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Extracting Forward-Looking Market-Implied Risk-Neutral Probability Distributions for Energy Spots from Energy Forwards and Options in the Unified Framework of the Non-Markovian Approach
Valery Kholodnyi Essen, Germany
13.05.2015
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Outline
28-Apr-15 2VTR/STR/Kholodnyi
• Introduction• Methodology
− The Non-Markovian Approach− Modeling Energy Spots− Modeling Energy Forwards− Modeling Energy Swaps− Modeling Energy European Options− Modeling Energy American Options− Extracting Forward-Looking Market-Implied Risk-Neutral Probability Distributions
• Examples of Oil, Gas and Power Markets • Conclusions
Outline
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Introduction
28-Apr-15 3
Introduction
Modern Energy Markets
As the energy markets are becoming deregulated worldwide, the modeling of the dynamics of energy prices is becoming one of the key problems in energy risk management, trading, and physical assets valuation.
The main features of the modern energy markets to be modeled are:
• Positive and negative prices
• Upwards and downwards spikes
• Daily, weekly, annual and meta-annual (business) cyclical patterns
• Linear and non-linear trends
One of the main difficulties in this modeling is to provide for different mechanisms for the reversion of energy spot prices to their long-term mean between spikes and during spikes, that is, for the decay of spikes.
The reason for these different mechanisms is that the dynamics of energy spot prices between spikes and during spikes are due to different fundamental drivers.
VTR/STR/Kholodnyi
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Introduction
28-Apr-15 4
Introduction
Modern Energy Markets
For example:
• Power spot prices in Germany in October 2009 and May 2013 fell to -500.02 €/MWh(off-peak) and to –3.61 €MWh (on-peak), respectively, compared with typical prices of around 50 €/MWh as a result of a relatively high supply versus demand due to the increased share of the installed intermittent renewable generation capacity
• Power spot prices in the US Midwest in June 1998 rose to $7,500/MWh compared with typical prices of around $30/MWh as a result of a relatively high demand versus supply due to unseasonably hot weather, planned and unplanned outages, and transmission constrains
In addition, the on-peak and off-peak power forward curves can not be modeled separately since the off-peak power forward prices can not be, in general, extracted from the corresponding on-peak and base power forward prices by the no-arbitrage argument due to the high asymmetry in the liquidity of the on-peak and base power forwards
VTR/STR/Kholodnyi
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Introduction
28-Apr-15 5
A Unified Modeling Framework
The benefits of a unified framework for modeling energy markets:
• Applicable across the instruments, commodities, regions and time periods
• Modeling of the joint dynamics of two or several different commodities− The same model is used for different commodities with possibly different
numerical values of the model’s parameters− The models for different commodities “talk” to each other
• Comparing the dynamics of different commodities and commodities themselves − The model’s parameters represent a unique “gene code” of a given commodity
• Efficient development, implementation and maintenance of the models− A single model to be developed, implemented and maintained
• Consistent business decision making across the instruments, commodities, the regions and time periods
Introduction
VTR/STR/Kholodnyi
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Introduction
28-Apr-15 6
Introduction
The Non-Markovian Approach
We present and further develop the non-Markovian approach (Kholodnyi, 2000) to modeling energy spot prices with spikes.
In contrast to the other approaches that are based on Markov stochastic processes we model energy spot prices with spikes as a non-Markovian stochastic process that allows for a unified modeling of positive and negative spot prices as well as upward and downward spikes directly as self-reversing jumps.
In this way different mechanisms are, in fact, responsible for the reversion of energy spot prices to their long-term mean between spikes and during spikes, that is, for the decay of spikes.
We show that this approach, in fact, represents a unified modeling framework applicable across the instruments, commodities, regions and time periods.
We use this approach to model energy forwards and options and extract the forward-looking market-implied risk-neutral probability distributions for the energy spot prices with trends, cyclical patterns and spikes from the related energy forward and options prices.
VTR/STR/Kholodnyi
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Introduction
28-Apr-15 7
Why a Non-Markovian Stochastic Process
We argue that employing a Markov stochastic process to model energy spot prices with spikes is ultimately making the same mechanism responsible for the reversion of energy prices to their long-term mean between spikes and during spikes, that is, for the decay of spikes.
Indeed, although a Markov stochastic process can produce a sharp upward/downward price movement as a suitable jump, it can not remember the magnitude of this sharp upward/downward price movement to separately produce a shortly followed sharp downward/upward price movement of approximately the same magnitude so that an upward/downward spike can form.
Introduction
VTR/STR/Kholodnyi
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The Non-Markovian Approach
28-Apr-15 8
The Main Features
• Provides a unified framework that allows for the consistent modeling of energy spots, forwards/swaps, and options on energy spots, forwards/swaps for single and multiple commodity energy markets
• The interpolation and extrapolation of market energy forward curves:− Re-construction of the highest granularity (for example, monthly, weekly, daily,
hourly or ¼-hourly) forward curves− Extension of the market forwards curve beyond their liquidity horizon
• Extracting not only historical but also forward-looking market-implied risk-neutral probability distributions for energy spots, forwards/swaps, and options
• Static and dynamic modeling, that is, the modeling of the energy forward curves for a single and multiple trading days
• Historical and forward-looking market-implied risk-neutral Monte Carlo simulations of energy spots, forwards/swaps, and options
• Can be viewed as a hybrid model that can be represented not only in terms of the stochastic drivers but also in terms of the fundamental drivers
VTR/STR/Kholodnyi
Methodology
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Methodology
Positive and Negative Energy Spots
Let be a mean-reverting process:
where:
• is the mean-reversion rate,
• is the long-term mean,
• is the volatility, and
• Wt is the Wiener process.
0)( >tη
)(ˆ tµ
0)( >tσ
,)()ˆ)(ˆ)((ˆ ttt dWtdtxttxd σµη +−=tx
Motivation
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Methodology
Positive and Negative Energy Spots
Then is a geometric mean-reverting process:
where:
• is the mean-reversion rate,
• is the log long-term mean,
• is the volatility, and
• Wt is the Wiener process.
txt es ˆˆ =
,)()ˆln)(ˆ)((ˆln ttt dWtdtsttsd σµη +−=
0)( >tη
)(ˆ tµ
0)( >tσ
Motivation
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Methodology
Positive and Negative Energy Spots
What if
where is an arbitrary representation function associated with .
)ˆ(ˆˆttt xρ=Ψ
)ˆ(ˆ xtρ tx
Motivation
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Methodology
Positive and Negative Energy Spots
For example (Kholodnyi, 2000), in the practically important special case the representation function is given by
where:
If A = 1, B = 0, α = 1, and β > 0, and hence .
We comment that at .
We also comment that is strictly increasing so that is well defined and
Finally, we comment that parameters A, B, α, and β can be time-dependent.
)ˆ(ˆ)ˆ(ˆ ,,, xx BAt βαρρ =
xxBA BeAex ˆˆ
,,, )ˆ(ˆ βαβαρ −−=
xBA ex ˆ
,,, )ˆ(ˆ =βαρ txt eˆˆ =Ψ
0)ˆ(ˆ ,,, =∗xBA βαρ )/(1)/ln(ˆ βα +∗ = ABx
)2
4ˆˆln(
1)ˆ(ˆ
21
,,, A
ABBA
+Ψ+Ψ=Ψ−
αρ αα
)ˆ(ˆ ,,, xBA βαρ )ˆ(ˆ 1,,, Ψ−βαρ BA
Motivation
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Methodology
Positive and Negative Energy Spots
Representation function for A = 1, B = 0.1, α = 1, and β =1:)ˆ(ˆ ,,, xBA βαρ
Representation Function
-3
-2
-1
0
1
2
3
-4 -3 -2 -1 0 1 2
x
Rep
rese
nta
tion
Fu
nct
ion
Representation Function
A Component
B Component
Motivation
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Methodology
Positive and Negative Energy Spots
Inverse representation function for A = 1, B = 0.1, α = 1, and β =1:)ˆ(ˆ 1,,, Ψ−βαρ BA
Inverse Representation Function
-5
-4
-3
-2
-1
0
1
2
3
-3 -2 -1 0 1 2 3 4 5
Psi
Inve
rse
Rep
rese
nta
tion
Fu
nct
ion
Representation Function
A Component
B Component
Motivation
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Methodology
Positive and Negative Energy Spots
Equivalently
where is an arbitrary representation function associated with .
We comment that
and
)ˆ(ˆttt sρ=Ψ
)ˆ(stρ ts
)ˆ(exp)ˆ(ˆ xx tt ρρ =)ˆ(lnˆ)ˆ( ss tt ρρ =
Motivation
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Methodology
Positive and Negative Energy Spots
For example (Kholodnyi, 2000), in the practically important special case the representation function is given by
where:
If A = 1, B = 0, α = 1, and β > 0, and hence .
