on to Index Option Products and Futures
FIGURE 29-1.
(Continued)
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Put-Call Ratio
FIGURE 29-2.
50-day index put-call ratio: 1985-91.
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527
528 Part V: Index Options and Futures
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Chapter 29: Introduction to Index Option Products and Futures
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Stock Index Hedging
Strategies
This chapter is devoted primarily to examining the various ways that one might hedge
a portfolio of stocks with index products. This portfolio might be a small one owned
by an individual investor or it might be as large as the entire S&P 500 Index. We
explore this strategy from the various viewpoints of the individual investor, the insti
tutional money manager, and the arbitrageur. This technique of hedging stocks with
index products has become quite popular and has also drawn some attention because
of the way it can cause short-term movements in the entire stock market. The rea
sons why these movements occur are also explained. Finally, we look at ways of sim
ulating a broad-based index by buying a group of stocks whose performance is geared
to that of the index itself.
MARKET BASKETS
One of the most popular strategies using index futures and options has been the tech
nique of buying stocks whose performance simulates the performance of a broader
index and hedging that purchase with the sale of overpriced futures or options based
on that index. The group of stocks that is purchased is commonly known as a "mar
ket basket" of stocks. This chapter describes how these baskets can be used to trade
against a very broad index, such as the S&P 500, or a far narrower index, such as the
Dow-Jones 30 Industrials (DJX), or even one as small as just a few stocks - perhaps
a portfolio held by any investor.
531
532 Part V: Index Options and Futures
The key to determining whether it will be profitable to trade some derivative
security - options or futures - against a set of stocks is generally the level of premi
um in the futures contract itself. That is, if the S&P 500 Index is at 405.00 and the
futures are trading at 408.00, then there is a premium of 3.00 - the futures contract
is trading 3.00 points higher than the index itself. The absolute level of the premium
is not what is important, but rather the relationship between the premium and the
fair value of the future. We will look at how to determine fair value shortly.
The futures are the leaders among the derivative securities, especially the S&P
500 futures. Whenever these become overpriced, other derivative securities will gen
erally follow suit. There are also futures on the NYSE Index, the Value Line Index,
and the Japanese Nikkei 225 Index. Moreover, there are cash-based index options on
the S&P 500 Index (as opposed to the futures options on that index) and the NYSE
Index. These other securities would include the QEX (S&P 100) options and NYSE
futures and options. It follows as well that when the S&P 500 futures become under
priced, the other derivative securities quickly fall into line. If the other derivative
securities don't follow suit, then there is an opportunity for spreading one market
against another. That type of spreading can frequently be profitable, and is discussed
in the next chapter.
The normal scenario is for most of the derivative securities to follow the lead of
the S&P 500 futures. When this happens, the only thing that is fairly priced is the
index itself - that is, stocks. Consequently, the logical way to hedge the derivative
security is to do it with stocks. The small investor might hedge his own individual
portfolio, although that would not be a perfect hedge since his own portfolio is not
composed of the exact same stocks as any index. If the index is small enough, such as
the 30-stock DJX, then one might buy all 30 stocks and sell the futures when they are
overpriced. This is a complete hedge and would, in fact, be an arbitrage. In the case
of a larger index such as the S&P 500, it would be possible only for the most profes
sional traders to buy all 500 stocks, so one might buy a smaller subset of the index in
hopes that this smaller set of stocks will mirror the performance of the index well
enough to simulate having bought the entire index. We take in-depth looks at both
types of hedging.
Even if the investor is not planning to use these hedging strategies, it is impor
tant for him to understand how they work These strategies have certain ramifications
for the way the entire stock market moves. In order to anticipate these movements,
a working knowledge of these hedging strategies is necessary. The first thing that one
must know in


ake in-depth looks at both
types of hedging.
Even if the investor is not planning to use these hedging strategies, it is impor
tant for him to understand how they work These strategies have certain ramifications
for the way the entire stock market moves. In order to anticipate these movements,
a working knowledge of these hedging strategies is necessary. The first thing that one
must know in order to implement any of these hedging strategies is how to determine
the fair value of a futures contract.
Chapter 30: Stock Index Hedging Strategies
FUTURES FAIR VALUE
533
The formula for calculating the fair value of the futures contract is extremely simple,
although one of the factors is a little difficult to obtain information on. First, let's look
at the simple futures fair value formula:
Simple Formula:
Futures fair value= Index x [l +Timex (Rate - Yield)]
where index is the current value of the index itself, rate is the current carrying rate
(typically, the broker loan rate), yield is the combined annual yield of all the stocks in
the index, and time is the time, in years, remaining until expiration of the contract.
