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1

Valuation and Hedging of Power-Sensitive Contingent Claims for Power with Spikes:

a Non-Markovian Approach

Return to Risk Limited website: www.RiskLimited.com

Valery A. Kholodnyi

February 25, 2004

Houston, Texas

http://www.risklimited.com/

2

Introduction

As the power markets are becoming deregulated worldwide, the modeling of the dynamics of power spot prices is becoming one of the key problems in the risk management, physical assets valuation, and derivative pricing.

One of the main difficulties in this modeling is to combine the following features:

• To provide a mechanism that allows for the absence of spikes in the prices of power-sensitive contingent claims while the power spot prices exhibit spikes, and

• To keep the dynamics of the prices of power-sensitive contingent claims consistent with the dynamics of the power spot prices.

3

Models for Power Spot Prices with Spikes

• Mean-Reverting Jump Diffusion Process (Ethier and Dorris, 1999; Clewlow, Strickland and Kaminski, 2000)

– the same mechanism is responsible for both the decay of spikes and the reversion of power prices to their equilibrium mean

• Mixture of Processes (Goldberg and Read, 2000; Ball and Torous, 1985)

– spikes and the regular, that is, inter-spike regime do not persist in time– relatively difficult to estimate parameters

• Regime Switching Process (Ethier, 1999; Duffie and Gray 1995)

– discreet time regime switching– inconsistent short term option values– relatively difficult to estimate parameters

4

The Non-Markovian Process for Power Spot Prices with Spikes

Motivation

• Different mechanisms should be responsible for:

– the reversion of power prices to their equilibrium mean in the regular, that is, inter-spike state

– the reversion of power prices to their long term mean in the spike state, that is, for the decay of spikes

• This is, in our opinion, due to the substantial difference in the scales of the deviations of power prices from their equilibrium mean in the spike and inter-spike states

– For example, power prices in the US Midwest in June 1998 rose to $7,500 per megawatt hour (MWh) compared with typical prices of around $30 per MWh

5


Main Features

• The spikes are modeled directly as self-reversing jumps, either multiplicative or additive, in continuous time

• The parameters that characterize spikes are frequency, duration, and magnitude

• The spikes parameters are directly observable from market data as well as admit structural interpretation

• The spike state and the regular, that is, inter-spike state do persist in time

6


Formal Definition

Define (Kholodnyi, 2000) the non-Markovian process for the power spot prices with spikes by

• t>0 is the power spot price at time t ,

• is the multiplicative magnitude of spikes at time t ,

• is the inter-spike power spot price at time t.

Assume that the spike process and inter-spike process are independent Markov processes.

,ˆttt

1t

0ˆ t

tt̂

7

Underlying Two-State Markov Process

Denote by Mt a two-state Markov process with continuous time t 0.

Denote the 22 transition matrix for the two-state Markov process Mt by

• Pss(T,t) and Prs(T,t) are transition probabilities from the spike state at time t to the spike and regular states at time T, and

• Psr(T,t) and Prr(T,t) are transition probabilities from the regular state at time t to the spike and regular states at time T.


),(),(

),(),(),(

tTPtTP

tTPtTPtTP

rrrs

srss

8

Generators of the Underlying Two-State Markov Process

The family of 22 matrices L = {L(t) : t 0} defined by

is said to generate the two-state Markov process Mt, and the 22 matrix

is called a generator.

In terms of the generators, P(T,t) is given by

( ) ( , ) ,T tdL t P T tdT

.)(

),(

Tt

dLetTP


)()(

)()()(

tLtL

tLtLtL

rrrs

srss

9

Decompositions of the Transition Probabilities of the Underlying Two-State Markov Process

It can be shown that

Moreover

where


.)(),(),(

,)(),(),(

')'(

')'()(

deLtPtTP

deLtPetTP

Tss

Tss

Tt ss

dLsr

T

t rrsr

dLsr

T

t rsdL

ss

),,(),(),( tTPtTPtTP rss

sssss

deLtPtTP

etTPT

ss

Tt ss

dLsr

T

t rsrss

dLsss

')'(

)(

)(),(),(

),(

10

Underlying Two-State Markov Process in the Time-Homogeneous Case

In the special case of a time-homogeneous two-state Markov process Mt the transition matrix P(T-t) and the generator L are given by

and

( )( ) ( )( )

( )( ) ( )( )( )

T t a b T t a b

T t a b T t a b

b ae b be

a b a b

a ae a be

a b a b

P T t

a bLa b


11

Construction of the Spike Process

t

Mt

Spike State Regular State

(t,

)

Time

Time

1

st rt


12

Formal Definition of the Spike Process

The transition probability density function for the spike process t as a Markov process is given by

where (x) is the Dirac delta function.