We comment that at .
We also comment that is strictly increasing so that is well defined and
Finally, we comment that parameters A, B, α, and β can be time-dependent.
)ˆ()ˆ( ,,, ss BAt βαρρ =
βαβαρ −−= sBsAsBA ˆˆ)ˆ(,,,
ssBA ˆ)ˆ(,,, =βαρ tt s=Ψ
0)ˆ(,,, =∗sBA βαρ )/(1)/(ˆ βα +∗ = ABs
)ˆ(,,, sBA βαρ )ˆ(1,,, Ψ−βαρ BA
αααρ /1
21
,,, )2
4ˆˆ()ˆ(
A
ABBA
+Ψ+Ψ=Ψ−
Motivation
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Methodology
Positive and Negative Energy Spots
Representation function for A = 1, B = 0.1, α = 1, and β =1:)ˆ(,,, sBA βαρ
Representation Function
-3
-2
-1
0
1
2
3
0 0,5 1 1,5 2 2,5 3
s
Rep
rese
nta
tion
Fu
nct
ion
Representation Function
A Component
B Component
Motivation
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Methodology
Positive and Negative Energy Spots
Inverse representation function for A = 1, B = 0.1, α = 1, and β =1:)ˆ(1,,, Ψ−βαρ BA
Inverse Representation Function
0
0,5
1
1,5
2
2,5
3
3,5
4
4,5
-3 -2 -1 0 1 2 3 4 5
Psi
Inve
rse
Rep
rese
nta
tion
Fu
nct
ion
Representation Function
A Component
B Component
Motivation
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Methodology
The Semilinear Black-Scholes Equation for American Options
Assume that the risk-neutral dynamics of the price st of an underlying security are given by the geometric Brownian motion:
where:
• r(s,t) is the continuously compounded interest rate,
• d(s,t) is the continuously compounded dividend yield in terms of the underlying security being a stock,
• σ(s,t) is the volatility, and
• Wt is the Wiener process.
,),()),(),(( ttttttt dWstsdtstsdtsrds σ+−=
Motivation
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Methodology
The Semilinear Black-Scholes Equation for American Options
The value of the European contingent claim on the underlying security with inception time t, expiration time T, and payoff g is given by the Black-Scholes equation:
For example, for European call and put options with strike X on the underlying security the payoffs are given by (s – X)+ and (X – s)+.
).(),(
,,0)()),(),((),(2
12
222
sgTsv
Ttvtrvs
stsdtsrvs
stsvt
=
<=−∂∂−+
∂∂+
∂∂
σ
Motivation
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Methodology
The Semilinear Black-Scholes Equation for American Options
It can be shown (Kholodnyi 1995) that the value of the American contingent claim on the underlying security with inception time t, expiration time T, and payoff g is given by the semilinear Black-Scholes equation:
where:
For example, for American call and put options with strike X on the underlying security the payoffs are given by (s – X)+ and (X – s)+ with the nonlinear terms of the form:
Motivation
),(),(
,,0),,()()),(),((),(2
12
222
sgTsv
Ttvstfvtrvs
stsdtsrvs
stsvt
T=
<=+−∂∂−+
∂∂+
∂∂
σ
).()()),(),((),(2
1),,(
2
222 vgHgtrg
sstsdtsrg
sstsg
tvstf ttttt −
−
∂∂−+
∂∂+
∂∂−=
+
σ
)).()(()),(),(())(,,(
)),()(()),(),(())(,,(
svsXHstsdXtsrsvstf
svXsHXtsrstsdsvstf
put
call
−−−=
−−−=++
++
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Methodology
The Semilinear Black-Scholes Equation for American Options
American call option with X = 50 for r = 0.1, d = 0.1, σ = 0.4
Motivation
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Methodology
The Semilinear Black-Scholes Equation for American Options
American put option with X = 50 for r = 0.1, d = 0.0, σ = 0.4
Motivation
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Energy Spot Process
Methodology
Definition
Define (Kholodnyi, 2000) the non-Markovian process for the energy spot prices with spikes by
where:
• is the energy spot price at time t,
• is the representation function,
• is the inter-spike process, and
• is the spike process.
Assume that the inter-spike process and spike process are independent Markov processes.
),ˆ( tttt s λρ=Ψ
tΨ),ˆ( λρ st
0ˆ >ts
0>tλ
ts tλ
VTR/STR/Kholodnyi
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Energy Spot Process
Methodology
Representation Function
For example (Kholodnyi, 2000), in the practically important special case the representation function is given by
where:
If A = 1, B = 0, α = 1, and β > 0, and hence .
We comment that at .
We also comment that is strictly increasing so that is well defined and
Finally, we comment that parameters A, B, α, and β can be time-dependent.
)ˆ(),ˆ( ,,, ss BAt λρλρ βα=
βαβαρ −−= sBsAsBA ˆˆ)ˆ(,,,
ssBA ˆ)ˆ(,,, =βαρ ttt sλ=Ψ
0)ˆ(,,, =∗sBA βαρ )/(1)/(ˆ βα +∗ = ABs
)ˆ(,,, sBA βαρ )ˆ(1,,, Ψ−βαρ BA
αααρ /1
21
,,, )2
4ˆˆ()ˆ(
A
ABBA
+Ψ+Ψ=Ψ−
![Page 26: Energy Finance Seminar - uni-due.de · • The interpolation and extrapolation of market energy forward curves: −Re-construction of the highest granularity (for example, monthly,](https://reader035.vdocuments.us/reader035/viewer/2022070708/5eb4823de2fea16f383e5247/html5/thumbnails/26.jpg)
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Energy Spot Process
Methodology
Representation Function
Representation function for A = 1, B = 0.1, α = 1, and β =1:)ˆ(,,, sBA βαρ
Representation Function
-3
-2
-1
0
1
2
3
0 0,5 1 1,5 2 2,5 3
s
Rep
rese
nta
tion
Fu
nct
ion
Representation Function
A Component
B Component
![Page 27: Energy Finance Seminar - uni-due.de · • The interpolation and extrapolation of market energy forward curves: −Re-construction of the highest granularity (for example, monthly,](https://reader035.vdocuments.us/reader035/viewer/2022070708/5eb4823de2fea16f383e5247/html5/thumbnails/27.jpg)
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Methodology
Spike 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
where:
• 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.
Energy Spot Process
=
),(),(
),(),(),(
tTPtTP
tTPtTPtTP
rrrs
srss
![Page 28: Energy Finance Seminar - uni-due.de · • The interpolation and extrapolation of market energy forward curves: −Re-construction of the highest granularity (for example, monthly,](https://reader035.vdocuments.us/reader035/viewer/2022070708/5eb4823de2fea16f383e5247/html5/thumbnails/28.jpg)
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Methodology
Spike Process
The family of 2×2 matrices L = {L(t) : t ≥ 0} defined by
is said to generate the two-state Markov process Mt, and the 2×2 matrix
is called a generator.
In terms of the generators, P(T,t) is given by
Energy Spot Process
( ) ( , ) ,T tdL t P T t
dT ==
=
)()(
)()()(
tLtL
tLtLtL
rrrs
srss
.)(
),(∫
=Tt
dLetTP
ττ
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Methodology
Spike Process
It can be shown that
Moreover
where
Energy Spot Process
.)(),(),(
,)(),(),(
')'(
')'()(
τττ
τττ
τ
τ
ττ
ττττ
deLtPtTP
deLtPetTP
Tss
Tss
Tt ss
dLsr
T
t rrsr
dLsr
T
t rsdL
ss
∫
∫∫
∫
∫=
+=
),,(),(),( tTPtTPtTP rss
sssss +=
.)(),(),(
),(
')'(
)(
τττ τ ττ
ττ
deLtPtTP
etTPT
ss
Tt ss
dLsr
T
t rsr
ss
dLsss
∫
∫
∫=
=
![Page 30: Energy Finance Seminar - uni-due.de · • The interpolation and extrapolation of market energy forward curves: −Re-construction of the highest granularity (for example, monthly,](https://reader035.vdocuments.us/reader035/viewer/2022070708/5eb4823de2fea16f383e5247/html5/thumbnails/30.jpg)
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Methodology
Spike Process
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
so that
Energy Spot Process
( )( ) ( )( )
( )( ) ( )( )( )
T t a b T t a b
T t a b T t a b
b ae b be
a b a b
a ae a be
a b a b
P T t− − + − − +
− − + − − +
+ − + + − + + +
− =
a bLa b
−=−
.)()( )())((
)( tTabatT
rss
tTasss e
ba
aebtTPetTP −−
+−−−− −
++=−=− and
![Page 31: Energy Finance Seminar - uni-due.de · • The interpolation and extrapolation of market energy forward curves: −Re-construction of the highest granularity (for example, monthly,](https://reader035.vdocuments.us/reader035/viewer/2022070708/5eb4823de2fea16f383e5247/html5/thumbnails/31.jpg)
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λt
Mt
Spike State Regular State
Ξ(t,λ
)
Time
Time
1
st rt
The Non-Markovian Process for Energy Spot Prices with Spikes
Spike Process
Methodology
VTR/STR/Kholodnyi
Energy Spot Process
VTR/STR/Kholodnyi
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Methodology
Spike Process
The transition probability density function for the spike process λt as a Markov process is given by
where δ(x) is the Dirac delta function.
