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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


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


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


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


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


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


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


Methodology


Then is a geometric mean-reverting process:

where:


• is the log long-term mean,



txt es ˆˆ =

,)()ˆln)(ˆ)((ˆln ttt dWtdtsttsd σµη +−=

0)( >tη

)(ˆ tµ

0)( >tσ

Motivation


Methodology


What if

where is an arbitrary representation function associated with .

)ˆ(ˆˆttt xρ=Ψ

)ˆ(ˆ xtρ tx

Motivation


Methodology


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


Methodology


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


A Component

B Component

Motivation


Methodology


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


A Component

B Component

Motivation


Methodology


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


Methodology



where:





)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


Methodology


Representation function for A = 1, B = 0.1, α = 1, and β =1:)ˆ(,,, sBA βαρ


-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


A Component

B Component

Motivation


Methodology


Inverse representation function for A = 1, B = 0.1, α = 1, and β =1:)ˆ(1,,, Ψ−βαρ BA


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


A Component

B Component

Motivation


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


,),()),(),(( ttttttt dWstsdtstsdtsrds σ+−=

Motivation


Methodology


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


Methodology


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

−−−=

−−−=++

++


Methodology


American call option with X = 50 for r = 0.1, d = 0.1, σ = 0.4

Motivation


Methodology


American put option with X = 50 for r = 0.1, d = 0.0, σ = 0.4

Motivation

© VERBUND AG, www.verbund.com Seite28-Apr-15 24

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


Energy Spot Process

Methodology



where:





)ˆ(),ˆ( ,,, ss BAt λρλρ βα=


ssBA ˆ)ˆ(,,, =βαρ ttt sλ=Ψ

0)ˆ(,,, =∗sBA βαρ )/(1)/(ˆ βα +∗ = ABs


αααρ /1

21

,,, )2

4ˆˆ()ˆ(

A

ABBA

+Ψ+Ψ=Ψ−


Energy Spot Process

Methodology




-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


A Component

B Component


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


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


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

∫

∫

∫=

=


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


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


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


Methodology

Inter-Spike Process

Consider a practically important special case when is a geometric mean-reverting process:

where:


• is the log long-term mean,



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


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


Energy Spot Process

Methodology

Inter-Spike Process


where:





)ˆ()ˆ( ,,, ss BAt βαρρ =


ssBA ˆ)ˆ(,,, =βαρ tt s=Ψ

0)ˆ(,,, =∗sBA βαρ )/(1)/(ˆ βα +∗ = ABs


αααρ /1

21

,,, )2

4ˆˆ()ˆ(

A

ABBA

+Ψ+Ψ=Ψ−


Energy Spot Process

Methodology

Inter-Spike Process



-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


A Component

B Component


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


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


A Component

B Component


Methodology

Inter-Spike Process

It is clear that if is a geometric mean-reverting process then is a mean-reverting process:

where:


• is the long-term mean,



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 +−=


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


Energy Spot Process

Methodology

Inter-Spike Process


where:





)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


Energy Spot Process

Methodology

Inter-Spike Process

Representation function for A = 1, B = 0.1, α = 1, and β =1:)ˆ(ˆ ,,, xBA βαρ


-3

-2

-1

0

1

2

3

-4 -3 -2 -1 0 1 2

x

Rep

rese

nta

tion

Fu

nct

ion


A Component

B Component


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


-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


A Component

B Component


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


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 ==


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


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


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∫∞

∗ =



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

−

∞

=

= ∫



Methodology


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 ∗



Methodology


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



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



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 =



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



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



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:


,)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



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((()( ∑ ++=∞

=



Methodology


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 :


).)

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

++

−+−−++

++

++−+++

+−=−

−+−

−+−

=

−+−

∑



Methodology


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 :


).)

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

+

−−−+

+

+−++

−−−+−=

−−

−−∞

=

−−−−−−

∑



Methodology


For the following typical values of the parameters:



Methodology


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


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:


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



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



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



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



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



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



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



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



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



Methodology


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



Methodology


For the following typical values of the parameters:



Methodology


Forward Prices

-50

0

50

100

150

200

0 1 2 3 4 5

Time

Fo

rwar

d P

pri

ce

With SpikesWithout Spikes



Methodology


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


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:


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


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.


Methodology

gTv

TtvtLvdt

d

=

<=+

)(

,,0)(

),()()()( trtLtLtL sr −+=

)(tLr )(tLs tλts


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:


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


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


Methodology

TgTv

TtvtFvtLvdt

d

=

<=++

)(

,,0),()(

( ).)(),( ννν −

+∂∂−=

+

tt gHtLgt

tF


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


Oil Market

Static Modeling

• Market: Brent Crude Oil

• Trade Date: 05-Jan-2011

• Contracts:

• 69 Contracts:

− 69 Monthly Futures Contracts

Oil Market



Oil Market

Forward Curves Modeling

Model to Market Comparison

Configuration: 8th Degree Legendre Polynomial, 0th Degree Trig Polynomial



Daily Forward Curve

Configuration: 8th Degree Legendre Polynomial, 0th Degree Trig Polynomial


Oil Market



PDF for Spots

Probability Distributions

Oil Market



CDF for Spots


Oil Market



QF for Spots


Oil Market



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



Oil Market

Forward Curves and Options Modeling

Market Data

Futures Contracts



Oil Market

Market Data

Call and Put Options






Forward Curve

Oil Market

%055.0||||

||~

||

2

2 =−

Market

MarketModel

F

FF




Oil Market

%05.1||||

||~

||

2

2 =−

Market

MarketModel

C

CC%03.1

||||

||~

||

2

2 =−

Market

MarketModel

P

PP


Options Prices




Oil Market

PDF for Spots




Oil Market

CDF for Spots




Oil Market

QF for Spots



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


Gas Market


Model to Market ComparisonConfiguration: 2nd Degree Legendre Polynomial, 3rd Degree Trig Polynomial


Gas Market


Daily Forward CurveConfiguration: 2nd Degree Legendre Polynomial, 3rd Degree Trig Polynomial



PDF for Spots

Gas Market



CDF for Spots

Gas Market



QF for Spots

Gas Market


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




Configuration: 2nd Degree Legendre Polynomial, 6th Degree Trig Polynomial

Power Market




Daily Forward Curve

Configuration: 2nd Degree Legendre Polynomial, 6th Degree Trig Polynomial

Power Market





PDF for Spots

Power Market




CDF for Spots

Power Market




QF for Spots

Power Market



Conclusions


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


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



References


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.


References


References

• 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.

energy finance seminar - uni-due.de · • the interpolation and extrapolation of market energy...

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