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Valuation and Hedging of Power-Sensitive
Contingent Claims for Power with Spikes:
a Non-Markovian Approach
Valery A. Kholodnyi
February 25, 2004
Houston, Texas
1
Introduction
As the power markets are becoming deregulated worldwide,
the modeling of the dynamics of power spot prices is
becoming one of the key problems in the risk management,
physical assets valuation, and derivative pricing.
One of the main difficulties in this modeling is to combine
the following features:
• To provide a mechanism that allows for the absence of
spikes in the prices of power-sensitive contingent claims
while the power spot prices exhibit spikes, and
• To keep the dynamics of the prices of power-sensitive
contingent claims consistent with the dynamics of the power
spot prices.
2
Models for Power Spot Prices
with Spikes
• Mean-Reverting Jump Diffusion Process (Ethier and
Dorris, 1999; Clewlow, Strickland and Kaminski, 2000)
– the same mechanism is responsible for both the decay of spikes
and the reversion of power prices to their equilibrium mean
• Mixture of Processes (Goldberg and Read, 2000; Ball and
Torous, 1985)
– spikes and the regular, that is, inter-spike regime do not persist in
time
– relatively difficult to estimate parameters
• Regime Switching Process (Ethier, 1999; Duffie and Gray
1995)
– discreet time regime switching
– inconsistent short term option values
– relatively difficult to estimate parameters
3
The Non-Markovian Process for
Power Spot Prices with Spikes
Motivation
• Different mechanisms should be responsible for:
– the reversion of power prices to their equilibrium mean in the
regular, that is, inter-spike state
– the reversion of power prices to their long term mean in the spike
state, that is, for the decay of spikes
• This is, in our opinion, due to the substantial difference in
the scales of the deviations of power prices from their
equilibrium mean in the spike and inter-spike states
– For example, power prices in the US Midwest in June 1998 rose
to $7,500 per megawatt hour (MWh) compared with typical prices
4
of around $30 per MWh
The Non-Markovian Process for
Power Spot Prices with Spikes
Main Features
• The spikes are modeled directly as self-reversing jumps,
either multiplicative or additive, in continuous time
• The parameters that characterize spikes are frequency,
duration, and magnitude
• The spikes parameters are directly observable from
market data as well as admit structural interpretation
• The spike state and the regular, that is, inter-spike state
do persist in time
5
The Non-Markovian Process for
Power Spot Prices with Spikes
Formal Definition
Define (Kholodnyi, 2000) the non-Markovian process for
the power spot prices with spikes by
ˆ ,
t  t 
t
• t>0 is the power spot price at time t ,
• t 1 is the multiplicative magnitude of spikes at time t ,
ˆ  0 is the inter-spike power spot price at time t.
• 
t
Assume that the spike process t and inter-spike
process ̂t are independent Markov processes.
6
The Non-Markovian Process for
Power Spot Prices with Spikes
Underlying Two-State Markov Process
Denote by Mt a two-state Markov process with continuous
time t  0.
Denote the 22 transition matrix for the two-state Markov
process Mt by
 Pss (T , t ) Psr (T , t ) 

P(T , t )  
 Prs (T , t ) Prr (T , t ) 
• Pss(T,t) and Prs(T,t) are transition probabilities from the
spike state at time t to the spike and regular states at time
T, and
• Psr(T,t) and Prr(T,t) are transition probabilities from the
regular state at time t to the spike and regular states at
time T.
7
The Non-Markovian Process for
Power Spot Prices with Spikes
Generators of the Underlying Two-State Markov
Process
The family of 22 matrices L = {L(t) : t  0} defined by
L(t)  d P(T ,t) T t ,
dT
is said to generate the two-state Markov process Mt, and
the 22 matrix
 Lss (t ) Lsr (t ) 

L(t )  
 Lrs (t ) Lrr (t ) 
is called a generator.
In terms of the generators, P(T,t) is given by
T L( )d

