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Return to Risk Limited website: www.RiskLimited.com. Valuation and Hedging of Power-Sensitive Contingent Claims for Power with Spikes: a Non-Markovian Approach. Valery A. Kholodnyi February 25, 2004 Houston, Texas. Introduction. - PowerPoint PPT Presentation
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1
Valuation and Hedging of Power-Sensitive Contingent Claims for Power with Spikes:
a Non-Markovian Approach
Return to Risk Limited website: www.RiskLimited.com
Valery A. Kholodnyi
February 25, 2004Houston, Texas
2
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.
3
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
4
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 of around $30 per MWh
5
The Non-Markovian Process for Power Spot Prices with SpikesMain 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
6
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
• t>0 is the power spot price at time t ,
• is the multiplicative magnitude of spikes at time t ,
• is the inter-spike power spot price at time t.
Assume that the spike process and inter-spike process are independent Markov processes.
,ˆttt
1t
0ˆ t
tt̂
7
Underlying Two-State Markov Process
Denote by Mt a two-state Markov process with continuous time t 0.
Denote the 22 transition matrix for the two-state Markov process Mt by
• 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.
The Non-Markovian Process for Power Spot Prices with Spikes
),(),(),(),(
),(tTPtTPtTPtTP
tTPrrrs
srss
8
Generators of the Underlying Two-State Markov Process
The family of 22 matrices L = {L(t) : t 0} defined by
is said to generate the two-state Markov process Mt, and the 22 matrix
is called a generator.
In terms of the generators, P(T,t) is given by
( ) ( , ) ,T tdL t P T tdT
.)(
),(
Tt
dLetTP
The Non-Markovian Process for Power Spot Prices with Spikes
)()()()(
)(tLtLtLtL
tLrrrs
srss
9
Decompositions of the Transition Probabilities of the Underlying Two-State Markov Process
It can be shown that
Moreover
where
The Non-Markovian Process for Power Spot Prices with Spikes
.)(),(),(
,)(),(),(
')'(
')'()(
deLtPtTP
deLtPetTPT
ss
Tss
Tt ss
dLsr
T
t rrsr
dLsr
T
t rsdL
ss
),,(),(),( tTPtTPtTP rss
sssss
deLtPtTP
etTPT
ss
Tt ss
dLsr
T
t rsrss
dLsss
')'(
)(
)(),(),(
),(
10
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
and
( )( ) ( )( )
( )( ) ( )( )( )
T t a b T t a b
T t a b T t a b
b ae b bea b a b
a ae a bea b a b
P T t
a bLa b
The Non-Markovian Process for Power Spot Prices with Spikes
11
Construction of the Spike Process
t
Mt
Spike State Regular State
(t,)
Time
Time
1
st rt
The Non-Markovian Process for Power Spot Prices with Spikes
12
Formal Definition of the Spike Process
The transition probability density function for the spike process t as a Markov process is given by
where (x) is the Dirac delta function.
The Non-Markovian Process for Power Spot Prices with Spikes
1 if)1(),(
)(),(),(
1 if
)1(),(
)(),(),(
)(
),,,(
')'(
')'(
)(
t
Trr
T
t
dL
srrrT
t
Trs
T
t
dL
srrsT
Tt
dL
Tt
tTP
deLtP
tTP
deLtP
e
TtTss
Tss
T
t ss
13
Inter-Spike Process
For example, can be a diffusion process defined by
where: is the drift, is the volatility, and W t is the Wiener process.
In the practically important special case of a geometric-mean reverting process we have
where: is the mean-reversion rate, is the equilibrium mean, and is the volatility.
