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<td width="82%" valign="top"><p ALIGN="left"><strong><font face="Arial" size="6"><b>PRICING
SELLER CHOICE<br>
</b></font><font face="Arial" size="4">BY JOHN LIU<br>
</font><font face="Arial" size="3">Senior Quantitative Analyst <br>
PSE&G<br>
</font></strong><font face="Arial" size="2">(<em>originally published by PMA OnLine
Magazine: 99/02</em>)</font><font SIZE="2"> </font></p>
<font SIZE="4"><p ALIGN="CENTER"></font><b><i><font FACE="Arial" SIZE="4"> </p>
<p></font></i></b><strong><font face="Arial">Introduction</font></strong></p>
<p><font face="Arial">The emerging US power market is one of the biggest commodity markets
in the world with annual transaction revenue of about $200 billions. The limited ability
of spatial and temporal arbitrage in power markets results in very volatile price swing in
forward and spot markets. Non-storability of electric power implicates once generated
electricity has to be consumed immediately. This also limits power market arbitrage across
time. The transmission cost and loss of electric power during transportation also restrain
the spatial arbitrage in the power markets. One tool can be used to hedge the volatile
price risk is to utilize physical and financial derivatives traded on the Exchanges and
OTC markets, such as future, forward, various option contracts. </font></p>
<p><font face="Arial">One of the interesting features in power derivative contracts on OTC
markets is the embedded option in the power derivative contracts or seller choice. The
detail specification of the Seller choice in the derivative contracts says that the seller
of power has the right to delivery contracted amount of power to a specific location at
the cheapest cost or a bus with the lowest market clear price to maximize the contract
profit. As we know some of delivery bus could have the zero or negative power price for
some hours because of congestion. Thus, the seller of power if delivered to a bus with the
negative price will get paid with not only the contract price but also the oppose sign of
the market clearing price at that bus. Since implementation of location marginal price
scheme in PJM pool on April 1, 1998, the volume of power derivatives with the seller
choice has dropped dramatically. One reason is that some market players could take the
better advantage of the seller choice but others do not. As the market becomes more
competitive, hazarding a guess in advance as to what and where exactly drives daily bus
prices in the PJM pool have just become more difficult for market participators. Did a
unit trip in the pool? Does PJM energy export out of the pool too much? Is load building,
or is a big player shifting back and forth between daily and hourly schedules? Another
reason is that the market lacks the proper valuation tool for the forward and option
contracts with the seller choice. In other words, if the embedded option in the power
derivative contracts can be properly priced, the fear of market players in the seller
choice should gradually disappear. The volume of power derivative contracts with the
seller choice may be revived. The power derivatives with the seller choice are also useful
tools to hedge and speculate location prices. </font></p>
<p><font face="Arial">The key question is that how much worth is the seller price and what
is impact of the seller choice on valuation of options. Intuitively, the price of forward
or future contracts with the seller choice is cheaper than one without the seller choice.
Similarly, the European call option with seller choice should be cheaper than one without
the seller choice and the European put option with seller choice should be more expensive
than one without the seller choice since the underlying asset with the seller choice is
cheaper than the one without it. In general, the delivery physical should be cheaper than
the contract with a specific delivery bus, such as Western Hub in PJM power market. In
this paper we will develop a stochastic price model to illustrate how to value the seller
choice and power derivatives with the seller choice properly. </font></p>
<p><font face="Arial"> <strong>The Seller Choice and The Location Marginal Price</strong></font></p>
<p><font face="Arial">PJM is one of the largest and most sophisticated power pool in North
American and the third largest power pool in the world. The serving regions include all or
part of States of Pennsylvania, New Jersey, Maryland, Delaware, Virginia, and the District
of Columbia. This region consists of 8.7% of US population and more than 7% of total
energy consumed in US power markets with more than 540 power plant units. Since the
implementation of the PJM Open Access Transmission Tariff on April 1, 1997, PJM has become
first regional, bid-based competitive wholesale power market and one of the most liquid
and active energy markets in the country. </font></p>
<p><font face="Arial">Location Marginal Price (LMP) is the marginal cost of supplying the
next increment of electricity power at a specific location on the power pool, including
both generation marginal cost of transmission congestion cost plus the cost of marginal
losses. In the absence of power delivery limitations, the price of energy in the entire
PJM power pool is equal to the cost of the most expensive generating resources that is
operating to meet the demand, including all energy transfer from other neighbor power
pools, such as Cinegry and Nepool. In this case all LMPs are the same and a market clear
price is set. Under some operating conditions, the next least-cost generator cannot be
used to meet increasing demand because of power delivery limitations or constraints on the
transmission system. When this occurs, a generator that is more expensive with a more
advantageous location relative to the transmission system limit must be operated in order
to meet demand. The common flow of energy within PJM pool is from West to East. When a
transmission system is constrained, the low cost energy from the west cannot flow to the
east. Consequently, a higher cost generator in the east must be dispatched to meet load.