P(T ,t)  e t
.
8
The Non-Markovian Process for
Power Spot Prices with Spikes
Decompositions of the Transition Probabilities of the
Underlying Two-State Markov Process
It can be shown that
T
t Lss ( ) d
Pss (T , t )  e
T
  Prs ( , t )Lsr ( )e
T
 Lss ( ') d '
t
T
Psr (T , t )   Prr ( , t )Lsr ( )e
T
 Lss ( ') d '
t
d ,
d .
Moreover
Pss (T , t )  Psss (T , t )  Pssr (T , t ),
where
T
t Lss ( ) d
P (T , t )  e
s
ss
T
P (T , t )   Prs ( , t )Lsr ( )e
r
ss
t
T
 Lss ( ') d '
d
9
The Non-Markovian Process for
Power Spot Prices with Spikes
Underlying Two-State Markov Process in the
Time-Homogeneous Case
In the special case of a time-homogeneous two-state
Markov process Mt the transition matrix P(T-t) and the
generator L are given by
P(T  t) 

 (T t )( a  b )
 b  ae

ab

 (T t )( a  b )
 a  ae

ab

b  be  (T t )( a b )
ab
a  be  (T t )( a b )
ab







and






L  a b
a b






10
The Non-Markovian Process for
Power Spot Prices with Spikes
t
(t,)
Construction of the Spike Process
1
Time
Mt
Spike State Regular State
ts
tr
Time
11
The Non-Markovian Process for
Power Spot Prices with Spikes
Formal Definition of the Spike Process
The transition probability density function for the spike
process t as a Markov process is given by
 t Lss ( ) d
 (t  T ) 
e
T
 T
L ( ') d '
  ( , T ) Prs ( , t ) Lsr ( )e  ss
d  if t  1
t
 (t , T , t , T )  
P (T , t ) (1  T )
 rs
T
T
 ( ,  ) P ( , t ) L ( )e  Lss ( ') d ' d 
T
rr
sr
if t  1
t
 P (T , t ) (1   )
T
 rr
T
where (x) is the Dirac delta function.
12
The Non-Markovian Process for
Power Spot Prices with Spikes
Inter-Spike Process
For example, ̂t can be a diffusion process defined by
ˆ   (
ˆ , t )dt   (
ˆ , t )dW ,
d
t
t
t
t
ˆ , t ) is the drift,  (
ˆ , t )  0 is the volatility, and
where:  (
t
t
Wt is the Wiener process.
In the practically important special case of a geometricmean reverting process we have
ˆ  (t )( (t )  ln 
ˆ )
ˆ dt   (t )
ˆ dW ,
d
t
t
t
t
t
where:  (t )  0 is the mean-reversion rate,  (t ) is the
equilibrium mean, and  (t )  0 is the volatility.
13
The Non-Markovian Process for
Power Spot Prices with Spikes
The Expected Time for t to be in the Spike and InterSpike States
The expected time t s for t to be in the spike state that
starts at time t is:


t s   (  t )e
  a ( ') d '
t
a( )d .
t
Similarly, the expected time t r for t to be in the inter-spike
state that starts at time t is:


t r   (  t )e
  b ( ') d '
t
b( )d .
t
In the special case of a time-homogeneous two-state
Markov process Mt:
t s  1 / a and tr  1 / b.
14
The Non-Markovian Process for
Power Spot Prices with Spikes
Interpretation of the Spike State of t as Spikes in
Power Prices
If the expected time t s for the non-Markovian process t to
be in the spike state is small relative to the characteristic
time of change of the process ̂ then the spike state of t
t
can be interpreted as spikes in power spot prices:
– t can exhibit sharp upward price movements shortly followed by
equally sharp downward prices movements of approximately the same
magnitude.
For example, if ̂ is a diffusion process then:
t
ˆ , t )t  1.
ˆ , t )t  1
 (
 2 (
and
s
s
In this case t s is the expected lifetime of a spike and t r 15
is the expected lifetime between two consecutive spikes.
The Non-Markovian Process for
Power Spot Prices with Spikes
Estimation of the Spike Parameters
• In the special case of a time-homogeneous two-state
Markov process the expected life-time of a spike is given by
ts  1 / a
• Similarly, the expected life-time between two consecutive
spikes is given by
tr  1 / b.
• The estimation of the probability density function (t,) for
the spike magnitude can be based on the standard
parametric or nonparametric statistical methods
– Scaling and asymptotically scaling distributions are of
16
a particular interest in practice
The Non-Markovian Process for
Power Spot Prices with Spikes
The Non-Markovian Process t as a Markov Process
with the Extended State Space
The state of the power market at any time t can be fully
characterized by a pair of the values of the processes t ,
and ̂t at time t.
Moreover, although the process t is non-Markovian it can
be, in fact, represented as a Markov process that for any
time t can be fully characterized by the values of the
processes t and ̂t at time t.
Equivalently, the non-Markovian process t can be
represented as a Markov process with the extended state
ˆ )
space that at any time t consists of all possible pairs (t , 
t
ˆ 0 .
with t  1 and 
17
t
European Contingent Claims on
Power in the Absence of Spikes
Valuing European Contingent Claims on Power as the
Discounted Risk-Neutral expected value of its payoff
Denote by
ˆ ),
Eˆ (t , T , g )  Eˆ (t , T , g )(
t
the value of the European contingent claim on power with
inception time t, expiration time T, and payoff g.
The value of this European contingent claim can be found
as the discounted risk-neutral expected value of its payoff:
T
ˆ )e
Eˆ (t , T , g )( 
t