1 if)1(),(

)(),(),(

1 if

)1(),(

)(),(),(

)(

),,,(

')'(

')'(

)(

t

Trr

T

t

dL

srrrT

t

Trs

T

t

dL

srrsT

Tt

dL

Tt

tTP

deLtP

tTP

deLtP

e

TtT

ss

T

ss

T

tss

13

Inter-Spike Process

For example, can be a diffusion process defined by

where: is the drift, is the volatility, and Wt is the Wiener process.

In the practically important special case of a geometric-mean reverting process we have

where: is the mean-reversion rate, is the equilibrium mean, and is the volatility.


t̂

,),ˆ(),ˆ(ˆtttt dWtdttd

,ˆ)(ˆ)ˆln)()((ˆttttt dWtdtttd

),ˆ( tt 0),ˆ( tt

0)( t )(t0)( t

14

The Expected Time for t to be in the Spike and Inter-Spike States

The expected time for t to be in the spike state that starts at time t is:

Similarly, the expected time for t to be in the inter-spike state that starts at time t is:

In the special case of a time-homogeneous two-state Markov process Mt:

st

.)()(')'(

dbett

t

db

rt

rt

.)()(')'(

daett

t

da

st

./1/1 btat rs and


15

Interpretation of the Spike State of t as Spikes in Power Prices

If the expected time for the non-Markovian process t to be in the spike state is small relative to the characteristic time of change of the process then the spike state of t can be interpreted as spikes in power spot prices:

– t can exhibit sharp upward price movements shortly followed by equally sharp downward prices movements of approximately the same

magnitude.

For example, if is a diffusion process then:

and

In this case is the expected lifetime of a spike and is the expected lifetime between two consecutive spikes.

st

t̂

t̂

1),ˆ(2 stt .1),ˆ( stt

st rt


16

Estimation of the Spike Parameters

• In the special case of a time-homogeneous two-state Markov process the expected life-time of a spike is given by

• Similarly, the expected life-time between two consecutive spikes is given by

• The estimation of the probability density function (t,) for the spike magnitude can be based on the standard parametric or nonparametric statistical methods

– Scaling and asymptotically scaling distributions are of a particular interest in practice

ats /1

./1 btr


17


The Non-Markovian Process t as a Markov Process with the Extended State Space

The state of the power market at any time t can be fully characterized by a pair of the values of the processes , and at time t.

Moreover, although the process t is non-Markovian it can be, in fact, represented as a Markov process that for any time t can be fully characterized by the values of the processes and at time t.

Equivalently, the non-Markovian process t can be represented as a Markov process with the extended state space that at any time t consists of all possible pairs with and .

t̂t

t t̂

)ˆ,( tt 1t 0ˆ t

18

Valuing European Contingent Claims on Power as the Discounted Risk-Neutral expected value of its payoff

Denote by

the value of the European contingent claim on power with inception time t, expiration time T, and payoff g.

The value of this European contingent claim can be found as the discounted risk-neutral expected value of its payoff:

where is the risk-neutral transition probability density function.

