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<td width="75%" valign="top"><p ALIGN="left"><font face="Arial" size="6"><b><strong>PRICING,
MODELING AND MANAGING PHYSICAL POWER DERIVATIVES</strong></b></font></p>
<p><font face="Arial" size="4">BY CORWIN JOY<br>
</font><font face="Arial" size="2">Caminus LLC<br>
Vice President of Financial Models<br>
5444 Westheimer, Suite 1430<br>
Houston, TX 77056<br>
(713) 626-1305</font><font face="Arial" size="3"><strong><em><br>
</em></strong><a href="mailto:[email protected]">[email protected]</a></font><font face="Arial" size="4"><br>
</font><font face="Arial" size="2">(<em>originally published by PMA OnLine Magazine: 99/01</em>)</font><font SIZE="2"> </font><font size="4"></p>
</font><font SIZE="2"><blockquote>
<blockquote>
<hr>
<p>As the Electricity market deregulates, it is becoming increasingly obvious that the use
of naive Black-Scholes for physical power options leads to severe pricing and hedging
errors. In what follows, we introduce a powerful new framework that follows in Black and
Scholes footsteps but is explicitly tied to the nature of physical electricity. This
framework allows us to price physical options with substantially increased accuracy and
opens the door for pricing retail electricity services.</p>
<hr>
</font>
</blockquote>
</blockquote>
<b><font face="Arial" size="3"><p></font><font size="3"><font FACE="Arial">I. Introduction</p>
</font></b><p>The thrust of this paper is to develop a framework for the pricing and
management of physical power options. There are three main difficulties in achieving this
goal. First, the dynamics of electricity markets are inherently different from most other
commodities due to the lack of physical storage and the strict transmission constraints
that exist on the system. Second, as the pace of deregulation increases existing market
dynamics are bound to change rapidly: making histories of regulated prices all but
useless. Finally, even the best set of market dynamics is worthless without a practical
strategy to manage any options that are sold under such a model. In the electricity market
in particular, careful consideration needs to be given to the best mix of physical and
financial assets needed to hedge obligations and appropriately control any associated
risks.</p>
<font FACE="Arial"><b><p>II. Goals</p>
<blockquote>
</b></font><ol>
<li type="I">Develop a framework for the pricing of physical power options.</li>
</ol>
<blockquote>
<ol type="a">
<li>The framework must be financially sound, and should use generally accepted financial
methods as much as possible given the special nature of electricity.<br>
</li>
<li>The framework must not be too heavily conditioned on historical data, since we expect
the market dynamics to evolve with deregulation.<br>
</li>
<li>The framework must reflect the physical nature of power, including the behavior of the
generation stack, and the ability to model plant outages.</li>
</ol>
</blockquote>
<ol>
<li type="I" value="2">Develop hedging strategies and systems to control the risks
associated with physical options.</li>
</ol>
<blockquote>
<ol>
<li type="a">Examine and describe what risks are even theoretically hedgeable under the
developed framework.<br>
</li>
<li>Survey available market contracts and liquidity available to lay off embedded risks.<br>
</li>
<li>Determine residual risk, and discuss additional contract terms that might be used to
control such events.</li>
</ol>
</blockquote>
</blockquote>
<font FACE="Arial"><b><p>III. Market</p>
</b></font><p>In analyzing the structure of the physical power market we chose as our
starting set of data the FERC filings for both load and system lambda (the marginal cost
of generation in $/MWH) from Jan 1, 1993 - Dec 31, 1996. There were two primary reasons
for using this data set as a starting point:</p>
<blockquote>
<ol>
<li type="I">This is the only real data we have. Historically prices have been regulated and