The Non-Markovian Process for Power Spot Prices with Spikes
t̂
,),ˆ(),ˆ(ˆtttt dWtdttd
,ˆ)(ˆ)ˆln)()((ˆttttt dWtdtttd
),ˆ( tt 0),ˆ( tt
0)( t )(t0)( t
14
The Expected Time for t to be in the Spike and Inter-Spike States
The expected time for t to be in the spike state that starts at time t is:
Similarly, the expected time for t to be in the inter-spike state that starts at time t is:
In the special case of a time-homogeneous two-state Markov process Mt:
st
.)()(')'(
dbett
t
db
rt
rt
.)()(')'(
daett
t
da
st
./1/1 btat rs and
The Non-Markovian Process for Power Spot Prices with Spikes
15
Interpretation of the Spike State of t as Spikes in Power PricesIf the expected time 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 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:
and
In this case is the expected lifetime of a spike and is the expected lifetime between two consecutive spikes.
st
t̂
t̂
1),ˆ(2 stt .1),ˆ( stt
st rt
The Non-Markovian Process for Power Spot Prices with Spikes
16
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
• Similarly, the expected life-time between two consecutive spikes is given by
• 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 a particular interest in practice
ats /1
./1 btr
The Non-Markovian Process for Power Spot Prices with Spikes
17
The Non-Markovian Process for Power Spot Prices with SpikesThe Non-Markovian Process t as a Markov Process with the Extended State SpaceThe state of the power market at any time t can be fully characterized by a pair of the values of the processes , and 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 and at time t.
Equivalently, the non-Markovian process t can be represented as a Markov process with the extended state space that at any time t consists of all possible pairs with and .
t̂t
t t̂
)ˆ,( tt 1t 0ˆ t
18
Valuing European Contingent Claims on Power as the Discounted Risk-Neutral expected value of its payoff
Denote by
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:
where is the risk-neutral transition probability density function.
)ˆ,ˆ,,( TtTtP
European Contingent Claims onPower in the Absence of Spikes
,ˆ)ˆ()ˆ,ˆ,,( )ˆ)(,,(ˆ0
)(
TTTt
dr
t dgTtPegTtE
T
t
),ˆ)(,,(ˆ),,(ˆtgTtEgTtE
19
Example: Geometric Mean-Reverting Process
It can be shown (Kholodnyi 1995) that
where:
European Contingent Claims onPower in the Absence of Spikes
.))()(
21)()((),(
,),(
,)(1),(ˆ
,)(1),(
')'(2
)(
')'(22
T
t
d
d
dT
t
T
t
deTtb
eTta
detT
Tt
drtT
Ttr
T
T
t
T
).ˆ)(,,(ˆ
ˆˆ
)ˆ()(2),(ˆ
)ˆ)(,,(ˆ
),(ˆ21)(),(),(
0),,(ˆ
0
))(,(ˆ)ˆln),(ˆln),((
21))(,(
2
2
2
TttTTtbTtat
BSTt
T
TT
tTTtTtbTtatTTtr
tMR
eegTtE
dge
tTTtegTtE
Tt
20
Example: Geometric Mean-Reverting Process
For example (Kholodnyi 1995):
where:
with:
European Contingent Claims onPower in the Absence of Spikes
),,ˆ,,(ˆ),ˆ,,(ˆ
),,ˆ,,(ˆ),ˆ,,(ˆ
),(ˆ21)(),(),(
0),,(ˆ
),(ˆ21)(),(),(
0),,(ˆ
2
2
XeeTtPXTtP
XeeTtCXTtC
TttTTtbTtat
BSTtt
MR
TttTTtbTtat
BSTtt
MR
),()(),,,(ˆ),()(),,,(ˆ
)))(,(())(,(,
))(,()))(,((,
dNeSdNXeXSTtP
dNXedNeSXSTtCtTTtr
ttTTtr
tBS
tTTtrtTTtrtt
BS
BS
BSBS
BS
BSBS
.21)(
,(
))(21()/ln(
2/
2
2
dyexN
tT
tTXSd
x y
BS
BSBSt
21
Notation
Denote by
the value of the European contingent claim on power with inception time t, expiration time T, and payoff
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
European Contingent Claims onPower in the Presence of Spikes
),ˆ)(,,()ˆ)(,,(),,( tttt gTtEgTtEgTtEt
).,ˆ()ˆ( TTTT gggT
).ˆ()ˆ( TTT ggT
22
General Case
The value E(t,T,g) can be found as the discounted risk-neutral expected value of the payoff g
where
is the the transition probability density function for t represented as a Markov process.