Example: Suppose the Zl'X Index is trading at 160.00, the broker loan rate is 10%,
the yield on the 500 stocks is 5%, and there are exactly 3 months remaining until
expiration of the futures contract. The time is .25, expressed in years, so the formu
la becomes
Future fair value= 160.00 x [l + .25 x (.10- .05)]
= 160.00 X (1 + .0125) = 162.00
Thus, the future should be trading at about 2 points above the value of the index
itself. This premium of the future over the index represents the savings in not having
to pay for and carry 500 stocks, less the loss of the dividends on the stocks ( the future
does not pay dividends). If the future should get very expensive - trading at 3.50 or
4 points over the index - then it would have to be considered very overvalued, and
an arbitrageur could move in to take advantage of that fact. Similarly, if the future
should trade cheaply, at less than a point over fair value, there might be an arbitrage
available in that case as well.
The fair value is really only a function of four things: the value of the index itself,
the time remaining until expiration, the current carrying rate, and the dividends
being paid by the stocks in the index until expiration. Notice that "the dividends paid
by the stock in the index until expiration" is not quite the same as the yield of the 500
stocks, which we used in the above simple formula. We will expand more on that dif
ference shortly.
Before doing that, however, let us look at how changes in the variables in the for
mula affect the fair value of the futures contract. More important, we are interested in
how changes in the variables affect the premium of the futures contract over the index
value. This is what one is primarily concentrating on when trading market baskets.
As the index value itself rises, the fair value premium rises. For example, if 2
points is the fair premium when the index is at 160, as in the above example, then 4
S34 Part V: Index Options and Futures
points would be the fair premium if the index were at 320 and all the other variables
remained the same. Conversely, as the index falls, the fair value of the premium
shrinks.
The premium rises and falls in direct correlation with the carrying rate as well
as with time remaining until expiration. Note that this statement is true for stock
options also, and for the same reason: The savings in carrying costs are greater when
rates are higher, or when one must hold for a longer time, or both. In the above
example, if one were to assume there were 6 months to expiration instead of 3, the
fair value of the premium would increase to 4 points from 2 points. Similarly, if the
time were decreased, the fair value would be smaller.
Some investors, primarily institutional investors, use the short-term T-bill rate
rather than the carrying rate in order to determine the futures fair value. The reason
they do that is to determine whether the money they have in cash is better off in T
bills or in an arbitrage strategy such as this. More will be said about this use of the T
hill rate later.
Dividends Have an Inverse Correlation to the Premium Value.
An increase in the overall yield of the index will shrink the fair value of the futures
contract. This is because the futures holder does not get the dividends and therefore
the future is not as valuable because of the loss of dividends. Conversely, if dividend
yields fall, then the fair value of the premium increases. This is not the whole story
on dividends, however.
Recall that a few paragraphs ago, it was pointed out that the yield and the
amount of dividends are not exactly the same thing. This is because stocks don't pay
their dividends in a uniform manner. Rather than paying a continuous yield as bonds
do, stocks normally pay their dividends in four lump sums a year. This means that the
yield variable in the simple formula shown above should be replaced by the actual
amount of dividends remaining until expiration. This fact makes the computation of
the fair value of the index a little more difficult. In order to do it accurately, one must
know the dividend amounts and ex-dividend dates of each of the stocks in the index.
Knowing all of this information is a far more formidable task than knowing the yield
of the index, since the yield is published weekly in several places. In fact, the servic
es of a computer are required in order to compute the actual dividend on the larger
indices, where 100 or more stocks are involved.
As a result, the actual formula changes slightly from the simple formula shown
above:
Actual Formula:
Futures fair value = Index x ( 1 + Time x Rate) - Dividends
Chapter 30: Stock Index Hedging Strategies S3S
In this formula, the dividends are taken to be the present worth of all the dividends
remaining until expira

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ual dividend on the larger
indices, where 100 or more stocks are involved.
As a result, the actual formula changes slightly from the simple formula shown
above:
Actual Formula:
Futures fair value = Index x ( 1 + Time x Rate) - Dividends
Chapter 30: Stock Index Hedging Strategies S3S
In this formula, the dividends are taken to be the present worth of all the dividends
remaining until expiration of the future.