)ˆ,ˆ,,( TtTtP

European Contingent Claims onPower in the Absence of Spikes

,ˆ)ˆ()ˆ,ˆ,,( )ˆ)(,,(ˆ0

)(

TTTt

dr

t dgTtPegTtE

T

t

),ˆ)(,,(ˆ),,(ˆtgTtEgTtE

19

Example: Geometric Mean-Reverting Process

It can be shown (Kholodnyi 1995) that

where:


.))(

)(

2

1)()((),(

,),(

,)(1

),(ˆ

,)(1

),(

')'(2

)(

')'(22

T

t

d

d

dT

t

T

t

deTtb

eTta

detT

Tt

drtT

Ttr

T

T

t

T

).ˆ)(,,(ˆ

ˆ

ˆ)ˆ(

)(2),(ˆ)ˆ)(,,(ˆ

),(ˆ2

1)(

),(),(0),,(ˆ

0

))(,(ˆ

)ˆln),(ˆln),((

2

1))(,(

2

2

2

TttTTtbTta

tBS

Tt

T

TT

tTTt

TtbTtatTTtr

tMR

eegTtE

dge

tTTt

egTtE

Tt

20

Example: Geometric Mean-Reverting Process

For example (Kholodnyi 1995):

where:

with:


),,ˆ,,(ˆ),ˆ,,(ˆ

),,ˆ,,(ˆ),ˆ,,(ˆ

),(ˆ2

1)(),(),(

0),,(ˆ

),(ˆ2

1)(),(),(

0),,(ˆ

2

2

XeeTtPXTtP

XeeTtCXTtC

TttTTtbTtat

BSTtt

MR

TttTTtbTtat

BSTtt

MR

),()(),,,(ˆ

),()(),,,(ˆ

)))(,(())(,(,

))(,()))(,((,

dNeSdNXeXSTtP

dNXedNeSXSTtC

tTTtrt

tTTtrt

BS

tTTtrtTTtrtt

BS

BS

BSBS

BS

BSBS

.2

1)(

,(

))(2

1()/ln(

2/

2

2

dyexN

tT

tTXSd

x y

BS

BSBSt

21

Notation

Denote by

the value of the European contingent claim on power with inception time t, expiration time T, and payoff

The payoff g can explicitly depend, in addition to the power price at time T, on the state, spike or inter-spike state, of the power price and the magnitude of the related spike.

If g depends only on the power price at time T we have

European Contingent Claims onPower in the Presence of Spikes

),ˆ)(,,()ˆ)(,,(),,( tttt gTtEgTtEgTtEt

).,ˆ()ˆ( TTTT gggT

).ˆ()ˆ( TTT ggT

22

General Case

The value E(t,T,g) can be found as the discounted risk-neutral expected value of the payoff g

where

is the the transition probability density function for t represented as a Markov process.


,)ˆ)(,,(ˆ),,,(

ˆ)ˆ(),,,()ˆ,ˆ,,()ˆ)(,,(

1

0 1

)(

TtTt

TTTTtTt

dr

t

dgTtETt

ddgTtTtPegTtE

T

T

T

t

t

),,,()ˆ,ˆ,,( TtTt TtTtP

23

The Case When (t,) is Time-Independent

The value E(t,T,g) is given by


1)ˆ)(,,(ˆ),(

)ˆ)(,,(ˆ)(),(

1

)ˆ)(,,(ˆ),(

)ˆ)(,,(ˆ)(),(

)ˆ)(,,(ˆ),(

)ˆ)(,,(

1

1

1

1

t

trr

TtTsr

t

trs

TtTrss

tsss

t

T

T

T

T

t

t

gTtEtTP

dgTtEtTP

gTtEtTP

dgTtEtTP

gTtEtTP

gTtE

if

if

24

The Case of Spikes with Constant Magnitude

Consider a special case of spikes with constant magnitude > 1, that is, when () is the delta function (- `).

The value E(t,T,g) is given by


)ˆ)(,,(ˆ),()ˆ)(,,(ˆ),()ˆ)(,,(

)ˆ)(,,(ˆ),()ˆ)(,,(ˆ),()ˆ)(,,(

11

1

trrtsrt

trstsst

gTtEtTPgTtEtTPgTtE

gTtEtTPgTtEtTPgTtE

t

t

25

Linear Evolution Equation for European Contingent Claims on Power with Spikes

It can be shown (Kholodnyi 2000) that the value E(t,T,g) of a European contingent claim on power with spikes is the solution of the following linear evolution equation

where and are the generators of and as Markov processes.


gTv

TtvtrvtvtLvdt

d

)(

,,0)()()(ˆ

)(ˆ tL )(tt̂ t

26


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:


.ˆ

ˆ)ˆln)()((ˆ

ˆ)(2

1)(ˆ

2

222

ttttL

)(ˆ tLt̂

1)1()()(),(

1)1()()()(),,(

''

'''

ttrrsrt

ttrsttsstt

tLtLt

tLtLt

if

if

)(t

27


In the special case of spikes with constant magnitude the generator (t) can be represented as the 22 matrix L*(t) transposed to the generator L(t) of the Markov process Mt.