Under a location marginal price model, the market clearing price of energy varies
depending on where the delivery bus is located. There are total three Hubs and 30 500KV
buses created in PJM pool. The Hub prices are the weighed average LMPs of the specified
buses within the Hub with a fixed and equal distribution for each bus. During the hot
summer and when transmission is constrained, the excess demand for power in a specific bus
can drive LMP skyrocket high.</font></p>
<p><font face="Arial"> <strong>A Valuation Model for the Seller Choice</strong></font></p>
<p><font face="Arial">As mentioned above, the value of the seller choice in power
derivative contracts should be incorporated into the price of derivative contracts. The
limitations of arbitrage in the electric power markets across time and space make the
power price much more volatile than that in other financial markets, such as metal,
natural gas, equity, bond, and foreign exchange markets. The excess demand in the hot
summer combining with outrage of power units can drive the power price hundreds times
higher than that in the normal summer time. The over forecast of power demand could cause
the hourly power price drop below zero. The power derivative contracts with the seller
choice insure that the seller of power could delivery the power to the bus with the lowest
location marginal price to maximize the profit. In the following, we will use a simple
stochastic model to illustrate how we can value the power derivative contracts with the
seller choice.</font></p>
<p><font face="Arial">For derivation simplification, we assume the market is perfect,
frictionless, and arbitrage free. We also assume the volatility and correlation are
constant. We further assume the well-known daily Black-Scholes spot price model as
following (Black-Scholes, 1973; Black, 1977).</font></p>
<p ALIGN="CENTER"><font face="Arial"><img SRC="../images/liu01.gif" WIDTH="126" HEIGHT="72"></font></p>
<p ALIGN="CENTER"> </p>
<p><font face="Arial">where <img SRC="../images/liu02.gif" WIDTH="18" HEIGHT="24">is spot
power price at time t at ith delivery bus (Hub), i= 1, 2, …, I, <img SRC="../images/liu03.gif" WIDTH="18" HEIGHT="24">is mean power price at ith delivery bus
(Hub), <img SRC="../images/liu04.gif" WIDTH="18" HEIGHT="24">is the volatility of power
price at ith delivery bus (Hub), and <img SRC="../images/liu05.gif" WIDTH="18" HEIGHT="24">is
standard Brownian motion or Wiener process for the price shock. Under the assumptions of
no-arbitrage, and perfect and complete markets, Girsanov theorem states that there exists
a uniquely risk-neutral measure or equivalent martingale measure such that all discounted
asset prices and contingent claims are martingale under risk-neutral probability space.