Example: In an example similar to the one given for the simple formula, suppose the
Z¥X Index is trading at 160.00, the broker loan rate is 10%, the present worth of the
dividends remaining until expiration is $1.89, and there are exactly 3 months remain
ing until expiration of the futures contract. The time is .25, expressed in years, so the
formula becomes
Future fair value = 160.00 x (1 + .25 x .10) 1.89
= 160.00 X (1 + .025) 1.89 = 162.11
In order to compute the present worth of the dividends of an index, it is neces
sary to know the amount of each stock's dividend as well as the payment date of the
dividend. To compute the index's dividend, one computes the present worth of each
dividend and multiplies that result by that stock's divisor in the index in order to give
the dividend the proper weight. The index's total dividend is the sum of each of these
individual stock computations. Each stock's divisor is merely the float of the stock
divided by the divisor of the index. In a price-weighted index, it is not necessary to
adjust the present worth of each dividend - merely add them together and divide by
the divisor. As an example, we will look at a hypothetical index composed of three
stocks in order to see how one computes the present worth of an index's dividends.
Example: Suppose a capitalization-weighted index is composed of three stocks: AAA,
BBB, and CCC. Furthermore, suppose that the amount of each of the individual
stocks' dividends and the time remaining until the dividend is paid are given in the
following table, as well as each stock's float. Finally, assume the divisor for the index
is 150,000,000.
Dividend Days until
Stock Amount Dividend Payout Float
AAA 1.00 35 50,000,000
BBB 0.25 60 35,000,000
CCC 0.60 8 120,000,000
Divisor: 150,000,000
In order to compute the present worth of a future amount, one uses the formula:
P t rth Future amount resen wo = ------
(1 + Rate)Time
where rate is the current short-term rate and time is expressed in years.
536 Part V: Index Options and Futures
Assume that the current interest rate is 10%. Then the present worth of AAA's
dividend would be:
Present worth AAA 1.00 =------(1 + .10)(35/360)
1.00 =----(1.10)(-0972)
1.00
1.0093
= 0.9908
The present worth of the dividend is always less than the actual dividend. The pres
ent worth of an amount is the amount of money that would have to be invested today
at the stated rate (10% in this example) to produce the future amount. That is, 99.08
cents invested at 10% would be worth exactly 1.00 in 35 days.
The present worth of the other two dividends is .2461 for BBB and .5987 for
CCC. The reader should verify for himself that these are indeed the correct amounts.
Notice that the present worth of a dividend is not much less than the actual value of
the dividend. However, in a larger index, where one is dealing with several hundred
dividends, the present worth may be significantly different from the actual sum of the
dividends, especially if short-term rates are high.
Since we made the assumption that this is a capitalization-weighted index, each
of these figures must be adjusted for the capitalization of the stock in order to give
each present worth the proper weight within the index. Thus, for AAA, the adjusted
dividend would be .9908 times 50,000,000 (AAA's float), divided by 150,000,000, the
divisor of the index. This would result in an adjusted dividend of .3303 for AAA.
When similar adjustments are made for BBB and CCC, their adjusted values become
.0574 and .4790, respectively.
Thus, the present worth of the dividend for the index would be the sum of the
three individual adjusted present worths, or .3303 + .0574 + .4790 = $0.8667.
Note: If the index were a price-weighted index, the index's dividend would be
the sum of these three present worths (.9908 + .2461 + .5987), divided by the divi
sor of that index.
The above fair value formula can be applied to options as well. For example, the
QEX Index does not have futures. However, the fair value calculations can be done
in the same manner, and the synthetic index then constructed by using puts and calls
can be compared with that fair value.
Chapter 30: Stock Index Hedging Strategies 537
Example: Suppose that OEX is trading at 364.50 and a September OEX future - if
one existed would have a fair value of 367.10. That is, the future would command
a premium of 2.60. Not only should a future trade with that theoretical premium, but
so should the "synthetic OEX" composed of puts and calls at the same strike. Hence,
the synthetic OEX constructed with options should trade at about 367.10 also.
That is, if the OEX Sep 365 call were selling for 4.60 and the Sep 365 put were
selling for 2.50, then the synthetic OEX constructed by the use of these two options
would be priced at 367.10. Recall that one determines the synthetic cost by adding
the strike, 365, to the call price, 4.60, and then subtracting the put price: 365 + 4.60
2.50 = 367.0. This synthetic price of 367.10 is literally the same as the theoretical
futures price of 367.10.
The same calculations can be applied to any index with listed options trading.