In turn, v and g can be represented as two-dimensional vector functions

Note that (t) represented as L*(t) can also be expressed in terms of the Pauli matrices. This gives rise to an analogy between the linear evolution equation for E(t,T,g) and the Schrodinger equation for a nonrelativistic spin 1/2 particle.


. and

11 ),,(

),,()(

T

T

t

t

g

gg

gTtE

gTtEtv

28

Ergodic Transition Probabilities for Mt

Assume that the spikes have constant magnitude and the underlying two-state Markov process Mt is time-homogeneous.

The transition probabilities for Mt can be represented as follows:

Pss(T,t) = s + O(e-(T - t)a), Psr(T,t) = s + O(e-(T - t)a),

Prs(T,t) = r + O(e-(T - t)a), Prr(T,t) = r + O(e-(T - t)a),

where:

s = b/(a + b) and r = a/(a + b)

are the ergodic transition probabilities.

Why Prices of European ClaimsOn Power Do Not Spike

29

Values of European Contingent Claims on Power Far From Expiration

The values Et=(t,T,g) and Et=1(t,T,g) of European

contingent claims on power coincide up to the terms of order O(e-(T - t)a) and hence can be combined into a single expression as follows (Kholodnyi 2000):

When T - t >> , Et=(t,T,g) and Et=1(t,T,g) differ

only by an exponentially small term.

As a result, prices of European contingent claims on power do not exhibit spikes while the power spot prices do.

ats /1

),()ˆ)(,,(ˆ)ˆ)(,,(ˆ)ˆ)(,,( )(1

atTtrtst eOgTtEgTtEgTtE


30

Values of European Contingent Claims on Power Far From Expiration

For example, (Kholodnyi 2000) the values of European call and put options with inception time t, expiration time T, and strike X are given by:


).(),ˆ,,(ˆ),ˆ,,(ˆ),ˆ,,(

),(),ˆ,,(ˆ),ˆ,,(ˆ),ˆ,,()(1

)(1

atTtrtst

atTtrtst

eOXTtPXTtPXTtP

eOXTtCXTtCXTtC

31

Example: Geometric Mean-Reverting Inter-Spike Process

It can be shown (Kholodnyi 2000) that the value E(t,T,g) of a European options with inception time t , expiration time T, and payoff g is given by

where:


),()ˆ)(,,(ˆ

)ˆ)(,,(ˆ)ˆ)(,,()(

1atT

tMR

r

tMR

st

eOgTtE

gTtEgTtE

).ˆ)(,,(ˆ)ˆ)(,,(ˆ ),(ˆ2

1)(

),(),(0),,(ˆ

2 TttTTtbTta

tBS

TttMR eegTtEgTtE

32


For example, (Kholodnyi 2000) the values of European call and put options with inception time t , expiration time T, and strike X are given by

where


),(),ˆ,,(ˆ),ˆ,,(ˆ),ˆ,,(

),(),ˆ,,(ˆ),ˆ,,(ˆ),ˆ,,()(1

)(1

atTt

MRrt

MRst

atTt

MRrt

MRst

eOXTtPXTtPXTtP

eOXTtCXTtCXTtC

).,ˆ,,(ˆ),ˆ,,(ˆ

),,ˆ,,(ˆ),ˆ,,(ˆ

),(ˆ2

1)(

),(),(0),,(ˆ

),(ˆ2

1)(

),(),(0),,(ˆ

2

2

XeeTtPXTtP

XeeTtCXTtC

TttTTtbTta

tBS

TttMR

TttTTtbTta

tBS

TttMR

33

Short-Lived Spikes

Consider the case of short-lived spikes, that is .