the small amount of spot electricity that was traded does not give an indicative benchmark
of market prices.<br>
</li>
<li>With the deregulation of electricity, spot prices are likely to become increasingly
volatile as larger amounts are traded on a more active basis. Two things will change only
slowly however: </li>
</ol>
<blockquote>
<ol>
<li type="a">The supply of available electricity (the fixed capital base of generation and
transmission). <br>
</li>
<li>The demand for electricity (early pilot studies indicate that most users demand is
relatively inelastic and will respond only slowly to consistent price pressures). </li>
</ol>
</blockquote>
<blockquote>
<p>Therefore, we feel that by making a careful study for the dynamics of the supply and
demand in this market we can develop a framework that will be able to adapt well to
changing market conditions.</p>
</blockquote>
</blockquote>
<font FACE="Arial"><b><p>IV. Approach</p>
</b></font><p>In order to capture the dynamics of the physical power market, and yet
retain compliance with widely accepted financial techniques, we have chosen as our
starting point the dynamics for the HJM extension of Black’s model for options on
futures:</p>
<p><img SRC="../images/form001.gif" WIDTH="156" HEIGHT="44"></p>
<p>where</p>
<p><img SRC="../images/form002.gif" WIDTH="49" HEIGHT="21"> = Futures price at time t for future expiring at T.</p>
<p><img SRC="../images/form003.gif" WIDTH="45" HEIGHT="21"> = Volatility at time t of F(t, T)</p>
<p><img SRC="../images/form004.gif" WIDTH="36" HEIGHT="21"> = Standard Brownian Motion</p>
<p>In examining the system lambda data, some fundamental observations become quickly
apparent.</p>
<p>(See Appendix A)</p>
<blockquote>
<ol>
<li>Current hour prices are strongly conditioned on the price in the previous hour.<br>
</li>
<li>Prices have a strong tendency towards mean reversion. Even if lambda starts out high, it
has a marked tendency to revert to a more normal level very quickly as the supply shortage
passes and the system operators struggle to bring the system back in balance and shut off
expensive peaking units as soon as possible.<br>
</li>
<li>As the load rises on the system, resources available to meet that load become fewer in
number so that the very same event in terms of an outage or a surge in demand has a
correspondingly greater impact at a higher system load level than it would at a lower
system load level. In financial terms this can be described by saying that as prices rise
the level of volatility will also rise. Furthermore, this also makes sense from a physical
point of view when we examine a graph of the marginal cost of generation as plotted versus
system load. As the system load increases, the marginal cost of generation increases
slowly at first, but then jumps sharply as short term gas fired or peaking units need to
be employed and the system becomes increasingly constrained in terms of power
availability.<br>
</li>
<li>The graph of system lambda is characterized by patterns which follow regular diurnal
cycles but have occasional sharp peaks where lambda surges well beyond these bands. The
first possible cause for such jumps would be a sharp change in the supply, as produced by
an unscheduled plant or transmission outage: we will address this further in what follows.
The second possible cause is a jump in demand: this we must also capture.</li>
</ol>
</blockquote>
<p>To apply these properties in our model for electricity, we generalize the Black-Scholes
framework by adding dependence of volatility on price level and the addition of jumps (see
See R. Merton, 1990, "Continuous Time Finance," Blackwell Publishers, p. 313 for
a discussion of general jump diffusion processes). Next, we model the jumps as Poisson
processes since this is the classical distribution used in the literature to model failure
events. Finally, we make all jumps of limited duration to reflect their transient nature
in the power system. This leads to the generalized dynamic:</p>