Energy Spot Process
=−
+∫Ξ
≠
−
+∫Ξ
+−∫
=Λ
∫
∫
1if)1(),(
)(),(),(
1if
)1(),(
)(),(),(
)(
),,,(
')'(
')'(
)(
t
Trr
T
t
dL
srrrT
t
Trs
T
t
dL
srrsT
Tt
dL
Tt
tTP
deLtP
tTP
deLtP
e
TtT
ss
T
ss
T
tss
λλδ
τττλτ
λ
λδ
τττλτ
λλδ
λλ
τ
τ
ττ
ττ
ττ
![Page 33: Energy Finance Seminar - uni-due.de · • The interpolation and extrapolation of market energy forward curves: −Re-construction of the highest granularity (for example, monthly,](https://reader035.vdocuments.us/reader035/viewer/2022070708/5eb4823de2fea16f383e5247/html5/thumbnails/33.jpg)
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Methodology
Inter-Spike Process
Consider a practically important special case when is a geometric mean-reverting process:
where:
• is the mean-reversion rate,
• is the log long-term mean,
• is the volatility, and
• Wt is the Wiener process.
We comment that can be a suitable multi-factor diffusion process.
ts
Energy Spot Process
,)()ˆln)(ˆ)((ˆln ttt dWtdtsttsd σµη +−=
0)( >tη
)(ˆ tµ
0)( >tσ
)(ˆ tµ
![Page 34: Energy Finance Seminar - uni-due.de · • The interpolation and extrapolation of market energy forward curves: −Re-construction of the highest granularity (for example, monthly,](https://reader035.vdocuments.us/reader035/viewer/2022070708/5eb4823de2fea16f383e5247/html5/thumbnails/34.jpg)
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Energy Spot Process
Methodology
Inter-Spike Process
Define (Kholodnyi, 2000) the inter-spike energy spot price process by
where:
• is the inter-spike power spot price process, and
• is the representation function associated with .
We comment that the stochastic differential equation for can be obtained with the help of the Ito’s lemma
We also comment that since the spike process is equal to unity between spikes, the power spot process coincides between spikes with the inter-spike power spot process so that
ts
)ˆ(ˆttt sρ=Ψ
tΨ
)ˆ(stρ
tΨ
tλtΨ
tΨ
1|),ˆ()ˆ( == λλρρ ss tt
![Page 35: Energy Finance Seminar - uni-due.de · • The interpolation and extrapolation of market energy forward curves: −Re-construction of the highest granularity (for example, monthly,](https://reader035.vdocuments.us/reader035/viewer/2022070708/5eb4823de2fea16f383e5247/html5/thumbnails/35.jpg)
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Energy Spot Process
Methodology
Inter-Spike Process
For example (Kholodnyi, 2000), in the practically important special case the representation function is given by
where:
If A = 1, B = 0, α = 1, and β > 0, and hence .
We comment that at .
We also comment that is strictly increasing so that is well defined and
Finally, we comment that parameters A, B, α, and β can be time-dependent.
)ˆ()ˆ( ,,, ss BAt βαρρ =
βαβαρ −−= sBsAsBA ˆˆ)ˆ(,,,
ssBA ˆ)ˆ(,,, =βαρ tt s=Ψ
0)ˆ(,,, =∗sBA βαρ )/(1)/(ˆ βα +∗ = ABs
)ˆ(,,, sBA βαρ )ˆ(1,,, Ψ−βαρ BA
αααρ /1
21
,,, )2
4ˆˆ()ˆ(
A
ABBA
+Ψ+Ψ=Ψ−
![Page 36: Energy Finance Seminar - uni-due.de · • The interpolation and extrapolation of market energy forward curves: −Re-construction of the highest granularity (for example, monthly,](https://reader035.vdocuments.us/reader035/viewer/2022070708/5eb4823de2fea16f383e5247/html5/thumbnails/36.jpg)
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Energy Spot Process
Methodology
Inter-Spike Process
Representation function for A = 1, B = 0.1, α = 1, and β =1:)ˆ(,,, sBA βαρ
Representation Function
-3
-2
-1
0
1
2
3
0 0,5 1 1,5 2 2,5 3
s
Rep
rese
nta
tion
Fu
nct
ion
Representation Function
A Component
B Component
![Page 37: Energy Finance Seminar - uni-due.de · • The interpolation and extrapolation of market energy forward curves: −Re-construction of the highest granularity (for example, monthly,](https://reader035.vdocuments.us/reader035/viewer/2022070708/5eb4823de2fea16f383e5247/html5/thumbnails/37.jpg)
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Energy Spot Process
Methodology
Inter-Spike Process
Inverse representation function for A = 1, B = 0.1, α = 1, and β =1:)ˆ(1,,, Ψ−βαρ BA
Inverse Representation Function
0
0,5
1
1,5
2
2,5
3
3,5
4
4,5
-3 -2 -1 0 1 2 3 4 5
Psi
Inve
rse
Rep
rese
nta
tion
Fu
nct
ion
Representation Function
A Component
B Component
![Page 38: Energy Finance Seminar - uni-due.de · • The interpolation and extrapolation of market energy forward curves: −Re-construction of the highest granularity (for example, monthly,](https://reader035.vdocuments.us/reader035/viewer/2022070708/5eb4823de2fea16f383e5247/html5/thumbnails/38.jpg)
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Methodology
Inter-Spike Process
It is clear that if is a geometric mean-reverting process then is a mean-reverting process:
where:
• is the mean-reversion rate,
• is the long-term mean,
• is the volatility, and
• Wt is the Wiener process.
We comment that can be a suitable multi-factor diffusion process.
ts
Energy Spot Process
0)( >tη
)(ˆ tµ
0)( >tσ
)(ˆ tµ
tt sx ˆlnˆ =
,)()ˆ)(ˆ)((ˆ ttt dWtdtxttxd σµη +−=
![Page 39: Energy Finance Seminar - uni-due.de · • The interpolation and extrapolation of market energy forward curves: −Re-construction of the highest granularity (for example, monthly,](https://reader035.vdocuments.us/reader035/viewer/2022070708/5eb4823de2fea16f383e5247/html5/thumbnails/39.jpg)
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Energy Spot Process
Methodology
Inter-Spike Process
Then (Kholodnyi, 2000) the inter-spike energy spot price process by
where:
• is the inter-spike power spot price process, and
• is the representation function associated with .
We comment that the stochastic differential equation for can be obtained with the help of the Ito’s lemma
We also comment that since the spike process is equal to unity between spikes, the power spot process coincides between spikes with the inter-spike power spot process so that
tΨ
tΨ
tλtΨ
tΨ
)ˆ(ˆˆttt xρ=Ψ
)ˆ(exp)ˆ(ˆ xx tt ρρ =
1|),ˆ(exp)ˆ(ˆ == λλρρ xx tt
tx
![Page 40: Energy Finance Seminar - uni-due.de · • The interpolation and extrapolation of market energy forward curves: −Re-construction of the highest granularity (for example, monthly,](https://reader035.vdocuments.us/reader035/viewer/2022070708/5eb4823de2fea16f383e5247/html5/thumbnails/40.jpg)
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Energy Spot Process
Methodology
Inter-Spike Process
For example (Kholodnyi, 2000), in the practically important special case the representation function is given by
where:
If A = 1, B = 0, α = 1, and β > 0, and hence .
We comment that at .
We also comment that is strictly increasing so that is well defined and
Finally, we comment that parameters A, B, α, and β can be time-dependent.