Let us now return to the broader subject at hand - trading market baskets of stocks
against futures.
PROGRAM TRADING
Two terms that conjure up images of the stock market crash in 1987 and other severe
price drops are "program trading" and "index arbitrage." Neither one by itself should
affect the stock market, since they are two-sided strategies - involving b


dex with listed options trading.
Let us now return to the broader subject at hand - trading market baskets of stocks
against futures.
PROGRAM TRADING
Two terms that conjure up images of the stock market crash in 1987 and other severe
price drops are "program trading" and "index arbitrage." Neither one by itself should
affect the stock market, since they are two-sided strategies - involving buying stocks
and selling futures. This two-sided aspect should have little effect on the market, the
oretically. However, in practice, it is often the case that trades are not executed simul
taneously, and the stock market takes a jump or a dive.
Program trading is nothing more than trading futures against a general stock
portfolio. Index arbitrage is trading futures against the exact stocks that comprise an
index.
Later discussions will assume that one is trying to create or simulate the index
itself in order to hedge it with futures. This is the arbitrage approach. However, there
are many other types of stock positions that may be hedged with the futures. These
might include a portfolio of one's own construction containing various stocks, or
might include a group of stocks from which one wants to remove "market risk."
Normally, one would not own the makeup of any index, but rather would have a
unique combination of stocks in his portfolio. Such an investor may want to use
futures to hedge what he does own.
One reason why an investor who owned stocks would want to sell index prod
ucts against them might be that he has turned bearish and would prefer to sell futures
rather than incur the costs involved with selling out his stock portfolio (and repur
chasing it later). Commission charges are quite small on futures transactions as com-
538 Part V: Index Options and Futures
pared to an equal dollar amount of stock. By selling the futures on an index - say, the
S&P 500- he removes the "market risk" from his portfolio (assuming the S&P 500
represents the "market"). What is left over after selling the futures is the "tracking
error." The discrepancy between the movement of the general stock market and any
individual portfolio is called "tracking error." This investor will still make money if his
portfolio outperforms the S&P 500, but he will find that he did not completely elim
inate his losses if his portfolio underperforms the index. Note that if the market goes
up, the investor will not make any money except for possible tracking error in his
favor.
REMOVING THE MARKET RISK FROM A PORTFOLIO
Stock portfolios are diverse in nature, not :p.ecessarily reflecting the composition of
the index underlying the futures contracts. The characteristics of the individual
stocks must be taken into account, for they may move more quickly or more slowly
than "the market." Let us spend a moment to define this characteristic of stocks that
is so important.
VOLATILITY VERSUS BETA
Recall that when we originally defined volatility for use in the Black-Scholes model,
we stated that Beta was not acceptable because it was strictly a measure of the cor
relation of a stock's performance to that of the stock market and was not a measure
of how fast the stock changed in price. Now we are concerned with how the stock's
movement relates to the market's as a whole. This is the Beta.
Unfortunately, Beta is not as readily available to the option strategist as is
volatility. Many option traders merely have to punch a button on their quote
machines and they can receive estimates of volatility. However, Beta estimates are
more difficult to obtain, and the ones that are available are often for very long time
periods, such as several years. These long-term Betas cannot be used for the pur
poses of the index hedging discussed in this chapter. Therefore, if one does not have
access to shorter-term Beta calculations, then he can approximate Beta by compar
ing an individual stock's volatility with the market's volatility.
Example: XYZ is a relatively volatile stock, having both an implied and historical
volatility of 36%. The overall stock market has a volatility of 15%. Therefore, one
could approximate the Beta of XYZ as
Beta approximation = 36/15 = 2.40
Chapter 30: Stock Index Hedging Strategies 539
There are certain situations in which this approximation would not work well,
because the stock has little or no correlation to the overall stock market (e.g., gold or
oil stocks). If one has a portfolio of stocks of that type, then he should make a serious
attempt to attain their Betas, for the Beta estimate method just described will not be
accurate. Such stocks may be volatile - that is, they change in price fairly rapidly -
but they may go in totally different directions from the overall stock market: They
would thus have high volatility, but low Beta. This is not conducive to the above
short-cut for approximating Beta from volatility.
The remaining examples in this chapter use the terms Beta and adjusted volatil
ity synonymously. Adjusted volatility is merely the approximation of Beta from
volatility as described above: the stock's volatility divided by the market's volatility.