European Contingent Claims onPower in the Presence of Spikes
,)ˆ)(,,(ˆ),,,(
ˆ)ˆ(),,,()ˆ,ˆ,,()ˆ)(,,(
1
0 1
)(
TtTt
TTTTtTt
dr
t
dgTtETt
ddgTtTtPegTtE
T
T
T
t
t
),,,()ˆ,ˆ,,( TtTt TtTtP
23
The Case When (t,) is Time-Independent
The value E(t,T,g) is given by
European Contingent Claims onPower in the Presence of Spikes
1)ˆ)(,,(ˆ),(
)ˆ)(,,(ˆ)(),(
1
)ˆ)(,,(ˆ),(
)ˆ)(,,(ˆ)(),(
)ˆ)(,,(ˆ),(
)ˆ)(,,(
1
1
1
1
t
trr
TtTsr
t
trs
TtTrss
tsss
t
T
T
T
T
t
t
gTtEtTP
dgTtEtTP
gTtEtTP
dgTtEtTP
gTtEtTP
gTtE
if
if
24
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
European Contingent Claims onPower in the Presence of Spikes
)ˆ)(,,(ˆ),()ˆ)(,,(ˆ),()ˆ)(,,(
)ˆ)(,,(ˆ),()ˆ)(,,(ˆ),()ˆ)(,,(
11
1
trrtsrt
trstsst
gTtEtTPgTtEtTPgTtE
gTtEtTPgTtEtTPgTtE
t
t
25
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
where and are the generators of and as Markov processes.
European Contingent Claims onPower in the Presence of Spikes
gTv
TtvtrvtvtLvdtd
)(
,,0)()()(ˆ
)(ˆ tL )(tt̂ t
26
Linear Evolution Equation for European Contingent Claims on Power with Spikes
In a practically important special case when is a geometric mean-reverting process the generator is given by
The generator is a linear integral operator with the kernel:
European Contingent Claims onPower in the Presence of Spikes
.ˆˆ)ˆln)()((ˆ
ˆ)(21)(ˆ
2
222
ttttL
)(ˆ tLt̂
1)1()()(),(1)1()()()(
),,( ''
'''
ttrrsrt
ttrsttsstt tLtLt
tLtLt
if if
)(t
27
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 22 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
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) and the Schrodinger equation for a nonrelativistic spin 1/2 particle.
European Contingent Claims onPower in the Presence of Spikes
. and
11 ),,(),,(
)(T
T
t
t
gg
ggTtEgTtE
tv
28
Ergodic Transition Probabilities for Mt
Assume that the spikes have constant magnitude and the underlying two-state Markov process Mt is time-homogeneous.
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)
are the ergodic transition probabilities.
Why Prices of European ClaimsOn Power Do Not Spike
29
Values of European Contingent Claims on Power Far From Expiration
The values Et=(t,T,g) and Et=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):
When T - t >> , Et=(t,T,g) and Et=1(t,T,g) differ
only by an exponentially small term.