 r ( ) d 
t
ˆ ,
ˆ ) g (
ˆ )d
ˆ ,
P
(
t
,
T
,

t
T
T
T

0
ˆ ,
ˆ ) is the risk-neutral transition probability
where P(t, T , 
t
T
18
density function.
European Contingent Claims on
Power in the Absence of Spikes
Example: Geometric Mean-Reverting Process
It can be shown (Kholodnyi 1995) that
Eˆ
MR
ˆ )
(t , T , g )( 
t
e
 r ( t ,T )(T  t )

2 (T  t )
ˆ (t , T )
0
ˆ
 Eˆ BS
ˆ ( t ,T ),0 (t , T , g )( t
where:
r (t , T ) 
1
T t

T
t
1
ˆ (t , T ) 
T t


e
ˆ  b ( t ,T )  ln 
ˆ )2
1 ( a ( t ,T ) ln 
t
T
2
2
ˆ ( t ,T )(T  t )
ˆ
d

ˆ ) T
g (
T
ˆ

T
1 2
ˆ
a ( t ,T ) b ( t , T ) ( T  t ) 2  ( t ,T )
e
e
).
r ( )d ,

T
t
T
 ( )e 
2
 2  ( ') d '
d ,
T
a(t , T )  e t
  ( ) d
b(t , T )  
T
t
,
1  2 ( )   ( ') d '
 ( )(  ( ) 
)e
d .
2  ( )
T
19
European Contingent Claims on
Power in the Absence of Spikes
Example: Geometric Mean-Reverting Process
For example (Kholodnyi 1995):
Cˆ
MR
ˆ , X )  Cˆ
(t , T , 
t
BS
ˆ ( t ,T ),0
ˆ
(t , T , 
t
ˆ , X )  Pˆ BS (t , T , 
ˆ
Pˆ MR (t , T , 
t
t
ˆ ( t ,T ),0
1 2
ˆ
a ( t , T ) b ( t ,T ) ( T  t ) 2  ( t , T )
e
e
1 2
ˆ
a ( t ,T ) b ( t ,T ) ( T  t ) 2  ( t ,T )
e
e
, X ),
, X ),
where:
CˆBSBS ,  BS (t , T , S t , X )  S t e (  BS  r (t ,T ))(T t ) N (d  )  Xe  r (t ,T )(T t ) N (d  ),
PˆBSBS ,  BS (t , T , St , X )  Xe  r (t ,T )(T t ) N (d  )  S t e (  BS  r (t ,T ))(T t ) N (d  ),
with:
1
ln( S t / X )  (  BS   2 BS )(T  t )
2
d 
,
 BS (T  t
N ( x) 
1
2

x
e y

2
/2
dy.
20
European Contingent Claims on
Power in the Presence of Spikes
Notation
Denote by
ˆ )  E (t , T , g )( 
ˆ
E (t , T , g )  Et (t , T , g )( 
t
t t , t )
the value of the European contingent claim on power with
inception time t, expiration time T, and payoff
ˆ )  g ( 
ˆ ,  ).
g  g T (
T
T
T
T
The payoff g can explicitly depend, in addition to the power
price at time T, on the state, spike or inter-spike state, of
the power price and the magnitude of the related spike.
If g depends only on the power price at time T we have
ˆ )  g ( 
ˆ ).
g T (
T
T
T
21
European Contingent Claims on
Power in the Presence of Spikes
General Case
The value E(t,T,g) can be found as the discounted riskneutral expected value of the payoff g
T
ˆ )  e t
Et (t , T , g )( 
t
 r ( ) d