THE PORTFOLIO HEDGE
In attempting to hedge a diverse portfolio, it is necessary to use the Beta or adjusted
volatility because one does not want to sell too many or too few futures. For exam
ple, if the portfolio were composed of nonvolatile stocks and one sold too many
futures against it, one could lose money if the market rallied, even if his portfolio out
performed the market. This would happen because the general market, being more
volatile, would rally farther than the nonvolatile portfolio. Ideally, one should sell
only enough futures so that there would be no gain or loss if the market rallied. There
would be only tracking error. Conversely, if one does not sell enough futures against
a volatile portfolio, then there is risk of loss if the market declines, since the portfo
lio would decline faster


market, being more
volatile, would rally farther than the nonvolatile portfolio. Ideally, one should sell
only enough futures so that there would be no gain or loss if the market rallied. There
would be only tracking error. Conversely, if one does not sell enough futures against
a volatile portfolio, then there is risk of loss if the market declines, since the portfo
lio would decline faster than the market.
The Beta or adjusted volatility of each stock is used in order to determine the
proper number of futures to sell against the portfolio. The dollar value (capitaliza
tion) of each stock in the index is adjusted by that stock's volatility to give an "adjust
ed capitalization" for each stock. Then, when all these are added together, one will
have determined how much "adjusted capitalization" must be hedged with futures.
The suggested method, described in the following example, uses an adjusted volatil
ity for each stock.
The steps to follow in determining how many futures to sell against a diverse
portfolio of stocks are as follows:
1. If you don't know the Beta, divide each stock's volatility by the market's (S&P
500) volatility. This is the stock's adjusted volatility.
2. Multiply the quantity of each stock owned by its price and then multiply by the
adjusted volatility from step 1. This gives the adjusted capitalization of the stock
in the portfolio.
540 Part V: Index Options and Futures
3. Add the results from step 2 together for each stock to get the total adjusted cap
italization of the portfolio.
4. Divide the sum from step 3 by the index price of the futures to be used and the
unit of trading for the futures ($500 per point for the S&P 500 futures) to deter
mine how many futures to sell.
Example: Suppose that one owns a portfolio of three diverse stocks: 3,000 GOGO,
an over-the-counter technology stock; 5,000 UTIL, a major public utility stock; and
2,000 OIL, a large oil company. The owner of this portfolio has become bearish on
the market and would like to sell futures against the portfolio. He needs to determine
how many futures to sell.
The prices and volatilities of these stocks are given in the following table. Assume
th't the volatility of the fictional ZYX Index is 15%. This is the "market's volatility" that
is divided into each stock's volatility to get its adjusted volatility (step 1, above).
Adjusted Adjusted
Volatility Quantity Capitolization
Stock Volatility (Step l) Price Owned (Step 2)
GOGO .60 ·4.00 25 3,000 $300,000
UTIL .12 0.80 60 5,000 240,000
OIL .30 2.00 45 2,000 180,000
Total adjusted capitalization: $720,000 (step 3)
Now suppose that the ZYX Index is trading at 178.65 and a 1-point move in the
futures is worth $500. Step 4 can now be calculated: $720,000 + 500 + 178.65, or 8.06
futures contracts. Thus, the sale of 8 futures contracts would adequately hedge this
diverse portfolio.
There is an important nuance in this simple example: The price of the index
should be used in aU hedging calculations, as opposed to using the price of the future.
There are many examples of hedging portfolios and market baskets with futures or
options in this chapter and the next. Regardless of the situation, the value of the index
should always be used to determine how much stock to buy or how much of the
derivative security to sell.
Note that the actual capitalization of the above example portfolio was only
$465,000 ($75,000 for GOGO, $300,000 for UTIL, and $90,000 for OIL). However,
the portfolio is more volatile than the general market because of the presence of the
two higher-volatility stocks. It is thus necessary to hedge $720,000 worth of "market,"
Chapter 30: Stock Index Hedging Strategies 541
or adjusted capitalization, in order to compensate for the higher volatility of the port
folio.
A similar process can be used for far larger portfolios. The estimate of volatili
ty is, of course, crucial in these calculations, but as long as one is consistent in the
source from which he is extracting his volatilities, he should have a reasonable hedge.
There is no way to judge the future performance of a portfolio of stocks versus the
ZYX Index. Thus, one has to expect a rather large tracking error. In this type of
hedge, one hopes to keep the tracking error down to a few percent, which could be
several points in the futures contracts over a long enough period of time. Of course,
the tracking error can work in one's favor also. The main point to recognize here is
that the vast majority of the risk of owning the portfolios has been eliminated by sell
ing the futures contracts. The upside profit potential of the portfolios has been elim
inated as well, but the premise was that the investor was bearish on the market.