Under the equivalent martingale measure, the diffusion model can be rewritten as:</font></p>
<p ALIGN="CENTER"><font face="Arial"><img SRC="../images/liu06.gif" WIDTH="118" HEIGHT="74"></font></p>
<p><font face="Arial">In this paper, we will mainly apply the equivalent martingale
approach to value the power derivatives. In some cases, the Monte-Carlo simulation
approach will be applied when the closed form solutions of derivative valuations are
unavailable. The essence of Girsanov theorem tells us the discounted cash flows of the
power derivative contracts are martingale under the equivalent martingale probability
space. It states as follows:</font></p>
<p ALIGN="CENTER"><font face="Arial"><img SRC="../images/liu07.gif" WIDTH="253" HEIGHT="66"></font></p>
<p ALIGN="CENTER"> </p>
<p><font face="Arial">where <img SRC="../images/liu08.gif" WIDTH="21" HEIGHT="22"> is the
payoff function of a power derivative contract. We assume short-term interest rate is
constant here and it is very easy to incorporate stochastic interest rate model into the
above valuation formula of the power derivatives. Clearly as long as we can properly model
the stochastic price process at each Bus, we could derive a joint distribution from them
and evaluate the above integration to obtain the contingent claim valuation. Another
robust method to derive the option valuation is through the Monte-Carlo simulation based
on the above formula. We can simulate a great number of stochastic paths for power prices
at each bus following the assumed power price models. Therefore, we obtain the contingent
claim price by averaging them.</font></p>
<p><font face="Arial">Suppose we have a contingent claim whose payoff function only
depends on underlying power prices of time T at each bus (Hub). </font></p>
<p ALIGN="CENTER"><font face="Arial"><img SRC="../images/liu09.gif" WIDTH="162" HEIGHT="22"></font></p>
<p ALIGN="CENTER"> </p>
<p><font face="Arial">Applying Ito Lemma and knowing that any contingent claim is a
martingale under the risk-neutral probability space from Girsanov theorem, the drift term
of <img SRC="../images/liu10.gif" WIDTH="26" HEIGHT="24"> is zero. Therefore, the value <img SRC="../images/liu11.gif" WIDTH="15" HEIGHT="24"> of a contingent claim at time t
satisfies the following stochastic partial differential equation (SPDE) (Willmott,
Dewynne, and Howison, 1993):</font></p>
<p ALIGN="CENTER"><font face="Arial"><img SRC="../images/liu12.gif" WIDTH="318" HEIGHT="49"></font></p>
<p ALIGN="CENTER"> </p>
<p><font face="Arial">The terminal conditions are</font></p>
<p ALIGN="CENTER"><font face="Arial"><img SRC="../images/liu13.gif" WIDTH="181" HEIGHT="26"></font></p>
<p><font face="Arial">The boundary conditions satisfied are</font></p>
<p ALIGN="CENTER"><font face="Arial"><img SRC="../images/liu14.gif" WIDTH="337" HEIGHT="53"></font></p>
<p> </p>
<p><font face="Arial">For example, the boundary conditions for the seller choice become</font></p>
<p> </p>
<p ALIGN="CENTER"><font face="Arial"><img SRC="../images/liu15.gif" WIDTH="528" HEIGHT="53"></font></p>
<p ALIGN="CENTER"> </p>
<p><font face="Arial">The boundary conditions for European call option are</font></p>
<p ALIGN="CENTER"><font face="Arial"><img SRC="../images/liu16.gif" WIDTH="500" HEIGHT="44"></font></p>
<p><font face="Arial">Theoretically, in order to value a contingent claim, you can always
solve the above stochastic partial stochastic differential equation (SPDE) numerically
either using finite difference or finite element approaches under the proper terminal and
boundary conditions. Another popular approach to solve the above SPDE using the
approximate method is to perform robust Monte-Carlo simulation. </font></p>
<p> </p>
<p><font face="Arial">Let's take a further look of the payoff functions for the popular
power derivatives. The payoff function for a daily forward contract with seller choice at
the time to maturity T has</font></p>
<p ALIGN="CENTER"><font face="Arial"><img SRC="../images/liu17.gif" WIDTH="158" HEIGHT="26"></font></p>
<p><font face="Arial">where <img SRC="../images/liu18.gif" WIDTH="21" HEIGHT="24"> is the
daily average price at the delivery bus (Hub) i.</font></p>
<p><font face="Arial">The payoff function for a standard European call option with seller
choice at T can be written as following.</font></p>
<p ALIGN="CENTER"><font face="Arial"><img SRC="../images/liu19.gif" WIDTH="11" HEIGHT="22"><img SRC="../images/liu20.gif" WIDTH="240" HEIGHT="22"></font></p>
<p ALIGN="CENTER"> </p>
<p><font face="Arial">where K is strike price of the call option and <img SRC="../images/liu21.gif" WIDTH="21" HEIGHT="24"> is the average daily close price of
electric power. This is a single daily European call option commonly traded in the power
markets.</font></p>
<p><font face="Arial">Now we can derive some useful closed form solutions in following if
we assume there are only two buses (Hubs) in a power pool or I=2. The payoff from a