 
0

1
ˆ ,
ˆ ) (t , T ,  ,  ) g (
ˆ )d d
ˆ
P(t , T , 
t
T
t
T
T
T
T
T

ˆ )d ,
   (t , T , t , T ) Eˆ (t , T , g T )( 
t
T
1
where
ˆ ,
ˆ )(t, T ,  ,  )
P(t, T , 
t
T
t
T
is the the transition probability density function for t
represented as a Markov process.
22
European Contingent Claims on
Power in the Presence of Spikes
The Case When (t,) is Time-Independent
The value E(t,T,g) is given by
ˆ )
 Psss (T , t ) Eˆ (t , T , g  )( 
t
t

 P r (T , t )   ( ) Eˆ (t , T , g )( 
ˆ ) d  if   1
ss
T

t
T
t
1
T

ˆ )  
ˆ )
Et (t , T , g )( 
Prs (T , t ) Eˆ (t , T , g T 1 )( 
t
t


ˆ ) d 
 Psr (T , t ) 1  (T ) Eˆ (t , T , g T )( 
t
T
if t  1

ˆ )
 Prr (T , t ) Eˆ (t , T , g T 1 )( 
t
23
European Contingent Claims on
Power in the Presence of Spikes
The Case of Spikes with Constant Magnitude
Consider a special case of spikes with constant magnitude
 > 1, that is, when () is the delta function (- `).
The value E(t,T,g) is given by
ˆ )  P (T , t ) Eˆ (t , T , g )( 
ˆ )  P (T , t ) Eˆ (t , T , g )( 
ˆ )
Et   (t , T , g )( 
t
ss

t
rs
1
t
ˆ )  P (T , t ) Eˆ (t , T , g )( 
ˆ )  P (T , t ) Eˆ (t , T , g )( 
ˆ )
Et 1 (t , T , g )( 
t
sr

t
rr
1
t
24
European Contingent Claims on
Power in the Presence of Spikes
Linear Evolution Equation for European Contingent
Claims on Power with Spikes
It can be shown (Kholodnyi 2000) that the value E(t,T,g) of a
European contingent claim on power with spikes is the
solution of the following linear evolution equation
d
v  Lˆ (t )v  (t )v  r (t )v  0,
dt
v(T )  g
t  T,
where Lˆ (t ) and (t ) are the generators of ̂t and
Markov processes.
 t as
25
European Contingent Claims on
Power in the Presence of Spikes
Linear Evolution Equation for European Contingent
Claims on Power with Spikes
In a practically important special case when ̂t is a
geometric mean-reverting process the generator Lˆ (t ) is
given by
2
1


2
2
ˆ
ˆ
ˆ
ˆ
L(t )   (t )
  (t )(  (t )  ln  )
.
2
ˆ
ˆ
2


The generator (t ) is a linear integral operator with the
kernel:
'
'

L
(
t
)

(



)

L
(
t
)

(
1


if t  1
'
ss
t
t
rs
t)
(t , t , t )  
'
'

(
t
,

)
L
(
t
)

L
(
t
)

(
1


if t  1
t
sr
rr
t)

26
European Contingent Claims on
Power in the Presence of Spikes
Linear Evolution Equation for European Contingent
Claims on Power with Spikes
In the special case of spikes with constant magnitude the
generator (t) can be represented as the 22 matrix L*(t)
transposed to the generator L(t) of the Markov process Mt.
In turn, v and g can be represented as two-dimensional
vector functions
 Et   (t , T , g ) 

v(t )  

E
(
t
,
T
,
g
)