Then for the ergodic transition probabilities we have

s = tch + o(tch) and r = 1 - tch + o(tch),

where

In turn, the value E(t,T,g) can be expressed as a correction to the value Ê(t,T,g):

rs tt

.// rsch ttabt

).()),,(ˆ),,(ˆ(),,(ˆ),,( 11 chch togTtEgTtEtgTtEgTtE


),,(ˆ gTtE ),,(ˆ gTtE ),,(ˆ gTtE

34


It can be shown (Kholodnyi 2000) that the values of European call and put options with strike X are given by

where


),()),ˆ,,(ˆ),ˆ,,(ˆ(

),ˆ,,(ˆ),ˆ,,(

),()),ˆ,,(ˆ),ˆ,,(ˆ(

),ˆ,,(ˆ),ˆ,,(

1

1

chtMR

tMR

ch

tMR

t

chtMR

tMR

ch

tMR

t

toXTtPXTtPt

XTtPXTtP

toXTtCXTtCt

XTtCXTtC

).,ˆ,,(ˆ),ˆ,,(ˆ

),,ˆ,,(ˆ),ˆ,,(ˆ

),(ˆ2

1)(

),(),(0),,(ˆ

),(ˆ2

1)(

),(),(0),,(ˆ

2

2

XeeTtPXTtP

XeeTtCXTtC

TttTTtbTta

tBS

TttMR

TttTTtbTta

tBS

TttMR

35

Power Forward Prices as Risk-Neutral Expected Power Spot Prices

Denote by

the power forward price at time t for the forward contract with maturity time T.

Power forward price can be found as the risk-neutral expected value of the power spot prices at time T:

),ˆ)(,(ˆ),(ˆtTtFTtF

T̂

),(ˆ TtF

.ˆˆ)ˆ,ˆ,,()ˆ)(,(ˆ0

TTTtt dTtPTtF

Power Forward Prices for PowerSpot Prices Without of Spikes

36


It can be shown (Kholodnyi 1995) that power forward prices are given by the following analytical expression:

where:

,ˆ)ˆ)(,(ˆ ),(),(),(ˆ2

1)( 2

Ttat

TtbTttT

t eeTtF

.))(

)(

2

1)()((),(

,),(

,)(1

),(ˆ

')'(2

)(

')'(22

T

t

d

d

dT

t

deTtb

eTta

detT

Tt

T

T

t

T


37

Example: Geometric Brownian Motion (GBM) for Power Forward Prices

The risk-neutral dynamics of is described by a geometric Brownian motion:

where:


.)()()(

ˆ

T

td

Fett

,)(),(ˆ),(ˆˆ dWtTtFTtFdF

),(ˆ TtF

38

General Case

Denote by

the power forward price at time t for the forward contract with maturity time T.

Power forward price F(t,T) can be found as the risk-neutral expected value of the power spot prices T at time T:

where is the risk-neutral average magnitudes of spikes

Power Forward Prices for PowerSpot Prices With Spikes

),ˆ)(,()ˆ)(,(),( tttt TtFTtFTtFt

),ˆ)(,(ˆ),(

ˆ)ˆ)(,,,()ˆ,ˆ,,()ˆ)(,(0 1

t

TTTTTtTtt

TtFTt

ddTtTtPTtF

t

t

),( Ttt

.),,,(),(1 TTTt dTtTt

t

39

The Case When (t,) is Time-Independent

The risk neutral average magnitude of spikes is given by

where is the risk-neutral conditional average magnitude of spikes given by

For example, if () is corresponds to a scaling probability distribution, that is, () = -1- , then


1),(),(

1),(),(),(),(

trrsr

trsrsst

sss

tTPtTP

tTPtTPtTPTt

t

if

if

.')'(1

d

.1,1

40

The Case of Spikes with Constant Magnitude

Consider a special case of spikes with constant magnitude > 1, that is, when () is the delta function (- `).

The risk neutral average magnitude of spikes is given by


1),(),(

),(),(),(

trrsr

trsss

tTPtTP

tTPtTPTt

t

if

if

41

Ergodic Transition Probabilities for Mt

Assume again that the spikes have constant magnitude and the underlying two-state Markov process Mt is time-homogeneous.