forward contract with the seller choice is</font></p>
<p ALIGN="CENTER"><font face="Arial"><img SRC="../images/liu22.gif" WIDTH="274" HEIGHT="22"></font></p>
<p><font face="Arial">You can see the purchased forward contract with the seller choice is
equivalent to a purchased forward at the delivery bus 2 and at the same time buy a spread
call option between bus 2 price and bus 1 price as long as two price processes are not the
same. Therefore, the payoff from the derivative contract with the seller choice is usually
lower than one without the seller choice since the second term in the above payoff
function is non-negative. Thus the price of power derivatives with seller choice is
cheaper than one without seller choice. Later we will derive a closed form formula for
forward contracts and European call option with the seller choice with only two buses. In
general, we have to run Monte-Carlo simulation or solve the above stochastic partial
differential equations numerically in order to obtain the value of seller choice with more
than two buses (Boyle, 1977; Clewlow and Carverhill, 1992). </font></p>
<p><font face="Arial">We will start with the standard European call options first. The
pricing formula for the standard European call option can be derived either from solving
PDE or evaluate the conditional expectation under the risk neutral measure directly (Black
and Scholes, 1973).</font></p>
<p ALIGN="CENTER"><font face="Arial"><img SRC="../images/liu23.gif" WIDTH="233" HEIGHT="144"></font></p>
<p ALIGN="CENTER"> </p>
<p><font face="Arial">The valuation formula for the spread option under Black-Scholes
model is first proposed by Margrabe (1978) and Rubinstein and Reiner (1992). Later, Geman
and Yor (1991) applied the elegant risk neutral approach to arrive the same formula in
much simpler way. A client may want to purchase a spread option to hedge the location
marginal price risk and speculate widening spread between two buses, say for some energy
delivery point, the LMP can go as low as zero or rise as high as price cap in the other
extreme. Thus, he can purchase a spread option based on two buses to hedge the LMP risks.
The price formula for the spread option is as follow.</font></p>
<p ALIGN="CENTER"><font face="Arial"><img SRC="../images/liu24.gif" WIDTH="271" HEIGHT="173"></font></p>
<p><font face="Arial">A pricing formula for a contingent claim based on the maximum of two
assets and a strike price under Black-Scholes diffusion model can be derived as following.</font></p>
<p ALIGN="CENTER"><font face="Arial"><img SRC="../images/liu25.gif" WIDTH="417" HEIGHT="106"></font></p>
<p ALIGN="CENTER"><font face="Arial"><img SRC="../images/liu26.gif" WIDTH="397" HEIGHT="397"></font></p>
<p><font face="Arial">where <img SRC="../images/liu27.gif" WIDTH="39" HEIGHT="22">is a
bivariate cumulate distribution function.</font></p>
<p><font face="Arial">Therefore, a contingent claim on minimum of two assets or a forward
contract with seller choice can be valued as following.</font></p>
<p ALIGN="CENTER"><font face="Arial"><img SRC="../images/liu28.gif" WIDTH="249" HEIGHT="225"></font></p>
<p> </p>
<p><font face="Arial">It is clear that not only spot prices of two assets but also
volatility and correlation between both two underlying assets appear in the valuation
formula. To limit the downside risk with multiple Buses, the client could buy an option
with the seller choice. It is easy to see that long a call option with seller choice is
equivalent to long two standard call options for underlying asset 1 and 2, respectively
and short one maximum of two underlying assets and strike price. The call options on
minimum of two assets or the call options with seller choice can be priced using the
following expression.</font></p>
<p ALIGN="CENTER"><font face="Arial"><img SRC="../images/liu29.gif" WIDTH="415" HEIGHT="210"></font></p>
<p ALIGN="CENTER"><font face="Arial"><img SRC="../images/liu30.gif" WIDTH="247" HEIGHT="399"></font></p>
<p><font face="Arial">where <img SRC="../images/liu42.gif" WIDTH="39" HEIGHT="22">is a
bivariate cumulate distribution function.</font></p>
<p><font face="Arial">The call-put parity for option with the seller choice is</font></p>
<p><font face="Arial"><img SRC="../images/liu31.gif" WIDTH="465" HEIGHT="22"></font></p>
<p><font face="Arial">Similarly, the power derivative contracts with the seller choice can
be extended to the case with multi-buses. But the numerical solution or Monte-Carlo
simulation is necessary to obtain the valuation of power derivatives with the seller
choice.</font></p>
<p><strong><font face="Arial">Numerical Examples</font></strong></p>
<p><font face="Arial">In this section, we will provide some numerical examples of how to
price the power derivatives with the seller choice under Black-Scholes model framework. We
estimate the model parameters using the historical daily power price data from West and PS