1
 t

and
 g T   
.
g  

g


1
 T 
Note that (t) represented as L*(t) can also be expressed in
terms of the Pauli matrices. This gives rise to an analogy
between the linear evolution equation for E(t,T,g) and27 the
Schrodinger equation for a nonrelativistic spin 1/2 particle.
Why Prices of European Claims
On Power Do Not Spike
Ergodic Transition Probabilities for Mt
Assume that the spikes have constant magnitude  and the
underlying two-state Markov process Mt is timehomogeneous.
The transition probabilities for Mt can be represented as
follows:
Pss(T,t) = s + O(e-(T - t)a),
Psr(T,t) = s + O(e-(T - t)a),
Prs(T,t) = r + O(e-(T - t)a),
Prr(T,t) = r + O(e-(T - t)a),
where:
s = b/(a + b)
and r = a/(a + b)
28
are the ergodic transition probabilities.
Why Prices of European Claims
On Power Do Not Spike
Values of European Contingent Claims on Power Far
From Expiration
The values Et=(t,T,g) and Et=1(t,T,g) of European
contingent claims on power coincide up to the terms of
order O(e-(T - t)a) and hence can be combined into a single
expression as follows (Kholodnyi 2000):
ˆ )   Eˆ (t, T , g )(
ˆ )   Eˆ (t, T , g )(
ˆ )  O(e (T t ) a ),
E(t, T , g )(
t
s

t
r
1
t
When T - t >> t s  1 / a , Et=(t,T,g) and Et=1(t,T,g) differ
only by an exponentially small term.
As a result, prices of European contingent claims on power
29
do not exhibit spikes while the power spot prices do.
Why Prices of European Claims
On Power Do Not Spike
Values of European Contingent Claims on Power Far
From Expiration
For example, (Kholodnyi 2000) the values of European call
and put options with inception time t, expiration time T, and
strike X are given by:
ˆ , X )   Cˆ (t , T , 
ˆ , 1 X )   Cˆ (t , T , 
ˆ , X )  O(e (T t ) a ),
C (t , T , 
t
s
t
r
t
ˆ , X )   Pˆ (t , T , 
ˆ , 1 X )   Pˆ (t , T , 
ˆ , X )  O(e (T t ) a ).
P(t , T , 
t
s
t
r
t
30
Why Prices of European Claims
On Power Do Not Spike
Example: Geometric Mean-Reverting Inter-Spike
Process
It can be shown (Kholodnyi 2000) that the value E(t,T,g) of a
European options with inception time t , expiration time T,
and payoff g is given by
ˆ )   Eˆ MR (t , T , g )( 
ˆ )
E (t , T , g )( 
t
s

t
MR
 (T  t ) a
ˆ
ˆ
 E (t , T , g )(  )  O(e
),
where:
r
1
ˆ )  Eˆ BS (t , T , g )( 
ˆ
Eˆ MR (t , T , g )( 
t
t
ˆ ( t ,T ),0
t
1 2
(
T

t
)
ˆ ( t ,T )
a ( t ,T ) b ( t ,T )
2
e
e
31
).
Why Prices of European Claims
On Power Do Not Spike
Example: Geometric Mean-Reverting Inter-Spike
Process
For example, (Kholodnyi 2000) the values of European call
and put options with inception time t , expiration time T, and
strike X are given by
ˆ , X )   Cˆ MR (t , T , 
ˆ , 1 X )   Cˆ MR (t , T , 
ˆ , X )  O(e  (T t ) a ),
C (t , T , 
t
s
t
r
t
ˆ , X )   Pˆ MR (t , T , 
ˆ , 1 X )   Pˆ MR (t , T , 
ˆ , X )  O(e (T t ) a ),
P(t , T , 
t
s
t
r
t
where
ˆ , X )  Cˆ BS (t , T , 
ˆ
Cˆ MR (t , T , 
t
t
ˆ ( t ,T ),0
ˆ , X )  Pˆ BS (t , T , 
ˆ
Pˆ MR (t , T , 
ˆ
t
t
 ( t ,T ),0
1 2
(
T