<p><img SRC="../images/form005.gif" WIDTH="217" HEIGHT="44"> (1)</p>
<p>where</p>
<p><img SRC="../images/form002.gif" WIDTH="49" HEIGHT="21"> = Price at time t for future delivery of power at
time T.</p>
<p><img SRC="../images/form006.gif" WIDTH="62" HEIGHT="21"> = Volatility at time t of F(t, T), conditioned on
price level F(t, T), and conditional on no arrivals of "jump" information.</p>
<p><img SRC="../images/form007.gif" WIDTH="24" HEIGHT="24"> = Independent Poisson processes, i, representing
jumps in system supply or demand.</p>
<p><img SRC="../images/form008.gif" WIDTH="20" HEIGHT="25">= mean number of jumps per unit
time for jump process i.</p>
<p><img SRC="../images/form009.gif" WIDTH="16" HEIGHT="24"> = <img SRC="../images/form010.gif" WIDTH="58" HEIGHT="24"> where <img SRC="../images/form011.gif" WIDTH="37" HEIGHT="24"> is the random
variable percentage change in the futures price if the Poisson event i occurs and E is the
expectation operator over the random variable <img SRC="../images/form012.gif" WIDTH="16" HEIGHT="24"></p>
<p><img SRC="../images/form013.gif" WIDTH="17" HEIGHT="24"> = Random variable representing the duration of the
jump due to Poisson process i, if a jump occurs</p>
<p>In other words, (1) could be written in the more cumbersome fashion</p>
<p><img SRC="../images/form014.gif" WIDTH="172" HEIGHT="44"> if no Poisson events occur</p>
<p><img SRC="../images/form015.gif" WIDTH="228" HEIGHT="44"> if one Poisson event, i,
occurs</p>
<p><img SRC="../images/form016.gif" WIDTH="293" HEIGHT="44"> if two Poisson events, i and
j, occur</p>
<p>… etc.</p>
<p>This equation can be converted to the risk neutral measure and discretized using the
standard first order Euler method as</p>
<p><img SRC="../images/form017.gif" WIDTH="324" HEIGHT="44"> (2)</p>
<p>where</p>
<p><font FACE="Symbol">e</font> = N(0, 1) iid r.v.</p>
<p><font FACE="Symbol">D</font> t = Simulation time step</p>
<p>Y(n) = 1 if n = 0 , Y(n) = <img SRC="../images/form018.gif" WIDTH="68" HEIGHT="40"> for
n >= 1 where the Y(i, j) are iid independent Poisson processes distributed with
parameter<img SRC="../images/form008.gif" WIDTH="20" HEIGHT="25">*t with random duration
d(i, j) in the event that a jump occurs. Specifically, if a jump of magnitude Y(i, j)
occurs at time t of duration d, then at time t+d we require a jump of magnitude 1/Y(i,j)
to represent the end of the event.</p>
<font FACE="Arial"><b><p>V. Benefits and Features of the Model</p>
<blockquote>
</b></font><ol>
<li>Extension of Black-Scholes process as applied to electricity markets and is familiar to
traders and auditors.<br>
</li>
<li>Volatility dependence on price level . b(. , ., F), reflects real world constraints of
load stack & observed data where price volatility tends to increase as we move up the
stack.<br>
</li>
<li>Volatility dependence on forward maturity date, b(t, T, .) allows us to capture term
structure of volatility and ensures that mean-reversion takes place through the decay of
the volatility curve (Samuelson’s hypothesis). This is allows us to reflect the long
term mean reversion properties of the system discussed earlier.<br>
</li>
<li>Jump process provides for explicit modeling of outage events. This is more realistic
from a physical point of view since plant outages happen in discrete steps and thus gives
us a more accurate dynamic. In addition, having a separate process for outages helps us to
price physical options where the payoff is tied to an outage event.</li>
</ol>
<font FACE="Arial"><b>
</blockquote>
<p>VI. Estimation of Parameters</p>
</b></font><p>Because of the many interacting pieces, estimation of these parameters is
somewhat complex.</p>
<i><p></i><strong>Volatility:</strong><i></p>
</i><p>To estimate the term structure of the volatility, b, we proceed in two stages.
First, the dependency of b on price, F, is estimated by comparing the percentage change in
volatility versus price level. This gives us a term structure for the volatility smile b(.