)ˆ(ˆ)ˆ(ˆ ,,, xx BAt βαρρ =
xxxBABA BeAeex ˆˆˆ
,,,,,, )()ˆ(ˆ βαβαβα ρρ −−==
xBA ex ˆ
,,, )ˆ(ˆ =βαρ txt eˆˆ =Ψ
0)ˆ(ˆ ,,, =∗xBA βαρ )/(1)/ln(ˆ βα +∗ = ABx
)2
4ˆˆln(
1)ˆ(ˆ
21
,,, A
ABBA
+Ψ+Ψ=Ψ−
αρ αα
)ˆ(ˆ ,,, xBA βαρ )ˆ(ˆ 1,,, Ψ−βαρ BA
![Page 41: Energy Finance Seminar - uni-due.de · • The interpolation and extrapolation of market energy forward curves: −Re-construction of the highest granularity (for example, monthly,](https://reader035.vdocuments.us/reader035/viewer/2022070708/5eb4823de2fea16f383e5247/html5/thumbnails/41.jpg)
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Energy Spot Process
Methodology
Inter-Spike Process
Representation function for A = 1, B = 0.1, α = 1, and β =1:)ˆ(ˆ ,,, xBA βαρ
Representation Function
-3
-2
-1
0
1
2
3
-4 -3 -2 -1 0 1 2
x
Rep
rese
nta
tion
Fu
nct
ion
Representation Function
A Component
B Component
![Page 42: Energy Finance Seminar - uni-due.de · • The interpolation and extrapolation of market energy forward curves: −Re-construction of the highest granularity (for example, monthly,](https://reader035.vdocuments.us/reader035/viewer/2022070708/5eb4823de2fea16f383e5247/html5/thumbnails/42.jpg)
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Energy Spot Process
Methodology
Inter-Spike Process
Inverse representation function for A = 1, B = 0.1, α = 1, and β =1:)ˆ(ˆ 1,,, Ψ−βαρ BA
Inverse Representation Function
-5
-4
-3
-2
-1
0
1
2
3
-3 -2 -1 0 1 2 3 4 5
Psi
Inve
rse
Rep
rese
nta
tion
Fu
nct
ion
Representation Function
A Component
B Component
![Page 43: Energy Finance Seminar - uni-due.de · • The interpolation and extrapolation of market energy forward curves: −Re-construction of the highest granularity (for example, monthly,](https://reader035.vdocuments.us/reader035/viewer/2022070708/5eb4823de2fea16f383e5247/html5/thumbnails/43.jpg)
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Energy Spot Process
Methodology
Monte Carlo Simulations
Geometric Mean-Reverting Process with Spikes
-50
0
50
100
150
200
0 100 200 300 400
Time Moments
Val
ue
of
the
Pro
cess
Spikes withMixtureMagnitude
VTR/STR/Kholodnyi
![Page 44: Energy Finance Seminar - uni-due.de · • The interpolation and extrapolation of market energy forward curves: −Re-construction of the highest granularity (for example, monthly,](https://reader035.vdocuments.us/reader035/viewer/2022070708/5eb4823de2fea16f383e5247/html5/thumbnails/44.jpg)
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Energy Spot Process
Methodology
Expected Time for Ψt to be in the Spike and Inter-Spike States
The expected time for Ψt to be in the spike state that starts at time t is:
Similarly, the expected time for Ψt to be in the inter-spike state that starts at time tis:
In the special case of a time-homogeneous two-state Markov process Mt:
st
rt
.)()(')'(
ττττ
ττdaett
t
da
st∫
∞∫−
−=
.)()(')'(
ττττ
ττdbett
t
db
rt∫
∞∫−
−=
./1 and /1 btat rs ==
![Page 45: Energy Finance Seminar - uni-due.de · • The interpolation and extrapolation of market energy forward curves: −Re-construction of the highest granularity (for example, monthly,](https://reader035.vdocuments.us/reader035/viewer/2022070708/5eb4823de2fea16f383e5247/html5/thumbnails/45.jpg)
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Energy Spot Process
Methodology
Interpretation of the Spike State of Ψt as Spikes in Energy Spot Prices
If the expected time for the non-Markovian process Ψt to be in the spike state is small relative to the characteristic time of change of the process then the spike state of Ψt can be interpreted as spikes in energy spot prices:
• Ψt can exhibit sharp upward/downward price movements shortly followed by equally sharp downward/upward prices movements of approximately the same magnitude which can be interpreted as upward/downward spikes.
For example, if is a diffusion process then:
and
In this case is the expected lifetime of a spike that starts at time t, and is the expected time between two consecutive spikes when the first spike ends at time t.
st
tΨ
tΨ
st rt
1),ˆ(2 <<Ψ sttσ .1),ˆ( <<Ψ sttµ
![Page 46: Energy Finance Seminar - uni-due.de · • The interpolation and extrapolation of market energy forward curves: −Re-construction of the highest granularity (for example, monthly,](https://reader035.vdocuments.us/reader035/viewer/2022070708/5eb4823de2fea16f383e5247/html5/thumbnails/46.jpg)
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Energy Spot Process
Methodology
The Non-Markovian Process Ψt as a Markov Process with the Extended State Space
The state of the energy market at any time t can be fully characterized by the values of the spike and inter-spike processes and at time t.
Moreover, although the process Ψt 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 Ψt can be represented as a Markov process with the extended state space that at any time t consists of all possible pairs with and .
tλ ts
tλ ts
)ˆ,( tt sλ0>tλ 0ˆ >ts
![Page 47: Energy Finance Seminar - uni-due.de · • The interpolation and extrapolation of market energy forward curves: −Re-construction of the highest granularity (for example, monthly,](https://reader035.vdocuments.us/reader035/viewer/2022070708/5eb4823de2fea16f383e5247/html5/thumbnails/47.jpg)
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Energy Forward Prices
Methodology
Inter-Spike Spot Process
Denote by
the energy forward price at time t for the forward contract with maturity time T.
Energy forward price can be found as the risk-neutral expected value of the inter-spike energy spot prices at time T :
where:
),ˆ)(,(ˆ),(ˆtTtFTtF Ψ=
),(ˆ TtF)ˆ(ˆ
TTT sρ=Ψ
)),ˆ()(,(ˆ)ˆ)(,(ˆ 1ttt TtFTtF Ψ=Ψ −∗ ρ
.ˆ)ˆ()ˆ,ˆ,,()ˆ)(,(ˆ0
TTTTtt sdsssTtPsTtF ρ∫∞
∗ =
![Page 48: Energy Finance Seminar - uni-due.de · • The interpolation and extrapolation of market energy forward curves: −Re-construction of the highest granularity (for example, monthly,](https://reader035.vdocuments.us/reader035/viewer/2022070708/5eb4823de2fea16f383e5247/html5/thumbnails/48.jpg)
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Energy Forward Prices
Methodology
Inter-Spike Spot Process: GMR Process
It can be shown (Kholodnyi 1995) that in the practically important special case of the representation function the energy forward prices are given by the following analytical expression:
where:
with
)ˆ(,,, sBA βαρ
)),ˆ()(,(ˆ)ˆ)(,(ˆ 1,,, tBAt TtFTtF Ψ=Ψ −∗βαρ
),ˆ)(,(ˆ)ˆ)(,(ˆ)ˆ)(,(ˆttt sTtFBsTtFAsTtF βα −
∗ −=
),(),())(,(ˆ
2
1
0
ˆ
ˆˆ)ˆ,ˆ,,()ˆ)(,(ˆ
22
Ttat
TtbtTTt
TTTtt
see
sdsssTtPsTtF
ωωωσ
ωω
−
∞
=
= ∫
![Page 49: Energy Finance Seminar - uni-due.de · • The interpolation and extrapolation of market energy forward curves: −Re-construction of the highest granularity (for example, monthly,](https://reader035.vdocuments.us/reader035/viewer/2022070708/5eb4823de2fea16f383e5247/html5/thumbnails/49.jpg)
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Energy Forward Prices
Methodology
Inter-Spike Spot Process: GMR Process
Moreover (Kholodnyi 1995) , in the practically important special case of α = β the energy forward prices are given by:
We comment that since the energy spot prices at time T can be negative for B > 0 , the energy forward prices and can also be negative for B > 0.