The transition probabilities for Mt can be represented as follows:

Pss(T,t) = s + O(e-(T - t)a), Psr(T,t) = s + O(e-(T - t)a),

Prs(T,t) = r + O(e-(T - t)a), Prr(T,t) = r + O(e-(T - t)a),

where:

s = b/(a + b) and r = a/(a + b)

are the ergodic transition probabilities.

Why Power Forward Prices Do Not Spike

42

Ergodic Average Magnitude of Spikes

The risk-neutral average magnitudes of spikes and coincide up to the terms of order O(e-(T - t)a).

Therefore, and can be combined into a single expression as follows:

where is the risk-neutral ergodic average magnitude of spikes given by


.rserg

erg

),( Ttt

),(1 Ttt

),( Ttt ),(1 Tt

t

),(),( )( atTerg eOTt

43

Power Forward Prices far From Maturity

The power forward prices Ft=(t,T) and Ft=1(t,T) coincide up

to the terms of order O(e-(T - t)a).

Therefore, Ft=(t,T) and Ft=1(t,T) can be combined into a

single expression as follows:

When T - t >> , Ft=(t,T) and Ft=1(t,T) differ only by

an exponentially small term.

As a result, power forward prices do not exhibit spikes while the power spot prices do.


).(),(ˆ),( )( atTerg eOTtFTtF

ats /1

44

Short-Lived Spikes

Consider the case of short-lived spikes, that is .

Then for the ergodic transition probabilities we have

s = tch + o(tch) and r = 1 - tch + o(tch),

where

For the average magnitude of spikes we have

In turn, F(t,T) can be expressed as a correction to


rs tt

.// rsch ttabt

).()1(1),( chch totTt ),(ˆ TtF

).(),(ˆ)1(),(ˆ),( chch toTtFtTtFTtF

45

Example: GBM for Power Forward Prices

Assume that the power forward prices follow a geometric Brownian motion.– this is, for example, the case when the power spot prices follow a geometric mean-reverting process.

Then power forward prices F(t,T) far from maturity also follow the same geometric Brownian motion.

This, for example, can be used for:

• the estimation of the volatility for the geometric Brownian motion for ,

• the estimation of the volatility and the mean-reversion rate for the geometric mean-reverting process for , and

• dynamic hedging of derivatives on forwards on power.

),(ˆ TtF

t̂

),(ˆ TtF

t̂


46

European Contingent Claims on Forwards on Power with Spikes

Geometric Mean-Reverting Inter-Spike Process and Spikes with Constant Magnitude

It can be shown (Kholodnyi 2000) that the value of a European contingent claim (on forwards on power for power with spikes) with inception time t, expiration time T, and payoff g is given by:

).()/)(,,(ˆ

)/)(,,(ˆ))(,,(

)(10),,(ˆ

0),,(ˆ

atTerg

BSTts

ergBS

Tts

eOFgTtE

FgTtEFgTtE

47


Geometric Mean-Reverting Inter-Spike Process and Spikes with Constant Magnitude

For example, (Kholodnyi 2000) the values of European call and put options (on forwards on power for power with spikes) with inception time t, expiration time T, and strike X are given by:

).(),,,(ˆ)/1(

))/(,,,(ˆ)/(),,,(

),(),,,(ˆ)/1(

))/(,,,(ˆ)/(),,,(

)(0),,(ˆ

10),,(ˆ

)(0),,(ˆ

10),,(ˆ

atTerg

BSTtergr

ergBS

Ttergs

atTerg

BSTtergr

ergBS

Ttergs

eOXFTtP

XFTtPXFTtP

eOXFTtC

XFTtCXFTtC

48


Geometric Mean-Reverting Inter-Spike Process and Short-Lived Spikes with Constant Magnitude

It can be shown (Kholodnyi 2000) that the value of a European contingent claim (on forwards on power for power with spikes) with inception time t, expiration time T, and payoff g can be represented as the following correction:

).())(,,()1(

))(,,(ˆ

))(,,(ˆ))(,,(

10),,(ˆ

10),,(ˆ

10),,(ˆ

chBS

Tt

BSTtch

BSTt

toFgTtF

FggTtEt

FgTtEFgTtE

49


Geometric Mean-Reverting Inter-Spike Process and Short-Lived Spikes with Constant Magnitude