, ., F) that is independent of the absolute level of volatility. Next, b(t, T) is fitted
to the observed market implied volatilities by examining the traded prices for options on
forward contracts and backing out the implied volatility as per the standard method for
HJM.</p>
<i><p></i><strong>Jumps - Traditional Estimation:</strong><i></p>
</i><p>Traditionally, the jump parameters for a Poisson jump-diffusion process are
estimated by using moment matching techniques. Following the outline in [Ball & Torus,
"On Jumps in Common Stock Prices and Their Impact on Call Option Pricing, <i>Journal
of Finance</i>, 40(1) - March 1985, p. 155-73] we could proceed as follows. Suppose that <img SRC="../images/form019.gif" WIDTH="102" HEIGHT="22">. Then, since the drift for futures
prices is zero, we obtain the density for the return on futures prices as</p>
<p><img SRC="../images/form020.gif" WIDTH="230" HEIGHT="46"></p>
<p>where </p>
<p><img SRC="../images/form021.gif" WIDTH="292" HEIGHT="24">.</p>
<p> </p>
<p>We then obtain sample moments, <img SRC="../images/form022.gif" WIDTH="21" HEIGHT="24">, from the historical
price data via</p>
<p><img SRC="../images/form023.gif" WIDTH="137" HEIGHT="45"></p>
<p>where <img SRC="../images/form024.gif" WIDTH="41" HEIGHT="21">is the change in the natural log of the security
price during time t, and T is the number of days in the estimation period. Note that
because jumps are transient on electricity systems, we must filter the above sample data
by excluding down jumps back to a "normal" state that occur after an up jump so
as not to double count any jumps that occur. Next, using the sample moments, sample
cumulants <img SRC="../images/form025.gif" WIDTH="22" HEIGHT="24">can be determined e.g.: [Kremer and Roenfeldt, <i>Warrant
Pricing: Jump-Diffusion vs. Black-Scholes</i>, JFQA 28(2), 255-271]</p>
<p><img SRC="../images/form026.gif" WIDTH="269" HEIGHT="101"></p>
<p>Such estimates are subject to severe numerical roundoff problems, however, and so the
preferred estimation method is the more careful scheme outlined in Press, Teukolsky,
Vetterling, Flannery, <i>Numerical Recipes in C 2<sup>nd</sup> ed.,</i> Cambridge
University Press, p. 612. </p>
<p>From these cumulants, we then solve equations for the parameter estimates as [Press, <i>A
Compound Events Model for Security Prices, </i>Journal of Business 40 (July 1967), 317-35]</p>
<p><img SRC="../images/form027.gif" WIDTH="214" HEIGHT="50"></p>
<p><img SRC="../images/form028.gif" WIDTH="52" HEIGHT="45"></p>
<p><img SRC="../images/form029.gif" WIDTH="116" HEIGHT="48"></p>
<p><img SRC="../images/form030.gif" WIDTH="221" HEIGHT="48"></p>
<p>In solving for <img SRC="../images/form031.gif" WIDTH="16" HEIGHT="17">, care must be
taken since there are usually two possible solutions in the interval of interest. Press
shows, however, that the equation for <img SRC="../images/form031.gif" WIDTH="16" HEIGHT="17"> has only two real roots of opposite sign. So, we simply choose the root which
gives <img SRC="../images/form032.gif" WIDTH="14" HEIGHT="18"> > 0.</p>
<i><p></i><strong>Jumps - Estimation for the Electricity Market:</strong><i></p>
</i><p>The key observation that makes this model extremely powerful for the physical power
market is that unlike the classical financial case shown above, we can calibrate the jumps
to actual physical events on the system. Since we have hour by hour system lambda,
simultaneous data for plant outages, and concurrent system load data we can calibrate
these events in a rather precise way. Thus, for example, the set of individual plants at a
location can be modeled as a log-binomial collection with the impact of each plant’s
outage on lambda calibrated to its MWH output. This gives a nice asymptotic tie-in to the
single jump process specified above since, via the Central Limit Theorem, for a large set
of log-binomially distributed jump events the distribution converges to <img SRC="../images/form019.gif" WIDTH="102" HEIGHT="22">.</p>
<font FACE="Arial"><b><p>VII. Hedging Issues</p>
</b></font><p>Even in theory it is impossible to be perfectly hedged versus jumps. The
reason for this is that no replicating position can protect you against such a
discontinuity and the best that you can hope for is that on average you are able to cover
your expected cost for these jumps. In practice, the problem is even worse because one
cannot easily rebalance in the physical asset and indeed there are cutoff dates for both
pre-schedules and monthly schedules that determine how much power you must block out as
your best estimate for these reserves. However, the key finding to note here is that the
presence of jumps can lead to significantly different deltas than what we would obtain
with a plain vanilla Brownian motion model. In addition, our calibration of events also
provides a powerful portfolio balancing tool that lets us know how many times we can