),(2
),())(,(ˆ2
1
),(2
),())(,(ˆ2
1
/12
)2
4ˆˆ(
)2
4ˆˆ(
))2
4ˆˆ)((,(ˆ)ˆ)(,(ˆ
22
22
TtattTtbtTTt
TtattTtbtTTt
ttt
A
ABeBe
A
ABeAe
A
ABTtFTtF
−−−
−
∗
+Ψ+Ψ−
+Ψ+Ψ=
+Ψ+Ψ=Ψ
ββσ
αασ
α
)ˆ(ˆ,,, TBAT sβαρ=Ψ
)ˆ)(,(ˆtTtF Ψ )ˆ)(,(ˆ
tsTtF ∗
![Page 50: Energy Finance Seminar - uni-due.de · • The interpolation and extrapolation of market energy forward curves: −Re-construction of the highest granularity (for example, monthly,](https://reader035.vdocuments.us/reader035/viewer/2022070708/5eb4823de2fea16f383e5247/html5/thumbnails/50.jpg)
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Energy Forward Prices
Methodology
Inter-Spike Spot Process: GMR Process
It can be shown (Kholodnyi 1995) that:
.)(ˆ)(),(
,),(
,)(1
),(ˆ
')'(
)(
')'(22
∫
∫
∫=
∫=
∫−
=
−
−
−
T
t
d
d
dT
t
deTtb
eTta
detT
Tt
T
T
t
T
ττµτη
ττσσ
τ
τ
ττη
ττη
ττη
![Page 51: Energy Finance Seminar - uni-due.de · • The interpolation and extrapolation of market energy forward curves: −Re-construction of the highest granularity (for example, monthly,](https://reader035.vdocuments.us/reader035/viewer/2022070708/5eb4823de2fea16f383e5247/html5/thumbnails/51.jpg)
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Energy Forward Prices
Methodology
Inter-Spike Spot Process: Time-Homogeneous GMR Process
It can be shown (Kholodnyi 1995) that in a practically important special case of a time-homogeneous geometric mean-reverting inter-spike process we have:
).1(ˆ),(
,),(
,)1()(2
),(ˆ
)(
)(
)(22
tT
tT
tT
eTtb
eTta
etT
Tt
−−
−−
−−
−==
−−
=
η
η
η
µ
ησσ
![Page 52: Energy Finance Seminar - uni-due.de · • The interpolation and extrapolation of market energy forward curves: −Re-construction of the highest granularity (for example, monthly,](https://reader035.vdocuments.us/reader035/viewer/2022070708/5eb4823de2fea16f383e5247/html5/thumbnails/52.jpg)
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Energy Forward Prices
Methodology
Inter-Spike Spot Process: GMR Process with Log-Linear Trends
Moreover (Kholodnyi 1995), consider a practically important special case of a geometric mean-reverting spot process with linear trends in the log volatility and the log long-term spot price :
and
where:
with ,
with .
,ˆˆ)(ln)(ˆ ttt TRσσσσ +==
,ˆˆ)(ˆln)(ˆ ttst TReq µµµ +==
tTRet σσσ ˆ)( = σσ ˆe=t
eqeqTRests µˆ)(ˆ = µˆ eseq =
![Page 53: Energy Finance Seminar - uni-due.de · • The interpolation and extrapolation of market energy forward curves: −Re-construction of the highest granularity (for example, monthly,](https://reader035.vdocuments.us/reader035/viewer/2022070708/5eb4823de2fea16f383e5247/html5/thumbnails/53.jpg)
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Energy Forward Prices
Methodology
Inter-Spike Spot Process: GMR Process with Log-Linear Trends
It can be shown (Kholodnyi 1995) that in the practically important special case of a geometric mean-reverting spot process with linear trends in the log volatility and the log long-term spot price we have:
where η(t) is time-independent and equal to η.
.)1(1
)(ˆ)1(ˆ),(
,),(
,)1())(ˆ(2
),(ˆ
)( )()()(
)(
))(ˆ(2ˆ22
tTtTTR
tT
tT
tT
TR
T
eteTeTtb
eTta
etT
eTt TR
TR
−−−−−−
−−
−+−
−−−+−=
=
−−+
=
ηηη
η
σησ
ηµµ
σησσ
![Page 54: Energy Finance Seminar - uni-due.de · • The interpolation and extrapolation of market energy forward curves: −Re-construction of the highest granularity (for example, monthly,](https://reader035.vdocuments.us/reader035/viewer/2022070708/5eb4823de2fea16f383e5247/html5/thumbnails/54.jpg)
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Energy Forward Prices
Methodology
Inter-Spike Spot Process: GMR Process with General Trends
Moreover (Kholodnyi 1995), consider a practically important special case of a geometric mean-reverting spot process with general trends in the volatility and the log long-term spot price :
and
∑∞
=
=0
22 )(n
nntt σσ
∑∞
=
=0
ˆ)(ˆn
nntt µµ
![Page 55: Energy Finance Seminar - uni-due.de · • The interpolation and extrapolation of market energy forward curves: −Re-construction of the highest granularity (for example, monthly,](https://reader035.vdocuments.us/reader035/viewer/2022070708/5eb4823de2fea16f383e5247/html5/thumbnails/55.jpg)
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Energy Forward Prices
Methodology
Inter-Spike Spot Process: GMR Process with General Trends
It can be shown (Kholodnyi 1995) that in the practically important special case of a geometric mean-reverting spot process with general trends in the volatility and the log long-term spot price we have:
where η(t) is time-independent and equal to η.
,)1()!(
!ˆ),(
,),(
,)2(
)1()!(
!1),(ˆ
0 01
)(
)(
0 01
)(22
∑ ∑
∑ ∑
∞
= =+
−−−−
−−
∞
= =+
−−−−
−−−
=
=
−−−−
=
n
n
mm
tTmnmnm
n
tT
n
n
mm
tTmnmnm
n
etT
mn
nTtb
eTta
etT
mn
n
tTTt
ηµη
ησσ
η
η
η
![Page 56: Energy Finance Seminar - uni-due.de · • The interpolation and extrapolation of market energy forward curves: −Re-construction of the highest granularity (for example, monthly,](https://reader035.vdocuments.us/reader035/viewer/2022070708/5eb4823de2fea16f383e5247/html5/thumbnails/56.jpg)
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Energy Forward Prices
Methodology
Inter-Spike Spot Process: GMR Process with Log-Linear Trends and CyclicalPatterns
Moreover (Kholodnyi 1995), consider a practically important special case of a geometric mean-reverting spot process with linear trends and cyclical patterns in the log volatility and the log long-term spot price
and
∑ +++=∞
=1 ˆˆ
))2
sin(ˆ)2
cos(ˆ(ˆˆ)(ˆm
sm
cmTR mt
Tmt
Ttt
µµ
πµπµµµµ
t
m
sm
cm
TRemtT
mtT
t σ
σσ
πσπσσσ 2
1
,2,222 )))2
sin()2
cos((()( ∑ ++=∞
=
![Page 57: Energy Finance Seminar - uni-due.de · • The interpolation and extrapolation of market energy forward curves: −Re-construction of the highest granularity (for example, monthly,](https://reader035.vdocuments.us/reader035/viewer/2022070708/5eb4823de2fea16f383e5247/html5/thumbnails/57.jpg)
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Energy Forward Prices
Methodology
Inter-Spike Spot Process: GMR Process with Log-Linear Trends and CyclicalPatterns
It can be shown (Kholodnyi 1995) that in the practically important special case of a geometric mean-reverting spot process with linear trends and cyclical patterns in the log volatility and the log long-term spot price :
where η(t) is time-independent and equal to η.
).)
2())(2(
))2
cos()2
()2
sin()(2())2
cos()2
()2
sin()(2(
)2
())(2(
))2
sin()2
()2
cos()(2())2
sin()2
()2
cos()(2(
(
)(2
1))(,(ˆ
2
ˆ
2
))((2
,2
2
ˆ
2
))((2
1
,22
))((2222
mT
emtT
mT
mtT
mTT
mT
mTT
mT
emtT
mT
mtT
mTT
mT
mTT
e
eetTTt
TR
tTTRTR
sm
TR
tTTRTRM
m
cm
T
TR
tTT
TR
TR
TR
TR
TR
µ
ησ
σσσσσσ
µ
ησ
σσσσσσσ
ησσ
πησ
πππησπππησσ
πησ
πππησπππησσ
ησσσ
++
−+−−++
++
++−+++
+−=−
−+−
−+−
=
−+−
∑
![Page 58: Energy Finance Seminar - uni-due.de · • The interpolation and extrapolation of market energy forward curves: −Re-construction of the highest granularity (for example, monthly,](https://reader035.vdocuments.us/reader035/viewer/2022070708/5eb4823de2fea16f383e5247/html5/thumbnails/58.jpg)
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Energy Forward Prices
Methodology
Inter-Spike Spot Process: GMR Process with Log-Linear Trends and CyclicalPatterns
It can be shown (Kholodnyi 1995) that in the practically important special case of a geometric mean-reverting spot process with linear trends and cyclical patterns in the log volatility and the log long-term spot price :
where η(t) is time-independent and equal to η.
).)
