The following article was written by two of my MBA
student interns, who used to shadow me at work on Fridays during 2001 - 2002 at
Progress Energy. --- Michael Guth
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MICHAEL A. S. GUTH, Ph.D., J.D.
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Financial Economics Homepage || Attorney at Law Homepage
Put-Call
Parity in the Gas Options Market
Natural
gas option markets, like other commodity option markets, fail to satisfy all of
the simplifying assumptions of the Black-Scholes pricing framework. In fact, gas markets satisfy only a few of
the key assumptions. Consequently, it would not be surprising to find that gas
option prices deviate from their Black-Scholes values. However, the put-call parity theorem, which
some may have viewed as an extension of Black-Scholes option pricing, seems to
hold with amazing precision in the gas options market.
The markets for natural gas and its derivatives have
undergone dramatic development over the past 10 years. The number of players
and traded volumes has increased to a point that gas markets are liquid and
efficient. One important indicator of gas market efficiency is the absence of
arbitrage opportunities, including mispricing of the relationship between put
and call options. The purpose of this article is to find theoretical and
empirical evidence of put-call parity in the gas options market.
What is a natural gas option?
A
natural gas option is an American option on a natural gas futures contract,
which gives the owner the right to buy or sell 10,000 MMBtu per option. These
options are traded on the NYMEX[1]
and require the delivery of the underlying futures contract when exercised at
any time prior to maturity. When a call futures option is exercised, the holder
acquires a long position in a gas futures contract plus a cash premium equal to
the difference of the last future settlement price and the strike price. In the
case of a put option, a short position in the futures contract is acquired plus
a cash amount equal to the difference of the strike price and the last future
settlement price.
In
the gas market, just as in most other commodity markets, the option is written
on a futures contract rather than on the commodity itself. Options on futures
are more popular than physical options, because it is much easier and cheaper
to deliver a futures contract than the physical gas asset. Both the lower transaction cost and the
convenience of cash settlement make futures options easier to use than physical
options for arbitrage, hedging, and speculation.
Table
1. Henry Hub Option Quotes. Calls March
2002. (1/18/2002)
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MOST |
|
OPEN |
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OPTION |
LAST |
OPEN |
HI |
LO |
RECENT |
CHG |
INTEREST |
VOLUME |
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SETTLE |
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NG H2C2050 |
0.313 |
0.313 |
0.313 |
0.313 |
0.313 |
0.027 |
100 |
100 |
|
NG H2C2250 |
0.25 |
0.25 |
0.25 |
0.25 |
N/A+ |
-0.003 |
4513 |
125 |
|
NG H2C2300 |
0.215 |
0.215 |
0.215 |
0.215 |
N/A+ |
0.033 |
3073 |
142 |
|
NG H2C2400 |
0.14 |
0.14 |
0.14 |
0.14 |
N/A+ |
-0.041 |
938 |
17 |
|
NG H2C2450 |
0.12 |
0.12 |
0.12 |
0.12 |
0.12 |
-0.022 |
526 |
526 |
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NG H2C2500 |
0.11 |
0.145 |
0.145 |
0.105 |
N/A+ |
-0.035 |
4971 |
25 |
The
classic model for pricing options on futures was developed by Fischer Black in
1976,[2]
which extended the popular Black-Scholes option pricing model to the case where
the underlying security is a futures contract. According to Black’s (1976)
model the option prices can be written as functions of variables as follows:
Call = c ( F, X, T, r, σ) Put = p ( F, X, T, r, σ),
where
F is the futures price, X is the strike price, T is time to
maturity, r is risk free rate, and σ is the volatility of the futures
price.
The
Black model applies to an abstracted world in which the following nine
assumptions hold:
1) The short-term interest rate is known and is
constant over time. Even though the Federal Reserve can change
short-term interest rates, this assumption is fairly innocuous.
2) The price of the
underlying asset follows a random walk in continuous time with a variance rate
proportional to the square of the asset price. Thus the distribution of
possible prices at the end of any finite interval is lognormal.
This
assumption implies that asset prices follow a stochastic Ito process, which is
a generalized Wiener process where the parameters of the drift and the noise
are functions of the underlying asset value:
dx = a(x, t)dt +
σ(x,t)z , where
x
– price of underlying asset, t – time, a – expected drift function, σ –
standard deviation of the noise, z – Wiener process variable dependent on a
random drawing from a standardized normal distribution. The variance rate is
proportional to the square of the asset price, because the variance is equal to
σ2, and σ is a linear function of the asset price,
according to the Ito process.
The
expected returns of the underlying asset are assumed to be normally distributed
and compounded continuously.
X1 = X0 ert, where
X0
and X1 are asset prices at time 0 and 1, r – continuously
compounded return per unit of time, t – time between 0 and 1. That is why the
prices are lognormally distributed.
3) The variance rate of the return on the asset is
constant. Natural
Gas prices violate the principles of the typical Brownian motion, since the
volatilities have a clear seasonal pattern.
Furthermore, natural gas volatilities change from month to month and
year to year.
4) The asset pays no dividends or other
distributions. This assumption holds
on the gas options market.
5) The option is of European type, and it can only be
exercised at maturity. Gas options
are American, they can be exercised at any time prior to maturity. Since there
is a chance that it will be optimal to exercise an American option prior to
maturity, they should be worth more than a European option.[3]
6) There are no transaction costs in buying or
selling the asset or the option. In
the gas options case there are some transaction costs, but they would only
cause minor deviations from the Black or Black-Scholes value.