t
)
ˆ ( t ,T )
a ( t , T ) b ( t ,T )
2
e
e
1 2
ˆ
a ( t ,T ) b ( t ,T ) ( T  t ) 2  ( t ,T )
e
e
, X ),
, X ).
32
Why Prices of European Claims
On Power Do Not Spike
Short-Lived Spikes
Consider the case of short-lived spikes, that is
t s  tr .
Then for the ergodic transition probabilities we have
s = tch + o(tch)
where
and r = 1 - tch + o(tch),
Eˆ (t , T , g )
tch  b / a  t s / tr .
In turn, the value E(t,T,g) can be expressed as a correction
to the value Ê(t,T,g):
E(t, T , g )  Eˆ (t, T , g1 )  tch ( Eˆ (t, T , g  )  Eˆ (t, T , g1 ))  o(tch ).
33
Why Prices of European Claims
On Power Do Not Spike
Example: Geometric Mean-Reverting Inter-Spike
Process
It can be shown (Kholodnyi 2000) that the values of
European call and put options with strike X are given by
ˆ , X )  Cˆ MR (t , T , 
ˆ ,X)
C (t , T , 
t
t
ˆ , 1 X )  Cˆ MR (t , T , 
ˆ , X ))  o(t ),
t ch (Cˆ MR (t , T , 
t
t
ch
ˆ , X )  Pˆ MR (t , T , 
ˆ ,X)
P (t , T , 
t
t
ˆ , 1 X )  Pˆ MR (t , T , 
ˆ , X ))  o(t ),
t ch (Pˆ MR (t , T , 
t
t
ch
where
ˆ , X )  Cˆ BS
ˆ
Cˆ MR (t , T , 
(
t
,
T
,

ˆ
t
t
 ( t ,T ),0
Pˆ
MR
ˆ , X )  Pˆ
(t , T , 
t
BS
ˆ ( t ,T ),0
ˆ
(t , T , 
t
1 2
ˆ
a ( t , T ) b ( t ,T ) ( T  t ) 2  ( t , T )
e
e
1 2
ˆ
a ( t ,T ) b ( t ,T ) ( T  t ) 2  ( t ,T )
e
e
, X ),
, X ).
34
Power Forward Prices for Power
Spot Prices Without of Spikes
Power Forward Prices as Risk-Neutral Expected Power
Spot Prices
Denote by
ˆ ),
Fˆ (t , T )  Fˆ (t , T )(
t
the power forward price at time t for the forward contract
with maturity time T.
Power forward price Fˆ (t , T ) can be found as the riskneutral expected value of the power spot prices ̂T at time
T:

ˆ )  P(t , T ,
ˆ ,
ˆ )
ˆ d
ˆ .
Fˆ (t , T )( 
t
t
T
T
T

0
35
Power Forward Prices for Power
Spot Prices Without of Spikes
Example: Geometric Mean-Reverting Inter-Spike
Process
It can be shown (Kholodnyi 1995) that power forward prices
are given by the following analytical expression:
ˆ )e
Fˆ (t , T )( 
t
where:
1
(T t ) ˆ 2 ( t ,T )
b ( t ,T )
2
1
ˆ (t , T ) 
T t
e

T
t
ˆ a (t ,T ) ,

t
T
 2 ( )e 
 2  ( ') d '
d ,
T
a (t , T )  e t
  ( ) d
b(t , T )  
T
t
,
1  2 ( )   ( ') d '
 ( )(  ( ) 
)e
d .
2  ( )
T
36
Power Forward Prices for Power
Spot Prices Without of Spikes
Example: Geometric Brownian Motion (GBM) for Power
Forward Prices
The risk-neutral dynamics of Fˆ (t , T ) is described by a
geometric Brownian motion:
dFˆ (t , T )  Fˆ (t , T ) Fˆ (t )dW ,
where:
T
 Fˆ (t )   (t )e
  ( ) d
t
.
37
Power Forward Prices for Power
Spot Prices With Spikes
General Case
Denote by
ˆ )  F (t , T )( 
ˆ
F (t , T )  Ft (t , T )( 
t
t t , t )
the power forward price at time t for the forward contract
with maturity time T.
Power forward price F(t,T) can be found as the risk-neutral
expected value of the power spot prices T at time T:
 