2(
))2
cos()2
()2
sin(())2
cos()2
()2
sin((
ˆ
)2
(
))2
sin()2
()2
cos(())2
sin()2
()2
cos((
ˆ(
))1(1
)((ˆ)1(ˆ),(
2
ˆ
2
)(
ˆˆˆˆˆˆ
2
ˆ
2
)(
ˆˆˆˆˆˆ
1
)()()(
mT
emtT
mT
mtT
mTT
mT
mTT
mT
emtT
mT
mtT
mTT
mT
mTT
eteTeTtb
tT
sm
tT
m
cm
tTtTTR
tT
µ
η
µµµµµµ
µ
η
µµµµµµ
ηηη
πη
πππηπππηµ
πη
πππηπππηµη
ηµµ
+
−−−+
+
+−++
−−−+−=
−−
−−∞
=
−−−−−−
∑
![Page 59: Energy Finance Seminar - uni-due.de · • The interpolation and extrapolation of market energy forward curves: −Re-construction of the highest granularity (for example, monthly,](https://reader035.vdocuments.us/reader035/viewer/2022070708/5eb4823de2fea16f383e5247/html5/thumbnails/59.jpg)
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Energy Forward Prices
Methodology
Inter-Spike Spot Process: GMR Process with Log-Linear Trends and CyclicalPatterns
For the following typical values of the parameters:
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Energy Forward Prices
Methodology
Inter-Spike Spot Process: GMR Process with Log-Linear Trends and CyclicalPatterns
Forward Prices
-20
0
20
40
60
80
100
120
140
160
0 1 2 3 4 5
Time
Fo
rwar
d P
pri
ce
Forward Price
![Page 61: Energy Finance Seminar - uni-due.de · • The interpolation and extrapolation of market energy forward curves: −Re-construction of the highest granularity (for example, monthly,](https://reader035.vdocuments.us/reader035/viewer/2022070708/5eb4823de2fea16f383e5247/html5/thumbnails/61.jpg)
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Energy Swap Prices
Inter-Spike Spot Process
It can be shown (Kholodnyi 1995) that energy swap price at time t for the swap with the delivery period [Tin ,Tfin ] is given by
where r(t) is the continuously compounded interest rate.
In the practically important special case of r(t) = 0 the preceding relationship takes the following form:
28-Apr-15 61VTR/STR/Kholodnyi
Methodology
,)ˆ)(,(ˆ)()ˆ)(,,(ˆ)(
1)(
dTeTtFdTeTTtFfinT
in
T
inT
finT
in
T
inT
T
dr
t
T
dr
tfinin ∫∫∫
Ψ∫
=Ψ−
−− ττττ
,)ˆ)(,(ˆ1)ˆ)(,,(ˆ dTTtF
TTTTtF
finT
inT
tinfin
tfinin ∫ Ψ−
=Ψ
![Page 62: Energy Finance Seminar - uni-due.de · • The interpolation and extrapolation of market energy forward curves: −Re-construction of the highest granularity (for example, monthly,](https://reader035.vdocuments.us/reader035/viewer/2022070708/5eb4823de2fea16f383e5247/html5/thumbnails/62.jpg)
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Energy Forward Prices
Methodology
Non-Markovian Spot Process
Denote by
the energy forward price at time t for the forward contract with maturity time T.
Energy forward price F(t,T) can be found as the risk-neutral expected value of the inter-spike energy spot prices at time T :
where:
)ˆ)(,(),( tTtFTtFt
Ψ= λ
),ˆ( TTTT s λρ=Ψ
))ˆ()(,()ˆ)(,( 1ttt TtFTtF
ttΨ=Ψ −∗ ρλλ
∫ ∫∞ ∞∗ Λ=0 0
ˆ),ˆ(),,,()ˆ,ˆ,,()ˆ)(,( TTTTTTtTtt dsdsTtssTtPsTtFt
λλρλλλ
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Energy Forward Prices
Methodology
Non-Markovian Spot Process: GMR Inter-Spike Process
It can be shown (Kholodnyi 2000) that in the practically important special case of the representation function the energy forward prices are given by:
where:
with being the ω-th moment of the spike process given by
)ˆ(),ˆ( ,,, ss BAt λρλρ βα=
))ˆ()(,()ˆ)(,( 1,,, tBAt TtFTtF
ttΨ=Ψ −∗
βαλλ ρ
)ˆ)(,(ˆ),()ˆ)(,(ˆ),()ˆ)(,( ttt sTtFTtBsTtFTtAsTtFttt ββ
λααλλ λλ −
−∗ −=
TTTt dTtTtt
λλλλλ ωωλ ∫
∞
Λ=0
),,,(),(
),( Ttt
ωλλ tλ
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Energy Forward Prices
Methodology
The Case When Ξ(t,λ) is Time-Independent
The ω-th moment of the spike process is given by
where is the ω-th moment of the conditional probability distribution for the multiplicative magnitude of spikes is given by
),( Ttt
ωλλ tλ
=+≠++=
1),(),(
1),(),(),(),(
trrsr
trsr
ssts
ss
tTPtTP
tTPtTPtTPTt
t λλλλλλ
ω
ωωωλ
if
if
.)(0∫∞
Ξ= λλλλ ωω d
ωλ
![Page 65: Energy Finance Seminar - uni-due.de · • The interpolation and extrapolation of market energy forward curves: −Re-construction of the highest granularity (for example, monthly,](https://reader035.vdocuments.us/reader035/viewer/2022070708/5eb4823de2fea16f383e5247/html5/thumbnails/65.jpg)
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Energy Forward Prices
Methodology
Pareto Probability Distributions
PDF of the “upward” Pareto distribution:
where and .
PDF of the “downward” Pareto distribution:
where and .
PDF of the mixture of the “upward” and “downward” Pareto distributions:
≤<>
=−−
+
min
min1
min,
1 if0
if)(
min λλλλλγλλ
γγ
λγP
0>γ 1min ≥λ
<≤<<
=−−
−
1 if0
0 if)(
max
max1
max, max λλ
λλλγλλγγ
λγP
).()()(maxminmaxmin ,,,,,,
λλλ λγλγλγλγ−+−+−+ += qPpPP
p
0>γ 10 max ≤< λ
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Energy Forward Prices
Methodology
Pareto Probability Distributions
Mixture of Upward and Downward Pareto Distributions
-0,5
0
0,5
1
1,5
2
2,5
3
0 1 2 3 4
Lambda
PD
F
Mixture
![Page 67: Energy Finance Seminar - uni-due.de · • The interpolation and extrapolation of market energy forward curves: −Re-construction of the highest granularity (for example, monthly,](https://reader035.vdocuments.us/reader035/viewer/2022070708/5eb4823de2fea16f383e5247/html5/thumbnails/67.jpg)
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Energy Forward Prices
Methodology
Spikes with Pareto Probability Distributions for their Magnitude
For example (Kholodnyi 2000), in the practically important special case when Ξ(λ) is equal to , the ω-th moment of the conditional probability distribution for the multiplicative magnitude of spikes is given by
where .
)(maxmin ,,,,
λλγλγ −+pP ωλ
ωω
ωλγλγ
ω
λωγ
γλωγ
γ
λλλλ
maxmin
0,,,,
)(maxmin
++
−=
=
−
−
+
+
∞
∫ −+
qp
dPp
+− <<− γωγ
![Page 68: Energy Finance Seminar - uni-due.de · • The interpolation and extrapolation of market energy forward curves: −Re-construction of the highest granularity (for example, monthly,](https://reader035.vdocuments.us/reader035/viewer/2022070708/5eb4823de2fea16f383e5247/html5/thumbnails/68.jpg)
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Energy Forward Prices
Methodology
Spikes with Constant Upward and Downward Magnitude
For example (Kholodnyi 2000), in the practically important special case when Ξ(λ) is equal to , the ω-th moment of the conditional probability distribution for the multiplicative magnitude of spikes is given by
where and .
ωλ)()( λλδλλδ −+− du qp
ωω
ωω
λλ
λλλλδλλδλ
du
uu
qp
dpp
+=
−+−= ∫∞
0
))()((
1>uλ 1<dλ
![Page 69: Energy Finance Seminar - uni-due.de · • The interpolation and extrapolation of market energy forward curves: −Re-construction of the highest granularity (for example, monthly,](https://reader035.vdocuments.us/reader035/viewer/2022070708/5eb4823de2fea16f383e5247/html5/thumbnails/69.jpg)
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Energy Forward Prices
Methodology
Spikes with Constant Magnitude
Consider a special case of spikes with constant magnitude , that is, when Ξ(λ)is the delta function δ(λ- λ`).
