Valuation and Hedging of PowerSensitive Contingent Claims for Power with Spikes: a NonMarkovian Appr

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Valuation and Hedging of Power-Sensitive
Contingent Claims for Power with Spikes a
Non-Markovian Approach
Valery A. Kholodnyi February 25, 2004 Houston,
Texas
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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.

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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

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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

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The Non-Markovian Process for Power Spot Prices
with Spikes

• Main Features
• The spikes are modeled directly as
  self-reversing jumps, either multiplicative or
  additive, in continuous time
• The parameters that characterize spikes are
  frequency,
• duration, and magnitude
• The spikes parameters are directly observable
  from market data as well as admit structural
  interpretation
• The spike state and the regular, that is,
  inter-spike state do persist in time

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The Non-Markovian Process for Power Spot Prices
with Spikes

• Formal Definition
• Define (Kholodnyi, 2000) the non-Markovian
  process for the power spot prices with spikes by
• ?tgt0 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.

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The Non-Markovian Process for Power Spot Prices
with Spikes

• Underlying Two-State Markov Process
• Denote by Mt a two-state Markov process with
  continuous time t ? 0.
• Denote the 2?2 transition matrix for the
  two-state Markov process Mt by
• Pss(T,t) 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.

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The Non-Markovian Process for Power Spot Prices
with Spikes
Generators of the Underlying Two-State Markov
Process The family of 2?2 matrices L L(t) t
? 0 defined by 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
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The Non-Markovian Process for Power Spot Prices
with Spikes
Decompositions of the Transition Probabilities of
the Underlying Two-State Markov Process It can
be shown that Moreover where
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The Non-Markovian Process for Power Spot Prices
with Spikes
Underlying Two-State Markov Process in the
Time-Homogeneous Case In the special case of a
time-homogeneous two-state Markov process Mt the
transition matrix P(T-t) and the generator L are
given by and
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The Non-Markovian Process for Power Spot Prices
with Spikes
Construction of the Spike Process
?t
?(t,?)
1
Time
Mt
Regular State
Spike State
Time
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The Non-Markovian Process for Power Spot Prices
with Spikes
Formal Definition of the Spike Process The
transition probability density function for the
spike process ?t as a Markov process is given
by where ?(x) is the Dirac delta function.
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The Non-Markovian Process for Power Spot Prices
with Spikes
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.
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The Non-Markovian Process for Power Spot Prices
with Spikes
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
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The Non-Markovian Process for Power Spot Prices
with Spikes

• Interpretation of the Spike State of ?t as Spikes
  in Power Prices
• If the expected time 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.

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The Non-Markovian Process for Power Spot Prices
with Spikes