As a result, prices of European contingent claims on power do not exhibit spikes while the power spot prices do.
ats /1
),()ˆ)(,,(ˆ)ˆ)(,,(ˆ)ˆ)(,,( )(1
atTtrtst eOgTtEgTtEgTtE
Why Prices of European ClaimsOn Power Do Not Spike
30
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:
Why Prices of European ClaimsOn Power Do Not Spike
).(),ˆ,,(ˆ),ˆ,,(ˆ),ˆ,,(
),(),ˆ,,(ˆ),ˆ,,(ˆ),ˆ,,()(1
)(1
atTtrtst
atTtrtst
eOXTtPXTtPXTtP
eOXTtCXTtCXTtC
31
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
where:
Why Prices of European ClaimsOn Power Do Not Spike
),()ˆ)(,,(ˆ)ˆ)(,,(ˆ)ˆ)(,,(
)(1
atTt
MRr
tMR
st
eOgTtE
gTtEgTtE
).ˆ)(,,(ˆ)ˆ)(,,(ˆ ),(ˆ21)(),(),(
0),,(ˆ
2 TttTTtbTtat
BSTtt
MR eegTtEgTtE
32
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
where
Why Prices of European ClaimsOn Power Do Not Spike
),(),ˆ,,(ˆ),ˆ,,(ˆ),ˆ,,(
),(),ˆ,,(ˆ),ˆ,,(ˆ),ˆ,,()(1
)(1
atTt
MRrt
MRst
atTt
MRrt
MRst
eOXTtPXTtPXTtP
eOXTtCXTtCXTtC
).,ˆ,,(ˆ),ˆ,,(ˆ
),,ˆ,,(ˆ),ˆ,,(ˆ
),(ˆ21)(),(),(
0),,(ˆ
),(ˆ21)(),(),(
0),,(ˆ
2
2
XeeTtPXTtP
XeeTtCXTtC
TttTTtbTtat
BSTtt
MR
TttTTtbTtat
BSTtt
MR
33
Short-Lived Spikes
Consider the case of short-lived spikes, that is .
Then for the ergodic transition probabilities we have
s = tch + o(tch) and r = 1 - tch + o(tch),
where
In turn, the value E(t,T,g) can be expressed as a correction to the value Ê(t,T,g):
rs tt
.// rsch ttabt
).()),,(ˆ),,(ˆ(),,(ˆ),,( 11 chch togTtEgTtEtgTtEgTtE
Why Prices of European ClaimsOn Power Do Not Spike
),,(ˆ gTtE ),,(ˆ gTtE ),,(ˆ gTtE
34
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
where
Why Prices of European ClaimsOn Power Do Not Spike
),()),ˆ,,(ˆ),ˆ,,(ˆ(
),ˆ,,(ˆ),ˆ,,(
),()),ˆ,,(ˆ),ˆ,,(ˆ(
),ˆ,,(ˆ),ˆ,,(
1
1
chtMR
tMR
ch
tMR
t
chtMR
tMR
ch
tMR
t
toXTtPXTtPt
XTtPXTtP
toXTtCXTtCt
XTtCXTtC
).,ˆ,,(ˆ),ˆ,,(ˆ
),,ˆ,,(ˆ),ˆ,,(ˆ
),(ˆ21)(),(),(
0),,(ˆ
),(ˆ21)(),(),(
0),,(ˆ
2
2
XeeTtPXTtP
XeeTtCXTtC
TttTTtbTtat
BSTtt
MR
TttTTtbTtat
BSTtt
MR
35
Power Forward Prices as Risk-Neutral Expected Power Spot Prices
Denote by
the power forward price at time t for the forward contract with maturity time T.