The ω-th moment of the spike process is given by
1≠λ
=+=+
=1),(),(
),(),(),(
trrsr
trsss
tTPtTP
tTPtTPTt
t λλλλλλ ω
ωωλ if
if
),( Ttt
ωλλ tλ
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Energy Forward Prices
Methodology
Non-Markovian Spot Process: GMR Inter-Spike Process
Moreover (Kholodnyi 2000), in the practically important special case of α = β the energy forward prices are given by:
We comment that since the power spot prices at time T can be negative for B > 0 , the power forward prices and can also be negative for B > 0.
),(2
),())(,(ˆ2
1
),(2
),())(,(ˆ21
/12
*
)2
4ˆˆ(),(
)2
4ˆˆ(),(
))2
4ˆˆ)((,(ˆ)ˆ)(,(ˆ
22
22
TtattTtbtTTt
TtattTtbtTTt
ttt
A
ABeeTtB
A
ABeeTtA
A
ABTtFTtF
t
t
tt
−−−−
−
+Ψ+Ψ−
+Ψ+Ψ=
+Ψ+Ψ=Ψ
αασαλ
αασαλ
αλλ
λ
λ
)ˆ(ˆ,,, TTBAT sλρ βα=Ψ)ˆ)(,(ˆ
tTtFt
Ψλ )ˆ)(,(ˆ *tsTtF
tλ
![Page 71: Energy Finance Seminar - uni-due.de · • The interpolation and extrapolation of market energy forward curves: −Re-construction of the highest granularity (for example, monthly,](https://reader035.vdocuments.us/reader035/viewer/2022070708/5eb4823de2fea16f383e5247/html5/thumbnails/71.jpg)
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Energy Forward Prices
Methodology
Non-Markovian Spot Process: GMR Inter-Spike Process
For the following typical values of the parameters:
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Energy Forward Prices
Methodology
Non-Markovian Spot Process: GMR Inter-Spike Process
Forward Prices
-50
0
50
100
150
200
0 1 2 3 4 5
Time
Fo
rwar
d P
pri
ce
With SpikesWithout Spikes
![Page 73: Energy Finance Seminar - uni-due.de · • The interpolation and extrapolation of market energy forward curves: −Re-construction of the highest granularity (for example, monthly,](https://reader035.vdocuments.us/reader035/viewer/2022070708/5eb4823de2fea16f383e5247/html5/thumbnails/73.jpg)
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Energy Forward Prices
Methodology
Non-Markovian Spot Process: GMR Inter-Spike Process
Forward Prices
-50
0
50
100
150
200
0 0,02 0,04 0,06 0,08 0,1 0,12 0,14
Time
Fo
rwar
d P
pri
ce
With SpikesWithout Spikes
![Page 74: Energy Finance Seminar - uni-due.de · • The interpolation and extrapolation of market energy forward curves: −Re-construction of the highest granularity (for example, monthly,](https://reader035.vdocuments.us/reader035/viewer/2022070708/5eb4823de2fea16f383e5247/html5/thumbnails/74.jpg)
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Energy Swap Prices
Non-Markovian Spot Process
It can be shown (Kholodnyi 1995 - 2000) that energy swap price at time t for the swap with the delivery period [Tin ,Tfin ] is given by
where r(t) is the continuously compounded interest rate.
In the practically important special case of r(t) = 0 the preceding relationship takes the following form:
28-Apr-15 74VTR/STR/Kholodnyi
Methodology
,)ˆ)(,()()ˆ)(,,()(
1)(
dTeTtFdTeTTtFfinT
in
T
inT
t
finT
in
T
inT
t
T
dr
t
T
dr
tfinin ∫∫∫
Ψ∫
=Ψ−
−− ττ
λ
ττ
λ
,)ˆ)(,(1
)ˆ)(,,( dTTtFTT
TTtFfinT
in
tt
T
tinfin
tfinin ∫ Ψ−
=Ψ λλ
![Page 75: Energy Finance Seminar - uni-due.de · • The interpolation and extrapolation of market energy forward curves: −Re-construction of the highest granularity (for example, monthly,](https://reader035.vdocuments.us/reader035/viewer/2022070708/5eb4823de2fea16f383e5247/html5/thumbnails/75.jpg)
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European Contingent Claims
Linear Evolution Equation
It can be shown (Kholodnyi 2000) that the value E(t,T,g)of a European contingent claim on energy spots with spikes is the solution of the following linear evolution equation:
where
with and being the generators of and as Markov processes.
28-Apr-15 75VTR/STR/Kholodnyi
Methodology
gTv
TtvtLvdt
d
=
<=+
)(
,,0)(
),()()()( trtLtLtL sr −+=
)(tLr )(tLs tλts
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European Contingent Claims
Linear Evolution Equation
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:
28-Apr-15 76VTR/STR/Kholodnyi
Methodology
)(tLr
)(tLs
ts
=−+Ξ>−+−
=Λ1)1()()(),(
1)1()()()(),,(
''
'''
ttrrsrt
ttrsttsstt
tLtLt
tLtLt
λλδλλλδλλδλλ if
if
.ˆ
ˆ)ˆln)(ˆ)(()ˆ
ˆ)((2
1)( 22
ssstt
ssttLr ∂
∂−+∂∂= µησ
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American Contingent Claims
Semilinear Evolution Equation
It can be shown (Kholodnyi 1995 - 2000) that the value A(t,T,g)of an American contingent claim on energy spots with spikes is the solution of the following semilinear evolution equation:
where F(t,v) is given by
28-Apr-15 77VTR/STR/Kholodnyi
Methodology
TgTv
TtvtFvtLvdt
d
=
<=++
)(
,,0),()(
( ).)(),( ννν −
+∂∂−=
+
tt gHtLgt
tF
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Extracting Forward-Looking Market-Implied Risk-Neutral Probabilities
The Optimization Problem
It can be shown (Kholodnyi 1995) that the risk-neutral parameters of the spot process can be extracted from the market forward curve by solving the following optimization problem:
where:
• M is the number of the trading days,
• Nm is the number of traded swaps at the trading day tm,
• p are the parameters of the model, and
• P is the set of admissible parameters.
28-Apr-15 78
Methodology
∑∑= =∈
−M
m
N
n
nfin
ninm
pModel
nfin
ninmMarket
Pp
m
TTtFTTtF1 1
2)()()()()( )),,(),,((min
VTR/STR/KholodnyiVTR/STR/Kholodnyi
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Oil Market
Static Modeling
• Market: Brent Crude Oil
• Trade Date: 05-Jan-2011
• Contracts:
• 69 Contracts:
− 69 Monthly Futures Contracts
Oil Market
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Oil Market
Forward Curves Modeling
Model to Market Comparison
Configuration: 8th Degree Legendre Polynomial, 0th Degree Trig Polynomial
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Daily Forward Curve
Configuration: 8th Degree Legendre Polynomial, 0th Degree Trig Polynomial
Forward Curves Modeling
Oil Market
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PDF for Spots
Probability Distributions
Oil Market
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CDF for Spots
Probability Distributions
Oil Market
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QF for Spots
Probability Distributions
Oil Market
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Oil Market
Static Modeling
• Market: WTI Crude Oil
• Trade Date: 28-Mar-2008
• Contracts:
• 2518 Contracts:
− 74 Monthly Futures Contracts
− 1252 American Style Call Options
− 1192 American Style Put Options
Oil Market
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Oil Market
Forward Curves and Options Modeling
Market Data
Futures Contracts
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Oil Market
Market Data
Call and Put Options
Forward Curves and Options Modeling
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Forward Curves and Options Modeling
Model to Market Comparison
Forward Curve
Oil Market
%055.0||||
||~
||
2
2 =−
Market
MarketModel
F
FF
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Forward Curves and Options Modeling
Oil Market
%05.1||||
||~
||
2
2 =−
Market
MarketModel
C
CC%03.1
||||
||~
||
2
2 =−
Market
MarketModel
P
PP
Model to Market Comparison
Options Prices
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Probability Distributions
Oil Market
PDF for Spots
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Probability Distributions
Oil Market
CDF for Spots
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Probability Distributions
Oil Market
QF for Spots
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Gas Market
Static Modeling
• Market: TTF Natural Gas
• Trade Date: 05-Sep-2011
• Contracts:
• 18 Contracts:
− 1 Day-Ahead
− 3 Monthly
− 4 Quarterly
− 6 Seasonal
− 4 Calendar
Gas Market
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Gas Market
Forward Curves Modeling
Model to Market ComparisonConfiguration: 2nd Degree Legendre Polynomial, 3rd Degree Trig Polynomial
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Gas Market
Forward Curves Modeling
Daily Forward CurveConfiguration: 2nd Degree Legendre Polynomial, 3rd Degree Trig Polynomial
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Probability Distributions
PDF for Spots
Gas Market
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Probability Distributions
CDF for Spots
Gas Market
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Probability Distributions
QF for Spots
Gas Market
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Power Market
Static Modeling
• Market: EEX Phelix Peak
• Trade Date: 06-Sep-2011
• Contracts:
• 12 Contracts:
− 1 Day-Ahead
− 2 Weekly
− 3 Monthly
− 3 Quarterly
− 3 Calendar
Power Market
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Model to Market Comparison
Configuration: 2nd Degree Legendre Polynomial, 6th Degree Trig Polynomial
Power Market
Forward Curves Modeling
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Daily Forward Curve
Configuration: 2nd Degree Legendre Polynomial, 6th Degree Trig Polynomial
Power Market
Forward Curves Modeling
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Probability Distributions
PDF for Spots
Power Market
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Probability Distributions
CDF for Spots
Power Market
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Probability Distributions
QF for Spots
Power Market
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Conclusions
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Conclusions
The Non-Markovian Approach:
• Unified modellig framework applicable across the instruments, commodities, regions and time periods
• The interpolation and extrapolation of market energy forward curves• Extracting not only historical but also forward-looking market-implied risk-neutral
probability distributions• Static and dynamic modeling of the energy forward curves• Historical and forward-looking market-implied risk-neutral Monte Carlo simulations • Can be viewed as a hybrid model
Applications:
• Risk Management • Trading • Physical Assets Valuation • Retail• Portfolio Optimization• Model Building, Implementation and Maintenance• Strategic Decision Making
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Conclusions
I thank my friends and former colleagues from Platts Analytics, Reliant Resources, TXU Energy Trading, Integrated Energy Services and Center for Quantitative Risk Analysis at Middle Tennessee State University for their attention to this work.