ˆ
ˆ ,
ˆ )(t , T ,  ,  )( 
ˆ
ˆ
Ft (t , T )( t )    P(t , T , 
t
T
t
T
T T )dT dT
0
1
ˆ ),
 t (t , T ) Fˆ (t , T )( 
t
where t (t , T ) is the risk-neutral average magnitudes of

spikes
38
 (t , T )   (t , T , t , T )T dT .
t
1
Power Forward Prices for Power
Spot Prices With Spikes
The Case When (t,) is Time-Independent
The risk neutral average magnitude of spikes is given by
Psss (T , t )t  Pssr (T , t )  Prs (T , t ) if t  1
t (t , T )  
if t  1
Psr (T , t )  Prr (T , t )
where  is the risk-neutral conditional average magnitude
of spikes given by

   (d ' ) '.
1
For example, if () is corresponds to a scaling probability
distribution, that is, () =  -1- , then
 

 1
,   1.
39
Power Forward Prices for Power
Spot Prices With Spikes
The Case of Spikes with Constant Magnitude
Consider a special case of spikes with constant magnitude
 > 1, that is, when () is the delta function (- `).
The risk neutral average magnitude of spikes is given by
Pss (T , t )  Prs (T , t ) if t  
t (t , T )  
Psr (T , t )  Prr (T , t ) if t  1
40
Why Power Forward Prices
Do Not Spike
Ergodic Transition Probabilities for Mt
Assume again that the spikes have constant magnitude 
and the underlying two-state Markov process Mt is timehomogeneous.
The transition probabilities for Mt can be represented as
follows:
Pss(T,t) = s + O(e-(T - t)a),
Psr(T,t) = s + O(e-(T - t)a),
Prs(T,t) = r + O(e-(T - t)a),
Prr(T,t) = r + O(e-(T - t)a),
where:
s = b/(a + b)
and r = a/(a + b)
41
are the ergodic transition probabilities.
Why Power Forward Prices
Do Not Spike
Ergodic Average Magnitude of Spikes
The risk-neutral average magnitudes of spikes t   (t , T )
and t 1 (t , T ) coincide up to the terms of order O(e-(T - t)a).
Therefore, t   (t , T ) and t 1 (t , T ) can be combined into a
single expression as follows:
 (t , T )  erg  O(e(T t ) a ),
where erg is the risk-neutral ergodic average magnitude
of spikes given by
erg   s    r .
42
Why Power Forward Prices
Do Not Spike
Power Forward Prices far From Maturity
The power forward prices Ft=(t,T) and Ft=1(t,T) coincide
up to the terms of order O(e-(T - t)a).
Therefore, Ft=(t,T) and Ft=1(t,T) can be combined into a
single expression as follows:
F (t , T )  erg Fˆ (t , T )  O(e  (T t ) a ).
When T - t >> t s  1 / a , Ft=(t,T) and Ft=1(t,T) differ only
by an exponentially small term.
As a result, power forward prices do not exhibit spikes
while the power spot prices do.
43
Why Power Forward Prices
Do Not Spike
Short-Lived Spikes
Consider the case of short-lived spikes, that is
t s  tr .
Then for the ergodic transition probabilities we have
s = tch + o(tch)
where
and r = 1 - tch + o(tch),
tch  b / a  t s / tr .
For the average magnitude of spikes we have
 (t , T )  1  tch (  1)  o(tch ).
In turn, F(t,T) can be expressed as a correction to Fˆ (t , T )
F (t , T )  Fˆ (t , T )  tch ( 1) Fˆ (t , T )  o(tch ).
44
Why Power Forward Prices
Do Not Spike
Example: GBM for Power Forward Prices
Assume that the power forward prices Fˆ (t , T ) follow a
geometric Brownian motion.
– this is, for example, the case when the power spot prices
̂t follow a geometric mean-reverting process.
Then power forward prices F(t,T) far from maturity also
follow the same geometric Brownian motion.
This, for example, can be used for:
• the estimation of the volatility for the geometric Brownian
motion for Fˆ (t , T ) ,
• the estimation of the volatility and the mean-reversion rate
for the geometric mean-reverting process for ̂t , and
45
• dynamic hedging of derivatives on forwards on power.
European Contingent Claims on
Forwards on Power with Spikes
Geometric Mean-Reverting Inter-Spike Process and
Spikes with Constant Magnitude
It can be shown (Kholodnyi 2000) that the value of a
European contingent claim (on forwards on power for
power with spikes) with inception time t, expiration time T,
and payoff g is given by:
E (t , T , g )( F )   s Eˆ BS
ˆ ( t ,T ),0 (t , T , g  )( F / erg ) 
 (T  t ) a
 s Eˆ BS
(
t
,
T
,
g
)(
F
/

