T ) − Pt in our
model, compare with other models
Introduction
The Model
Electricity
forward
prices
Constant
coefficients
What’s next
Luciano Campi
Risk neutral dynamics of spot and forward electricity prices
CEREMADE, Laboratoire FiME (Paris-Dauphine)
What’s next ?
Pricing of options on forward electricity : it can be reduced
to pricing of basket options of commodities
Risk neutral
dynamics of
spot and
forward
electricity
prices
Luciano
Campi
Simulations and estimation of parameters in progress ...
Make the model more complex, e.g. add stochastic
convenience yields and interest rate, more than one failure
at the time ...
Study the risk premium π(t, T ) = Ft (T ) − Pt in our
model, compare with other models
Introduction
The Model
Electricity
forward
prices
Constant
coefficients
What’s next
Luciano Campi
Risk neutral dynamics of spot and forward electricity prices
CEREMADE, Laboratoire FiME (Paris-Dauphine)
What’s next ?
Pricing of options on forward electricity : it can be reduced
to pricing of basket options of commodities
Risk neutral
dynamics of
spot and
forward
electricity
prices
Luciano
Campi
Simulations and estimation of parameters in progress ...
Make the model more complex, e.g. add stochastic
convenience yields and interest rate, more than one failure
at the time ...
Study the risk premium π(t, T ) = Ft (T ) − Pt in our
model, compare with other models
Introduction
The Model
Electricity
forward
prices
Constant
coefficients
What’s next
Luciano Campi
Risk neutral dynamics of spot and forward electricity prices
CEREMADE, Laboratoire FiME (Paris-Dauphine)
What’s next ?
Pricing of options on forward electricity : it can be reduced
to pricing of basket options of commodities
Risk neutral
dynamics of
spot and
forward
electricity
prices
Luciano
Campi
Simulations and estimation of parameters in progress ...
Make the model more complex, e.g. add stochastic
convenience yields and interest rate, more than one failure
at the time ...
Study the risk premium π(t, T ) = Ft (T ) − Pt in our
model, compare with other models
Introduction
The Model
Electricity
forward
prices
Constant
coefficients
What’s next
Luciano Campi
Risk neutral dynamics of spot and forward electricity prices
CEREMADE, Laboratoire FiME (Paris-Dauphine)