I thank my friends and colleges from Verbund for their warm welcome and attention to this talk.
I thank the organizers of the Energy Finance Seminar for their kind invitation to give this talk.
I thank my wife Larisa and my sons Nikita and Ilya for their love, patience and care.
I also thank Simon Eberle for his help with the computations.
Acknowledgements
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References
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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.• D. Duffie and S. Gray, Volatility in Energy Prices, In Managing Energy Price Risk, Risk Publications, London, UK, 1995.• V. Kholodnyi and J. Price, Foreign Exchange Option Symmetry, World Scientific, River Edge, New Jersey, 1998.• V. 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 and S. Burchett, Energy Operational Risk, Risk Publications, London, UK, To Appear.• 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, Valuation and Hedging of European Contingent Claims on Power with Spikes: a Non-Markovian Approach, Journal of Engineering
Mathematics, 49(3) (2004) 233-252.• V. Kholodnyi, Modeling Power Forward Prices for Power with Spikes: a Non-Markovian Approach, Journal of Nonlinear Analysis, 63 (2005) 958-
965.• V. Kholodnyi, Valuation and Hedging of Contingent Claims on Power with Spikes: a Non-Markovian Approach, Journal of Derivatives: Use,
Trading and Regulation, 11(4) (2006) 308-333.• V. Kholodnyi, The Non-Markovian Approach to the Valuation and Dynamic Hedging of Contingent Claims on Power with Spikes, International
Journal of Ecology and Development, 5(6) (2006) 44-62.• V. Kholodnyi, The Non-Markovian Approach to the Valuation and Hedging of European Contingent Claims on Power with Scaling Spikes,
Journal of Nonlinear Analysis: Hybrid Systems, 2(2) 2008, 285-309.• V. Kholodnyi, Modeling Power Forward Prices for Power Spot Prices with Trends and Spikes in the Framework of the Non-Markovian Approach,
In S. Sambandham and etc, Editors, Proceedings of the 5th International Conference on Dynamic Systems and Applications, Dynamic Publishers, 2008.
• V. Kholodnyi and N. Kholodnyi, Numerical Investigation of the Implied Volatility for European Call and Put Options on Forwards on Power with Spikes in the Framework of the Non-Markovian Approach, In S. Sambandham and etc, Editors, Proceedings of the 5th International Conference on Dynamic Systems and Applications, Dynamic Publishers, 2008.
• V. Kholodnyi, The non-Markovian approach to the valuation and hedging of European contingent claims on power with spikes of Pareto distributed magnitude, In Advances in Mathematical Problems in Engineering, Aerospace and Sciences, S. Sivasundaram (ed.), Cambridge Scientific Publishers, Cambridge, UK, pp. 275–308, 2008.
• V. Kholodnyi, Modeling power forward prices for power spot prices with upward and downward spikes in the framework of the non-Markovianapproach, Journal of Mathematics in Engineering, Science and Aerospace, Vol. 2, No. 2, pp. 105-120, 2011.
• V. Kholodnyi, Modeling Power Forward Prices for Positive and Negative Power Spot Prices with Upward and Downward Spikes in the Framework of the Non-Markovian Approach, In F. Benth, V. Kholodnyi and P. Laurence, Editors, Quantitative Energy Finance, Springer, New York, 2013.
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• V. Kholodnyi, A Nonlinear Partial Differential Equation for American Options in the Entire Domain of the State Variable, Journal of Nonlinear Analysis, 30 (8) (1997) 5059-5070.
• V. Kholodnyi, A Semilinear Evolution Equation for General Derivative Contracts, In J.F. Price, Editor, Derivatives and Financial Mathematics, Nova Science Publishers, Inc., Commack, New York, 1997, 119 –138.
• V. Kholodnyi, Universal Contingent Claims in a General Market Environment and Multiplicative Measures: Examples and Applications, Journal of Nonlinear Analysis, 62 (2005) 1437-1452.
• V. Kholodnyi, Universal Contingent Claims in a General Market Environment and Multiplicative Measures, Proceedings of the 4th International Conference on Dynamical Systems and Applications, Dynamic Publishers, Atlanta, Georgia, 2005, 206-212.
• V. Kholodnyi, The Semilinear Evolution Equation for Universal Contingent Claims: Examples and Applications, In R.P Agarwal and K. Perera, Editors, Proceedings of the Conference on Differential and Difference Equations and Applications, Hindawi, New York, 2006, 519 -528.
• V. Kholodnyi, Universal Contingent Claims and Valuation Multiplicative Measures with Examples and Applications, Journal of Nonlinear Analysis, 69 (3) (2008) 880-890.
• V.A. Kholodnyi, The Semilinear Evolution Equation for Universal Contingent Claims in a General Market Environment with Examples and Applications, In S. Sivasundaram and etc, Editors, Advances in Dynamics and Control: Theory, Methods and Applications, Cambridge Scientific Publishers, Cambridge, UK, 2011, 107 - 121.
• V. Kholodnyi, Valuation and Dynamic Replication of Contingent Claims in the Framework of the Beliefs-Preferences Gauge Symmetry,European Physical Journal B, 27 (2) (2002) 229-238.
• V. Kholodnyi, Beliefs-Preferences Gauge Symmetry and Dynamic Replication of Contingent in a General Market Environment, Journal of the Dynamics of Continuous, Discrete and Impulsive Systems B, 10 (1) (2003) 81-94.
• V. Kholodnyi and M. Lukic, Random Field Formulation for the Term Structure of Interest Rates, In J.F. Price, Editor, Derivatives and Financial Mathematics, Nova Science Publishers, Inc., Commack, New York, 1997, 139-143.
• V. Kholodnyi, A Nonlinear Partial Differential Equation for American Options in the Entire Domain of the State Variable, IES Preprint, 1995.• 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, A Non-Markovian Process for Power Prices with Spikes and Valuation of Contingent Claims on Power, Preprint, TXU Energy
Trading, 2000.• V. Kholodnyi, Modeling Power Forward Prices for Power with Spikes, TXU Preprint, 2000. • V. Kholodnyi, Valuation and Dynamic Replication of Contingent Claims on Power with Spikes in the Framework of the Beliefs-Preferences
Gauge Symmetry, Preprint, TXU Energy Trading, 2000.• V. Kholodnyi, Valuation of a Swing Option in the Mean-Reverting Market Environment, Preprint, TXU Energy Trading, 2000.• V. Kholodnyi, Valuation of a Swing Option on Power with Spikes, Preprint, TXU Energy Trading, 2000.• V. Kholodnyi, Valuation of a Spark Spread Option on Power with Spikes, Preprint, TXU Energy Trading, 2000.• V. Kholodnyi, Valuation of European Contingent Claims on Power at Two Distinct Points on the Grid with Spikes in Both Power Prices, Preprint,
TXU Energy Trading, 2000.• V. Kholodnyi, Valuation of a Transmission Option on Power with Spikes, Preprint, TXU Energy Trading, 2000.