Note that if the futures are overpriced when one enacts his bearishly-oriented
portfolio hedge, he will gain an additional advantage. This will act to offset some neg
ative tracking error, should such tracking error occur. However, there is no guaran
tee that overpriced futures will be available at the time that the investor or portfolio
manager decides to tum bearish. It is better to sell the futures and establish the
hedge at the time one turns bearish, rather than to wait and hope that they will
acquire a large premium before one sells them.
HEDGING PORTFOLIOS WITH INDEX OPTIONS
As mentioned earlier, one could substitute options for futures wherever appropriate.
If he were going to sell futures, he could sell calls and buy puts instead. In this sec
tion, we are also going to take a more sophisticated look at using index options against
stock portfolios.
First, let us examine how the investor from the previous example might use
index options to hedge his portfolio.
Example: Suppose that an investor owns the same portfolio as in the previous exam
ple: 3,000 COCO, 5,000 UTIL, and 2,000 OIL. He decides to hedge with index
UVX, which has options worth $100 per point. Assume that the volatility of the UV


ing index options against
stock portfolios.
First, let us examine how the investor from the previous example might use
index options to hedge his portfolio.
Example: Suppose that an investor owns the same portfolio as in the previous exam
ple: 3,000 COCO, 5,000 UTIL, and 2,000 OIL. He decides to hedge with index
UVX, which has options worth $100 per point. Assume that the volatility of the UVX
is 15%. This investor would then compute his total adjusted capitalization in the same
manner as in the previous example, again arriving at a figure of $720,000.
Suppose that the UVX Index is at 175.60. This investor would want to hedge his
$720,000 of adjusted capitalization with 4,100 "shares" of UVX ($720,000 + 175.60).
Since a 1-point move in UVX options is worth $100, this means that one would sell
41 UVX calls and buy 41 UVX puts. He would probably use the 175 strike or possi-
542 Part V: Index Options and Futures
bly the 180 strike, since those strikes are the ones whereby the calls have the least
chance for early assignment.
Where short options are involved, as with the calls in the above example, one
must be aware of the possibility of early assignment exposing the portfolio.
Consequently, if the marketplace has an equal premium on the futures and the "syn
thetic" UVX, one should sell the futures in that case, because there is no possibility
of unwanted assignment. However, if the options represent a synthetic price that is
more expensive than the futures, then using the options may be more attractive.
Example: Suppose that our same investor has decided to hedge his portfolio with its
$720,000 of adjusted capitalization. He is indifferent as to whether to use the ZYX
futures or the UV:X options. He will use whichever one affords him the better oppor
tunity. The following table depicts the prices of the securities that he is considering,
as well as their fair values.
Current Fair Index
Security Price Value Price
ZYX Jun Future 180.50 180.65 178.65
UVX Jun 175 Call 5 5 175.60
UVX Jun 175 Put 2 21/2 175.60
UVX Jun 180 Call 21/2 21/2 175.60
UVX Jun 1 80 Put 41/2 5 175.60
This investor essentially has three choices: (1) to use the ZYX futures, (2) to use
the UVX options with the 175 strike, or (3) to use the UV:X options with the 180
strike. Notice that the ZYX future is trading 15 cents below its fair value (180.50 vs.
180.65). The UV:X Index fair value, as shown by the fair values of the options, is
177.50. This can be computed by adding the call price to the strike and subtracting
the put price. In the case of either strike, the fair values indicate a UV:X Index fair
value of 177.50.
However, the actual markets are slightly out of line. When using the actual
prices, one sees that he can sell the UV:X Index synthetically for 178.00 whether he
uses the l 75's or the 180's. Thus, by using the UV:X options he can sell the UVX
"future" synthetically for ½ point over fair value, while the ZYX futures would have
to be sold at 15 cents under fair value. Thus, the options appear to be a better choice
since 65 cents ( the 50 cents that the UV:X options are overvalued plus the 15 cents
that the futures are undervalued) is probably enough of an edge to offset the possi
bility of early assignment.
Chapter 30: Stock Index Hedging Strategies 543
With the futures having been eliminated as a possibility, the investor must now
choose which strike to use. Since he will be selling calls and buying puts, and since
either strike allows him to synthetically sell the UVX "future" at 178, he should
choose the 180 strike. This should be his choice because the 180 calls are out-of-the
money and thus less likely to be the object of an early assignment.