oversell our plant to cover a group of options we have sold at any desired level of
confidence. This can be a key competitive advantage for the aggressive pricing of options
in the deregulating marketplace. </p>
<font FACE="Arial"><b><p>VIII. Customer Behavior</p>
</b></font><p>In the financial world, most of the options that have been developed assume
exercise for a particular volume. In contrast, the gas and electricity markets allow
customers to "swing" in the amount of volume that they take. As an example, we
might sell a customer a physical power contract to take 100 MWH/H at COB with the right to
take up to 20 additional MWH in any hour for the month of June 1998 at the price of
$25/MWH. How much is this option worth? The first questions that arise in this context are
as follows:</p>
<blockquote>
<ol>
<li>To what degree is the customer exercising the option economically versus they just need
the power?<br>
</li>
<li>More generally, is the customer’s exercise behavior correlated with price levels?<br>
</li>
<li>How elastic is the customer’s demand: would higher price levels or a pass through
of prices affect their behavior?<br>
</li>
<li>How will the American style features of the above contract affect its value and optimal
exercise time?</li>
</ol>
</blockquote>
<p>In addition, most contracts like the one above have additional American exercise
features and boundaries which limit either the daily, weekly or monthly swings that can be
exercised. As a first attempt to answer these questions, we note that several studies have
been done throughout the US by utilities attempting determine customers’ price
elasticity. The result of these studies is clear: without strong, consistent and long term
pricing incentives that punish expensive customer behavior, there is almost no impact of
price on the amount of electricity taken by most customer classes [K.H. Tiedemann, <i>Time-Of-Use
Rates, Demand Charges and Residential Peak Energy Demand, </i>1996 EPRI Conference on
Innovative Approaches to Electricity Pricing, p. 16-1]. Only the largest customers are
really price sensitive and even they have a limited ability to shift their load. A
simplifying assumption then, is that customers are inelastic: they do not or cannot
exercise these options economically and have a simple correlation of load with price
level. What is more, correlating customer takes with price levels helps us to cherry pick
the most profitable customers. As our starting point, we take the famous
"Quanto" option in finance whose name came from an abbreviation for Option
Quantity Unknown. The idea behind a Quanto option, originally developed for foreign
stocks, is that the payoff is given by</p>
<p><img SRC="../images/form033.gif" WIDTH="272" HEIGHT="25"></p>
<p>where</p>
<p><img SRC="../images/form034.gif" WIDTH="18" HEIGHT="21"> = A fixed foreign currency conversion rate.</p>
<p><img SRC="../images/form035.gif" WIDTH="24" HEIGHT="18"> = The final price of the foreign stock, in foreign
currency</p>
<p><img SRC="../images/form036.gif" WIDTH="20" HEIGHT="17"> = The strike price in foreign
currency.</p>
<p><img SRC="../images/form037.gif" WIDTH="17" HEIGHT="17"> = The strike price in domestic terms.</p>
<p>For example, a US investor buying a British stock Quanto option with a strike of <img SRC="../images/form036.gif" WIDTH="20" HEIGHT="17"> = �20, a fixed conversion rate of <img SRC="../images/form034.gif" WIDTH="18" HEIGHT="21"> = 1.50 ($/�) and a final stock price of <img SRC="../images/form035.gif" WIDTH="24" HEIGHT="18"> = �28 would receive a payoff of </p>
<p><img SRC="../images/form038.gif" WIDTH="249" HEIGHT="25"></p>
<p>Notice here that the option writer’s exposure is determined by two unknowns. The
first unknown is the final value of the stock price. The second unknown is how much
foreign currency they will need to convert into domestic currency at the fixed exchange
rate <img SRC="../images/form034.gif" WIDTH="18" HEIGHT="21">. Hence the "Quantity Unknown" is the
amount of foreign currency to convert at option payoff. The analogy for the energy market
is now obvious: think of <img SRC="../images/form036.gif" WIDTH="20" HEIGHT="17"> as the
strike price for the power swing option, <img SRC="../images/form035.gif" WIDTH="24" HEIGHT="18"> as the spot
price for power in a particular hour, and the "unknown amount of foreign
currency" as the unknown amount of power that the user will take. (Note that we are
ignoring the American features of this option for the moment.)</p>
<p>Under the Black-Scholes type framework, we can obtain a closed form formula that gives
a price for a single swing option under the assumption that the customer’s load is
log-normally distributed and correlated with price levels. This leads to the standard
Quanto formula for the call price [Eric Reiner, 1992, "Quanto Mechanics,"
published in "From Black Scholes to Black Holes," Risk Magazine Press (1992), p.