For example, (Kholodnyi 2000) the values of European call and put options (on forwards on power for power with spikes) with inception time t, expiration time T, and strike X can be represented as the following corrections:

).(),,,()1(

),,,(ˆ),,,(ˆ

),,,(ˆ),,,(

),(),,,()1(

),,,(ˆ),,,(ˆ

),,,(ˆ),,,(

0),,(ˆ,

0),,(ˆ1

0),,(ˆ

0),,(ˆ

0),,(ˆ,

0),,(ˆ1

0),,(ˆ

0),,(ˆ

chBS

Ttp

BSTt

BSTtch

BSTt

chBS

Ttc

BSTt

BSTtch

BSTt

toXFTtF

XFTtPXFTtPt

XFTtPXFTtP

toXFTtF

XFTtCXFTtCt

XFTtCXFTtC

50

Extensions of the Model

• Both positive and negative spikes as well as spikes of more complex shapes can be considered• European contingent claims on power with spikes and another commodity that does not exhibit spikes can also be valued. Those include fuel and weather sensitive derivatives such as spark spread options and full requirements contracts • European options on power at two distinct points on the grid with spikes in both power prices can also be valued. Those include transmission options• Contingent claims of a general type such as universal contingent claims on power with spikes can be valued with the help of the semilinear evolution equation for universal contingent claims (Kholodnyi, 1995). Those include Bermudan and American options.

51

Acknowledgements

I thank my friends and former colleagues from Reliant Resources, TXU Energy Trading, and Integrated Energy Services for their attention to this work.

I thank my friends and colleges from the College of Basic and Applied Sciences, in general, and the Department of Mathematical Sciences and the Center for Quantitative Risk Analysis, in particular, of Middle Tennessee State University for their warm welcome and attention to this presentation.

I thank the organizers of the Energy Finance and Credit Summit 2004 for their kind invitation and support of this presentation.

I thank my wife Larisa and my son Nikita for their love, patience and care.

52

References

• R. Ethier and G. Dorris, Do not Ignore the Spikes, EPRM, July-August, 1999, 31-33.• L. Clewlow, C. Strickland and V. Kaminski, Jumping the Gaps, EPRM, December, 2000, 26-27.• R. Goldberg and J. Read, Dealing with a Price-Spike World, EPRM, July-August, 2000, 39-41.• C. Ball and W. Torous, On Jumps in Common Stock Prices and Their Impact on Call Option Pricing, Journal of Finance, XL (1), March 1985, 155-173• R. Ethier, Estimating the Volatility of Spot Prices in restructured Electricity Markets and the Implications for Option Values, Cornell University, 1999.• D. Duffie and S. Gray, Volatility in Energy Prices, In Managing Energy Price Risk, Risk Publications, London, UK, 1995.• V. Kholodnyi, Introduction to the Beliefs-Preferences Gauge Symmetry, Elsevier Science, Amsterdam, Holland, 2003, To appear.• V. Kholodnyi and J. Price, Foreign Exchange Option Symmetry, World Scientific, River Edge, New Jersey, 1998.• V.A Kholodnyi and J.F. Price, Foundations of Foreign Exchange Option Symmetry, IES Press, Fairfield, Iowa, 1998.• V. Kholodnyi, Beliefs-Preferences Gauge Symmetry Group and Dynamic Replication of Contingent Claims in a General Market Environment, IES Press, Research Triangle Park, North Carolina, 1998.• V. Kholodnyi, A Non-Markov Method, EPRM, March, 2001, 20-24.• V. Kholodnyi, Analytical Valuation in a Mean-Reverting World, EPRM, August, 2001, 40-45.• V. Kholodnyi, Analytical Valuation of a Full Requirements Contract as a Real Option by the Method of Eigenclaims, In E. I. Ronn, Editor, Real Options and Energy Management, Risk Publications, 2002.• V. Kholodnyi, On the Linearity of Bermudan and American Options in Partial Semimodules, IES Preprint, 1995.• V. Kholodnyi, The Stochastic Process for Power Prices with Spikes and Valuation of European Contingent Claims on Power, TXU Preprint, 2000.• V. Kholodnyi, Modeling Power Forward Prices for Power with Spikes, TXU Preprint, 2001.

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