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

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The Non-Markovian Process for Power Spot Prices
with Spikes
The Non-Markovian Process ?t as a Markov Process
with the Extended State Space The state of the
power market at any time t can be fully
characterized by a pair of the values of the
processes , 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
.
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European Contingent Claims on Power in the
Absence of Spikes
Valuing European Contingent Claims on Power as
the Discounted Risk-Neutral expected value of its
payoff Denote by 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.
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European Contingent Claims on Power in the
Absence of Spikes
Example Geometric Mean-Reverting Process It can
be shown (Kholodnyi 1995) that where
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European Contingent Claims on Power in the
Absence of Spikes
Example Geometric Mean-Reverting Process For
example (Kholodnyi 1995) where with
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European Contingent Claims on Power in the
Presence of Spikes
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
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European Contingent Claims on Power in the
Presence of Spikes
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.
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European Contingent Claims on Power in the
Presence of Spikes
The Case When ?(t,?) is Time-Independent The
value E(t,T,g) is given by
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European Contingent Claims on Power in the
Presence of Spikes
The Case of Spikes with Constant Magnitude
Consider a special case of spikes with constant
magnitude ? gt 1, that is, when ?(?) is the delta
function ?(?- ?). The value E(t,T,g) is given
by
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European Contingent Claims on Power in the
Presence of Spikes
Linear Evolution Equation for European Contingent
Claims on Power with Spikes It can be shown
(Kholodnyi 2000) that the value E(t,T,g) of a
European contingent claim on power with spikes is
the solution of the following linear evolution
equation where and are the
generators of and as Markov processes.
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European Contingent Claims on Power in the
Presence of Spikes
Linear Evolution Equation for European Contingent
Claims on Power with Spikes In a practically
important special case when is a geometric
mean-reverting process the generator is
given by The generator is a linear
integral operator with the kernel
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European Contingent Claims on Power in the
Presence of Spikes
Linear Evolution Equation for European Contingent
Claims on Power with Spikes In the special case
of spikes with constant magnitude the generator
?(t) can be represented as the 2?2 matrix L(t)
transposed to the generator L(t) of the Markov
process Mt. In turn, v and g can be represented
as two-dimensional vector functions 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.
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Why Prices of European Claims On Power Do Not
Spike
Ergodic Transition Probabilities for Mt Assume
that the spikes have constant magnitude ? and the
underlying two-state Markov process Mt is
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.
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Why Prices of European Claims On Power Do Not
Spike
Values of European Contingent Claims on Power Far
From Expiration The values E?t?(t,T,g) and
E?t1(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 gtgt , E?t?(t,T,g) and
E?t1(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.
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Why Prices of European Claims On Power Do Not
Spike
Values of European Contingent Claims on Power Far
From Expiration For example, (Kholodnyi 2000) the
values of European call and put options with
inception time t, expiration time T, and strike X
are given by
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Why Prices of European Claims On Power Do Not
Spike
Example Geometric Mean-Reverting Inter-Spike
Process It can be shown (Kholodnyi 2000) that
the value E(t,T,g) of a European options with
inception time t , expiration time T, and payoff
g is given by where
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Why Prices of European Claims On Power Do Not
Spike
Example Geometric Mean-Reverting Inter-Spike
Process For example, (Kholodnyi 2000) the values
of European call and put options with inception
time t , expiration time T, and strike X are
given by where
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Why Prices of European Claims On Power Do Not
Spike
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)
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Why Prices of European Claims On Power Do Not
Spike
Example Geometric Mean-Reverting Inter-Spike
Process It can be shown (Kholodnyi 2000) that
the values of European call and put options with
strike X are given by where
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Power Forward Prices for Power Spot Prices
Without of Spikes
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
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Power Forward Prices for Power Spot Prices
Without of Spikes
Example Geometric Mean-Reverting Inter-Spike
Process It can be shown (Kholodnyi 1995) that
power forward prices are given by
the following analytical expression where
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Power Forward Prices for Power Spot Prices
Without of Spikes
Example Geometric Brownian Motion (GBM) for
Power Forward Prices The risk-neutral dynamics of
is described by a geometric
Brownian motion where
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Power Forward Prices for Power Spot Prices With
Spikes
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
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Power Forward Prices for Power Spot Prices With
Spikes
The Case When ?(t,?) is Time-Independent The risk
neutral average magnitude of spikes is given
by 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
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Power Forward Prices for Power Spot Prices With
Spikes
The Case of Spikes with Constant Magnitude
Consider a special case of spikes with constant
magnitude ? gt 1, that is, when ?(?) is the delta
function ?(?- ?). The risk neutral average
magnitude of spikes is given by
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Why Power Forward Prices Do Not Spike
Ergodic Transition Probabilities for Mt Assume
again that the spikes have constant magnitude ?
and the underlying two-state Markov process Mt is
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.
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Why Power Forward Prices Do Not Spike
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
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Why Power Forward Prices Do Not Spike
Power Forward Prices far From Maturity The power
forward prices F?t?(t,T) and F?t1(t,T) coincide
up to the terms of order O(e-(T -
t)a). Therefore, F?t?(t,T) and F?t1(t,T) can be
combined into a single expression as
follows When T - t gtgt ,
F?t?(t,T) and F?t1(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.
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Why Power Forward Prices Do Not Spike
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
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Why Power Forward Prices Do Not Spike

• 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.

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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
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European Contingent Claims on Forwards on Power
with Spikes
Geometric Mean-Reverting Inter-Spike Process and
Spikes with Constant Magnitude For example,
(Kholodnyi 2000) the values of European call and
put options (on forwards on power for power with
spikes) with inception time t, expiration time T,
and strike X are given by
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European Contingent Claims on Forwards on Power
with Spikes
Geometric Mean-Reverting Inter-Spike Process and
Short-Lived Spikes with Constant Magnitude It can
be shown (Kholodnyi 2000) that the value of a
European contingent claim (on forwards on power
for power with spikes) with inception time t,
expiration time T, and payoff g can be
represented as the following correction
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European Contingent Claims on Forwards on Power
with Spikes
Geometric Mean-Reverting Inter-Spike Process and
Short-Lived Spikes with Constant Magnitude For
example, (Kholodnyi 2000) the values of European
call and put options (on forwards on power for
power with spikes) with inception time t,
expiration time T, and strike X can be
represented as the following corrections
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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.

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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.
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