Power forward price can be found as the risk-neutral expected value of the power spot prices at time T:
),ˆ)(,(ˆ),(ˆtTtFTtF
T̂),(ˆ TtF
.ˆˆ)ˆ,ˆ,,()ˆ)(,(ˆ0
TTTtt dTtPTtF
Power Forward Prices for PowerSpot Prices Without of Spikes
36
Example: Geometric Mean-Reverting Inter-Spike Process
It can be shown (Kholodnyi 1995) that power forward prices are given by the following analytical expression:
where:
,ˆ)ˆ)(,(ˆ ),(),(),(ˆ21)( 2
Ttat
TtbTttT
t eeTtF
.))()(
21)()((),(
,),(
,)(1),(ˆ
')'(2
)(
')'(22
T
t
d
d
dT
t
deTtb
eTta
detT
Tt
T
T
t
T
Power Forward Prices for PowerSpot Prices Without of Spikes
37
Example: Geometric Brownian Motion (GBM) for Power Forward Prices
The risk-neutral dynamics of is described by a geometric Brownian motion:
where:
Power Forward Prices for PowerSpot Prices Without of Spikes
.)()()(
ˆ
T
td
F ett
,)(),(ˆ),(ˆˆ dWtTtFTtFd F
),(ˆ TtF
38
General CaseDenote by
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:
where is the risk-neutral average magnitudes of spikes
Power Forward Prices for PowerSpot Prices With Spikes
),ˆ)(,()ˆ)(,(),( tttt TtFTtFTtFt
),ˆ)(,(ˆ),(
ˆ)ˆ)(,,,()ˆ,ˆ,,()ˆ)(,(0 1
t
TTTTTtTtt
TtFTt
ddTtTtPTtF
t
t
),( Ttt
.),,,(),(1 TTTt dTtTt
t
39
The Case When (t,) is Time-IndependentThe risk neutral average magnitude of spikes is given by
where is the risk-neutral conditional average magnitude of spikes given by
For example, if () is corresponds to a scaling probability distribution, that is, () = -1- , then
Power Forward Prices for PowerSpot Prices With Spikes
1),(),(1),(),(),(
),(trrsr
trsrsst
sss
tTPtTPtTPtTPtTP
Ttt
if
if
.')'(1
d
.1,1
40
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
Power Forward Prices for PowerSpot Prices With Spikes
1),(),(
),(),(),(
trrsr
trsss
tTPtTPtTPtTP
Ttt
if
if
41
Ergodic Transition Probabilities for Mt
Assume again that the spikes have constant magnitude and the underlying two-state Markov process Mt is time-homogeneous.
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)
are the ergodic transition probabilities.
Why Power Forward Prices Do Not Spike
42
Ergodic Average Magnitude of Spikes
The risk-neutral average magnitudes of spikes and coincide up to the terms of order O(e-(T - t)a).
Therefore, and can be combined into a single expression as follows:
where is the risk-neutral ergodic average magnitude of spikes given by
Why Power Forward Prices Do Not Spike
.rserg
erg
),( Ttt
),(1 Ttt
),( Ttt ),(1 Tt
t
),(),( )( atTerg eOTt
43
Power Forward Prices far From Maturity
The power forward prices Ft=(t,T) and Ft=1(t,T) coincide up
to the terms of order O(e-(T - t)a).
Therefore, Ft=(t,T) and Ft=1(t,T) can be combined into a
single expression as follows:
When T - t >> , Ft=(t,T) and Ft=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.
Why Power Forward Prices Do Not Spike
).(),(ˆ),( )( atTerg eOTtFTtF
ats /1
44
Short-Lived Spikes
Consider the case of short-lived spikes, that is .
Then for the ergodic transition probabilities we have
s = tch + o(tch) and r = 1 - tch + o(tch),
where
For the average magnitude of spikes we have
In turn, F(t,T) can be expressed as a correction to
Why Power Forward Prices Do Not Spike
rs tt
.// rsch ttabt
).()1(1),( chch totTt ),(ˆ TtF
).(),(ˆ)1(),(ˆ),( chch toTtFtTtFTtF
45
Example: GBM for Power Forward PricesAssume that the power forward prices follow a geometric Brownian motion.– this is, for example, the case when the power spot prices 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 ,• the estimation of the volatility and the mean-reversion rate for the geometric mean-reverting process for , and• dynamic hedging of derivatives on forwards on power.