7) It is possible to borrow any fraction of the price
of the underlying asset to buy it or to hold it, at the short-term interest
rate. This is a somewhat drastic assumption. Gas is traded in units of 10,000 MMBtu. A trader cannot delta hedge easily over the futures exchange and
buy and sell, e.g., 654 MMBtu. However,
most counterparties in the OTC market would probably be willing to trade any
rounded 1000 MMBtu volume, e.g., 2000, 3000, 4000 MMBtu.
8) There are no penalties to short selling. This seems like a safe
assumption, however, we have to admit that short selling entails repayment over a fixed horizon. The real world
repayment horizon may be shorter than the Black-Scholes idealized repayment
horizon, which means the short position might have to be closed before option
values returned in principle to the Black or Black-Scholes values.
9) Trading is
continuous. This assumption is unrealistic,
since gas futures are not traded continuously throughout the
day. Unlike foreign exchange and securities
markets that are active around the clock, it would be hard to find
counterparties for gas futures at midnight, or even past 5 PM.
Despite the fact that
some of the key assumptions of the Black and Black-Scholes model do not hold
fully for gas options markets, the Black formula is widely used to price gas
options. The model is far from perfect,
but in the absence of any better option pricing formula, gas marketers and
traders use the Black model to value gas options.
Put-Call Parity
The efficiency of the gas options market can be tested
through the put-call parity. It defines
a relationship between the prices of a call and a put option, which is derived
through the replication of payoffs at maturity. Let’s look at the following two
portfolios:
Portfolio 1: A call option + cash equal to PV(X)
= C + X e –rT
Portfolio 2: A put option + a long futures contract + cash equal to PV(F0) =
P + F0 e –rT
The
payoffs for these two portfolios are shown in Table 2.
Table 2. Portfolio Pay-off Structure at
Maturity
Strike Price (X) =
3 , Current Futures Price (F0) = 2.5
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Portfolio 1 |
Portfolio 2 |
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Futures Price |
Call |
Strike |
Total |
Put |
Long Futures |
Current Futures Price (F0) |
Total |
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4 |
1 |
3 |
4 |
0 |
4 - 2.5 |
2.5 |
4 |
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3.5 |
0.5 |
3 |
3.5 |
0 |
3.5 – 2.5 |
2.5 |
3.5 |
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3 |
0 |
3 |
3 |
0 |
3 - 2.5 |
2.5 |
3 |
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2.5 |
0 |
3 |
3 |
0.5 |
2.5 – 2.5 |
2.5 |
3 |
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2 |
0 |
3 |
3 |
1 |
2 – 2.5 |
2.5 |
3 |
As
seen from the “Total” columns for Portfolio 1 and Portfolio 2, the two
portfolios are identical at maturity (Table 2). Therefore, they should have
equivalent value at any time prior to maturity: value of Portfolio1 = value of Portfolio2.
Therefore, call + X e
–rT = put + F0 e –rT ([4])
Despite
the fact that some of the key assumptions of the Black option pricing model do
not hold for natural gas markets, the put-call parity relationship does
hold. As shown in Table 3, we analyzed
empirical data on gas puts and calls in the period 1998 – 2001. Slight
differences between the values of the two portfolios can be attributed to the
transaction costs and probably a few
“stale” prices.[5]
Table 3. Empirical Evidence of Put - Call
Parity
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Date |
Call |
PV(strike) |
Put |
Futures |
PV(Futures) |
Difference between Portfolios |
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12/27/1999 |
0.0010 |
2.3 |
0.029 |
2.27 |
2.269629 |
0.0020 |
|
1/5/2000 |
0.1550 |
2.182 |
0.154 |
2.2 |
2.181842 |
0.0010 |
|
1/20/2000 |
0.0900 |
2.547 |
0.08 |
2.56 |
2.557422 |
0.0000 |
Whenever
there is a significant difference between Portfolio 1 and 2, there exists an
arbitrage opportunity, and a risk-free profit could be generated by replicating
a mispriced option.
Let’s
assume that 1/5/2000 call was priced at $0.2 instead at $0.1550, then the
following arbitrage could be performed:
·
Short the Call: sell
Call at $0.2
·
Buy Put at $0.029,
enter into a long futures contract at $2.2, invest PV of $2.2 into a risk- free
asset, and borrow $2.2 (in this case the Futures price is equal to the Strike,
so the borrowing and investing offset each other)
At maturity we will be able to pocket the difference
of $0.045 per option contract.
It seems logical that there is an interdependent link
between the Black-Scholes pricing formulas and the Put-Call parity. In reality,
however, the parity relationship is completely independent of the Black-Scholes
framework. Put-Call parity existed long before the development of modern option
pricing theory. In the 19th century, for example, some innovative
stock brokers were using the parity relationship to charge higher than the
legally allowed 7% interest rate by entering into contracts with options and
assets in order to replicate a higher interest rate structure.[6]
Even if the gas option prices were completely
mispriced according to the Black-Scholes theory, they should still follow
put-call parity. Otherwise, a risk-free
arbitrage opportunity would exist, and some arbitrageur would take advantage of
the market inefficiency. The empirical evidence proves that the Put-Call parity
exists in the gas options market all the time, even though the market option
prices may not be equal to the Black-Scholes prices.