HEDGING WITH INDEX PUTS
Let us now move on to discuss ways of hedging in which a complete hedge is not
established, but rather some risk is taken. The main difference between options and
futures is that futures lock in a price, while options lock in a worst-case price (at
greater cost) but leave room for further profit potential. To see this, consider a long
stock portfolio hedged by short futures. In this case, one eliminates his upside profit
potential except for positive tracking error. However, if he buys put options instead,
he expends money - thereby incurring a greater cost to himself than if he had used
futures - but he still has profit potential if the market rallies.
One could hedge a long stock portfolio with options by either buying index puts
or selling index calls. Buying the puts is generally the more attractive strategy, espe
cially if the puts are cheap. In order to properly establish the hedge, it is not only nec
essary to adjust the dollars of stock in accordance with the Beta, but the deltas of the
options must be taken into account as well. The following example will demonstrate
the use of puts to hedge a portfolio of diverse stocks.
Example: Assume that an investor has the same portfolio of three stocks that was
used in a previous example: 3,000 GOGO, 5,000 UTIL, and 2,000 OIL. He has
become somewhat bearish on the market in general and would like to hedge some of
his downside risk. However, he decides to use puts for the hedge just in case there is
a further rally in the market.
The table from the earlier example is reprinted below, showing the adjusted
volatilities and capitalizations for each stock in the portfolio. The total adjusted cap
italization of the portfolio is $720,000, as before.
Adjusted Adjusted
Volatility Quantity Capitalization
Stock Volatility (Step l) Price Owned (Step 2)
GOGO .60 4.00 25 3,000 $300,000
UTIL .12 0.80 60 5,000 240,000
OIL .30 2.00 45 2,000 180,000
Total adjusted capitalization: $720,000 (step 3)
544 Part V: Index Options and Futures
There are two ways that one might want to approach hedging this $720,000
portfolio with puts.
1. As disaster insurance: Buy enough (out-of-the-money) puts so that the portfolio
would be 100% hedged below th


p 2)
GOGO .60 4.00 25 3,000 $300,000
UTIL .12 0.80 60 5,000 240,000
OIL .30 2.00 45 2,000 180,000
Total adjusted capitalization: $720,000 (step 3)
544 Part V: Index Options and Futures
There are two ways that one might want to approach hedging this $720,000
portfolio with puts.
1. As disaster insurance: Buy enough (out-of-the-money) puts so that the portfolio
would be 100% hedged below the striking price of the puts.
2. As a hedge against current market movements: Buy enough puts so that all cur
rent portfolio movements are hedged.
Example - Method 1: In this method, the portfolio manager is looking for disaster
insurance. He is not so much concerned with hedging current market movements as
he is with preventing a major loss if the market should collapse. The manager often
uses an out-of-the-money put for disaster insurance.
Assume that he is going to use the UVX Index puts, which are worth $100 per
point. The March 170 puts, trading at 1, are going to be used in the hedge. The index
is currently at 178.00.
He would therefore divide his portfolio's adjusted capitalization ($720,000) by
the value of the striking price of the puts to be used. In this case, the value of the
striking price is $17,000 (100 x 170).
Puts to buy = $720,000 I $17,000 = 42.3
Cost of 42 puts: $4,200
Striking value of 42 puts: $714,000 (42 x $17,000)
The cost of buying 42 puts is $4,200. This can be thought of as an insurance pre
mium, paid to buy $714,000 worth of insurance. He will have market risk on his port
folio between the current price of the index (178.00) and the striking price (170.00).
The 42 puts would hedge a little of the drop in his portfolio during that 8-point drop
in the index, but their full protective value would not be felt until they were in-the
money. It is not an exact hedge, of course, since the UVX Index may perform differ
ently from the portfolio once UVX drops below 170. However, this put purchase will
definitely remove a great deal of the market risk of further drops.
Example - Method 2: In this method, the portfolio manager is attempting to hedge
the current value of his portfolio. He wants no further downside losses in his portfo
lio at all. He would generally buy at- or in-the-money puts in this case and would use
the put's delta in order to construct a complete hedge.
Again assume that he is going to use the UVX Index puts, which are worth $100
per point. In this case, however, with the index at 178.00, he is considering the March
180 puts, trading at 4½, with a delta of -0.60 to be used in the hedge.