zone in PJM pool and then examine the impact of seller choice on power forward and option
contract valuation. The power derivative contracts in West Hub are the most actively
traded. PJM power pool officially implemented the locational marginal price from April 1,
1998. We have West and East Hubs daily power price data from April 1, 1998 to December 31,
1998. Fig. 1 displays two on-peak price data series in West Hub and PS zone. The average
spread between PS zone and West hub for this period is $0.64. The estimated daily
volatility for PS zone and West Hub are 91.19% and 93.33%, respectively. The correlation
between PS zone and West hub is 99.67%. </font></p>
<p><font face="Arial">Diagrams Fig. 2.1 to 2.4 show the values of a forward contract with
the seller choice vary with time to maturity, correlation, volatility, and spot price. We
choose an example of a forward contract with the seller choice with two buses. The value
of the forward contract increases non-linearly from $34.99 to $39.99 as S1 rises from $35
to $45 (see Fig. 2.1). Whereas the value of the forward contract decreases from $40 to
$39.18 as the time to maturity varies from one day to one year (see Fig. 2.2). As the
correlation between two zones increases from 50% to 100%, the value of forward contract
rises from $30.13 to $39.99. Similarly, as PS zone volatility varies from 50% to 100%, the
value of forward contract initially increases linearly from $35.51 to $39.54 at volatility
of 92% then decreases to $39.26. Clearly, the underlying asset price, time to maturity,
and volatility of two assets nonlinearly affect the value of forward contract. In most
cases, the value of forward contract is over-priced.</font></p>
<p><font face="Arial">The similar results can be found in call option contract with the
seller choice. In our example, the average over-priced value varies from $0.5317 to
$4.3465. Fig. 3.1 to Fig. 3.5 show the comparison of the value of a daily call option
strip with the seller choice varies with the spot price, option strike, time to maturity,
correlation, volatility to a standard call option. The option value increases from $6.8208
to $9.8475 as spot price in PS zone rises from $35 to $45 whereas the standard call option
price changes from $6.8221 to $12.8084. On the other hand, call option price drops
linearly from $11.6313 to $8.0653 as the strike price increases from $35 to $45. The
average over-priced value of the call option is $0.7583. Similarly, the call option with
the seller choice increases non-linearly from $0.5595 to $12.7567 as time to maturity
changes from one day to one year whereas the standard call option varies from $1.0512 to
$13.7289. The call option value increases from $3.9048 to $9.8465 as the correlation
between two zones rises from 50% to 100% whereas the call value increases from $5.5051 to
$9.6841 as the volatility in PS zone changes from 50% to 100%. The comparison results
imply that the call option is more over-priced for both highly volatility markets, such as
daily option markets compared to monthly option markets. On the other hand, the call
option with seller choice is less over-priced for the highly correlated buses, such as two
closest buses in both East and West Hubs.</font></p>
<p><strong><font face="Arial">Conclusion</font></strong></p>
<p><font face="Arial">In this paper, we developed a valuation model for pricing the power
derivative contracts with the seller choice under Black-Scholes model framework. The
seller choice or the embedded option in the power derivative contracts is very common in
OTC markets. Since the implementation of LMP in PJM pool, the volume of power derivatives
with the seller choice has been dropped dramatically. The principal reason is that we do
not have the proper valuation model to price the seller choice and incorporate the
valuation of the embedded option or the seller choice into power derivative valuation. Our
results indicate that the power derivative contracts with the seller choice in the markets
are overall over-priced. For example, the call option with the seller choice is over
priced range from $0.5317 to $4.3465 depending on underlying volatility and correlation.
The power derivatives with the seller choice can provide useful tools for market players
to hedge their location basis risk in the highly volatile power markets. This paper hopes
to shorten the gap and revive the interest of trading power derivative contracts with the
seller choice. The methodology of valuation model for the power derivatives with the
seller choice developed in this paper can be extended to multi-buses case. The
multi-factor mean-reversion price model and the stochastic volatility model can also be
easily incorporated into the current valuation model. For the American options, the
multi-dimension lattice methods are required to value the options with the early
exercises.</font></p>
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