)

O
(
e
).
ˆ ( t ,T ),0
1
erg
46
European Contingent Claims on
Forwards on Power with Spikes
Geometric Mean-Reverting Inter-Spike Process and
Spikes with Constant Magnitude
For example, (Kholodnyi 2000) the values of European call
and put options (on forwards on power for power with
spikes) with inception time t, expiration time T, and strike X
are given by:
1
C (t , T , F , X )   s ( / erg )Cˆ BS
(
t
,
T
,
F
,
(

/

)
X)
ˆ ( t ,T ),0
erg
 (T  t ) a
 r (1 / erg )Cˆ BS
(
t
,
T
,
F
,

X
)

O
(
e
),
ˆ ( t ,T ),0
erg
P(t , T , F , X )   s ( / erg ) PˆˆBS(t ,T ),0 (t , T , F , ( / erg ) 1 X ) 
 r (1 / erg ) PˆˆBS(t ,T ),0 (t , T , F , erg X )  O(e  (T t ) a ).
47
European Contingent Claims on
Forwards on Power with Spikes
Geometric Mean-Reverting Inter-Spike Process and
Short-Lived Spikes with Constant Magnitude
It can be shown (Kholodnyi 2000) that the value of a
European contingent claim (on forwards on power for
power with spikes) with inception time t, expiration time T,
and payoff g can be represented as the following
correction:
E (t , T , g )( F )  Eˆ BS
ˆ ( t ,T ),0 (t , T , g1 )( F ) 

t ch Eˆ BS
ˆ ( t ,T ),0 (t , T , g   g1 )( F ) 

(  1) FBS
ˆ ( t ,T ),0 (t , T , g1 )( F )  o(t ch ).
48
European Contingent Claims on
Forwards on Power with Spikes
Geometric Mean-Reverting Inter-Spike Process and
Short-Lived Spikes with Constant Magnitude
For example, (Kholodnyi 2000) the values of European call
and put options (on forwards on power for power with
spikes) with inception time t, expiration time T, and strike X
can be represented as the following corrections:
C (t , T , F , X )  Cˆ BS
ˆ ( t ,T ),0 (t , T , F , X ) 

1
ˆ BS
t ch Cˆ BS
ˆ ( t ,T ),0 (t , T , F ,  X )  Cˆ ( t ,T ),0 (t , T , F , X ) 

(  1) FBS
c ,ˆ ( t ,T ),0 (t , T , F , X )  o(t ch ),
P(t , T , F , X )  PˆˆBS( t ,T ),0 (t , T , F , X ) 

t ch PˆˆBS(t ,T ),0 (t , T , F , 1 X )  PˆˆBS(t ,T ),0 (t , T , F , X ) 

(  1) FBSp ,ˆ (t ,T ),0 (t , T , F , X )  o(t ch ).
49
Extensions of the Model
• Both positive and negative spikes as well as spikes of
more complex shapes can be considered
• European contingent claims on power with spikes and
another commodity that does not exhibit spikes can also be
valued. Those include fuel and weather sensitive derivatives
such as spark spread options and full requirements
contracts
• European options on power at two distinct points on the
grid with spikes in both power prices can also be valued.
Those include transmission options
• Contingent claims of a general type such as universal
contingent claims on power with spikes can be valued with
the help of the semilinear evolution equation for universal
contingent claims (Kholodnyi, 1995). Those include
50
Bermudan and American options.
Acknowledgements
I thank my friends and former colleagues from Reliant
Resources, TXU Energy Trading, and Integrated Energy
Services for their attention to this work.
I thank my friends and colleges from the College of Basic
and Applied Sciences, in general, and the Department of
Mathematical Sciences and the Center for Quantitative Risk
Analysis, in particular, of Middle Tennessee State University
for their warm welcome and attention to this presentation.
I thank the organizers of the Energy Finance and Credit
Summit 2004 for their kind invitation and support of this
presentation.
I thank my wife Larisa and my son Nikita for their love,
patience and care.
51
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