152]</p>
<p><img SRC="../images/form039.gif" WIDTH="418" HEIGHT="64"></p>
<p>where</p>
<p><img SRC="../images/form040.gif" WIDTH="290" HEIGHT="50"></p>
<p>The above formula provides a particularly simple model for pricing retail options where
the customer swing is driven by price levels. In practice, much more sophisticated models
of customer behavior can be developed. We would advocate this formula as a way of
calibrating numerical results obtained from the more realistic jump diffusion process and
an in house model of customer behavior.</p>
<font FACE="Arial"><b><p>IX. Additional Issues</p>
</b></font><p>Although the described jump diffusion model is very powerful, significant
practical caveats remain. First, how liquid is the physical pool at which the option has
been sold? This will dictate to what degree sellers need to have physical assets on hand
and how easily they can adjust their positions by purchasing spot power. Second, what are
the liabilities and obligations if the seller over/under delivers and how can these be
managed from both a risk and reputation point of view? Finally, what kind of financial
system is needed to properly track and manage the risks in these types of deals? While
vital, such questions tend to be addressed differently within each trading organization
and are best dealt with on a case by case basis.</p>
<font FACE="Arial"><b><p>X. Summary and Conclusions</p>
</b></font><p>This paper gives an introduction to some of the fundamental considerations
in pricing physical power options at the retail level and outlines a simple framework to
address these issues. To make the discussion concrete, a particular jump diffusion process
was built to capture a number of the special features in the power market. In addition, we
showed how properly understanding and modeling these features leads to significant
advantages in hedging, trading and pricing both retail and wholesale deals.</p>
</font><font FACE="Arial" SIZE="3"><b><p>Appendix A</b></font><ol>
<li><font size="3">Graph of system lambda Current Hour vs. Prior Hour at a typical power
pool. Note that as the price in the Prior Hour rises the volatility of the lambda for the
next hour also increases.</font></li>
</ol>
<p align="center"><img src="../images/pricfig1.gif" alt="pricfig1.gif (4814 bytes)" WIDTH="451" HEIGHT="321"></p>
<ol>
<li type="disc" value="2"><font size="3">Plot of Next Day vs. Prior Day system lambda at the
same hour. Note the much weaker correlation, hinting at a reversion effect. This is
confirmed by plots of Hour on Hour system lambda for prior hour prices at the 70<sup>th</sup>
percentile and 90<sup>th</sup> percentile. Note that the 90<sup>th</sup> percentile has
some expected reversion due to sample bias since the percentiles and next/prior hours are
all chosen from the same sample data.</font></li>
</ol>
<p align="center"><img src="../images/pricfig2.gif" alt="pricfig2.gif (7065 bytes)" WIDTH="461" HEIGHT="335"></p>
<p align="center"><img src="../images/pricfig3.gif" alt="pricfig3.gif (10156 bytes)" WIDTH="676" HEIGHT="437"></p>
<p align="center"><img src="../images/pricfig4.gif" alt="pricfig4.gif (7690 bytes)" WIDTH="579" HEIGHT="392"></p>
<ol>
<li type="disc" value="3"><font size="3">Plots of system lambda versus Load. </font></li>
</ol>
<p><img src="../images/pricfig5.gif" alt="pricfig5.gif (15019 bytes)" WIDTH="579" HEIGHT="421"></p>
<p><img src="../images/pricfig6.gif" alt="pricfig6.gif (9266 bytes)" WIDTH="585" HEIGHT="403"></p>
<p><img src="../images/pricfig7.gif" alt="pricfig7.gif (7976 bytes)" WIDTH="559" HEIGHT="403"></p>
<p><img src="../images/pricfig8.gif" alt="pricfig8.gif (10886 bytes)" WIDTH="499" HEIGHT="465"></td>
</tr>
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