),(ˆ TtF
t̂
),(ˆ TtF
t̂
Why Power Forward Prices Do Not Spike
46
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:
).()/)(,,(ˆ)/)(,,(ˆ))(,,(
)(10),,(ˆ
0),,(ˆ
atTerg
BSTts
ergBS
Tts
eOFgTtE
FgTtEFgTtE
47
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:
).(),,,(ˆ)/1(
))/(,,,(ˆ)/(),,,(
),(),,,(ˆ)/1(
))/(,,,(ˆ)/(),,,(
)(0),,(ˆ
10),,(ˆ
)(0),,(ˆ
10),,(ˆ
atTerg
BSTtergr
ergBS
Ttergs
atTerg
BSTtergr
ergBS
Ttergs
eOXFTtP
XFTtPXFTtP
eOXFTtC
XFTtCXFTtC
48
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:
).())(,,()1(
))(,,(ˆ))(,,(ˆ))(,,(
10),,(ˆ
10),,(ˆ
10),,(ˆ
chBS
Tt
BSTtch
BSTt
toFgTtF
FggTtEt
FgTtEFgTtE
49
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:
).(),,,()1(
),,,(ˆ),,,(ˆ),,,(ˆ),,,(
),(),,,()1(
),,,(ˆ),,,(ˆ),,,(ˆ),,,(
0),,(ˆ,
0),,(ˆ1
0),,(ˆ
0),,(ˆ
0),,(ˆ,
0),,(ˆ1
0),,(ˆ
0),,(ˆ
chBS
Ttp
BSTt
BSTtch
BSTt
chBS
Ttc
BSTt
BSTtch
BSTt
toXFTtF
XFTtPXFTtPt
XFTtPXFTtP
toXFTtF
XFTtCXFTtCt
XFTtCXFTtC
50
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 Bermudan and American options.
51
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.
52
References• R. Ethier and G. Dorris, Do not Ignore the Spikes, EPRM, July-August, 1999, 31-33.• L. Clewlow, C. Strickland and V. Kaminski, Jumping the Gaps, EPRM, December, 2000, 26-27.• R. Goldberg and J. Read, Dealing with a Price-Spike World, EPRM, July-August, 2000, 39-41.• C. Ball and W. Torous, On Jumps in Common Stock Prices and Their Impact on Call Option Pricing, Journal of Finance, XL (1), March 1985, 155-173• R. Ethier, Estimating the Volatility of Spot Prices in restructured Electricity Markets and the Implications for Option Values, Cornell University, 1999.• D. Duffie and S. Gray, Volatility in Energy Prices, In Managing Energy Price Risk, Risk Publications, London, UK, 1995.• V. Kholodnyi, Introduction to the Beliefs-Preferences Gauge Symmetry, Elsevier Science, Amsterdam, Holland, 2003, To appear.• V. Kholodnyi and J. Price, Foreign Exchange Option Symmetry, World Scientific, River Edge, New Jersey, 1998.• V.A Kholodnyi and J.F. Price, Foundations of Foreign Exchange Option Symmetry, IES Press, Fairfield, Iowa, 1998.• V. Kholodnyi, Beliefs-Preferences Gauge Symmetry Group and Dynamic Replication of Contingent Claims in a General Market Environment, IES Press, Research Triangle Park, North Carolina, 1998.• V. Kholodnyi, A Non-Markov Method, EPRM, March, 2001, 20-24.• V. Kholodnyi, Analytical Valuation in a Mean-Reverting World, EPRM, August, 2001, 40-45.• V. Kholodnyi, Analytical Valuation of a Full Requirements Contract as a Real Option by the Method of Eigenclaims, In E. I. Ronn, Editor, Real Options and Energy Management, Risk Publications, 2002.• V. Kholodnyi, On the Linearity of Bermudan and American Options in Partial Semimodules, IES Preprint, 1995.• V. Kholodnyi, The Stochastic Process for Power Prices with Spikes and Valuation of European Contingent Claims on Power, TXU Preprint, 2000.• V. Kholodnyi, Modeling Power Forward Prices for Power with Spikes, TXU Preprint, 2001.
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