Chapter 30: Stock Index Hedging Strategies 545
In this case, the number of puts is determined by using the same formula as in
the above example and then also dividing by the absolute value of the delta:
Puts to buy = $720,000 / (100 X 180) / 0.60
= 67
Cost of protection: 67 x $450 = $30,150
In this case, the portfolio manager is spending much more for the puts, but for
his additional expense, he acquires immediate protection for his portfolio.
Furthermore, there is some intrinsic value to the puts he bought (2 points, or $13,400
on 67 of them). If the UVX Index drops at all, these puts will immediately begin to
hedge his entire portfolio against loss. Of course, if the market rises, he loses his
much more expensive insurance cost.
When one uses options instead of futures to hedge his position, he must make
adjustments when the deltas of the options change. This was not the case when futures
were used; perhaps with futures, one might recalculate the adjusted capitalization of
the portfolio occasionally, but that would not be expected to affect the quantity of
futures to any great degree. With put options, however, the changing delta can make
the position delta short when the market declines, or can make it delta long if the mar
ket rises. This situation is akin to being long a straddle the position becomes delta
short as the market declines and becomes delta long as the market rises.
Basically, the adjustments would be same as those that a long straddle holder
would make. If the market rallied, the position would be delta long because the delta
of the puts would have shrunk and they would not be providing the portfolio with as
much adjusted dollar protection as it needs. The investor might roll the puts up to a
higher strike, a move that essentially locks in some of his stock profits. Alternatively,
he could buy more puts at the current (low) strike to increase his protection.
Conversely, if the market had declined immediately after the position was
established, the investor will find himself delta short. The delta of the long puts will
have increased and there will actually be too much protection in place. His adjust
ment alternatives are still the same as those of a long straddle holder - he might sell
some of the puts and thereby take a profit on them while still providing the required
protection for the stock portfolio. Also, he might roll the puts down to a lower strike,
although that is a less desirable alternative.
HEDGING WITH INDEX CALLS
Another strategy to protect a stock portfolio is to establish a ratio write using short
calls against the long stock. This is the opposite of using puts for protection, in that
it is more equivalent to being short a straddle.
546 Part V: Index Options and Futures
Example: In the last example, the March 180 put had a delta of -0.60. The March
180 call should then have a delta of 0.40. If the portfolio manager wanted to hedge
his portfolio by ratio writing calls against it, he could use the same formula as in the
previous example:
Calls to sell = $720,000 I (100 x 180) I 0.40 = 100
He would sell 100 calls to hedge his portfolio.
Adjustments would be made in much the same manner as those that a straddle
seller would make. If the market rises, the delta of the calls will increase and the posi
tion will be delta short. One would probably buy calls in that case. Follow-up ac


use the same formula as in the
previous example:
Calls to sell = $720,000 I (100 x 180) I 0.40 = 100
He would sell 100 calls to hedge his portfolio.
Adjustments would be made in much the same manner as those that a straddle
seller would make. If the market rises, the delta of the calls will increase and the posi
tion will be delta short. One would probably buy calls in that case. Follow-up action
for an actual short straddle might dictate buying in some of the underlying security
rather than buying the calls, but that is not a realistic alternative in this case, since the
sample portfolio is probably stable.
If, on the other hand, the market declined after the short calls were sold against
the portfolio, the position would become delta long as the delta of the calls shrinks.
The normal action in that case would be to roll the calls down and reestablish the
proper amount of protection for the portfolio.
Overall, hedging the portfolio with short index calls does not present as attrac
tive a position as hedging with long index puts. This is due mostly to the nature of
what the portfolio manager is trying to accomplish, as opposed to the relative merits
of long and short straddles. As was pointed out in previous chapters, straddle selling,
while risky, is an excellent strategy on a statistical basis. However, in this section we
are not dealing with a strategist who is going to go out and buy stocks and then write
index calls against them. Rather, we have an existing portfolio and the portfolio man
ager is becoming bearish on the market. Thus, the stock portfolio is a fixed entity and
the index options or futures are being built around it for protection.
Long puts serve the purpose of protection far better than short calls, for the fol
lowing reasons. First, the types of adjustments that need to be made by a straddle
seller often involve buying stock or at least buying relatively deep in-the-money calls.
A portfolio manager or investor holding a portfolio stock may not need or want to get
involved in a multi-optioned position. Second, with calls there is large risk to the
upside in case of a large market rally. Someone holding a portfolio of stocks might be
willing to forego upside profits ( as in the sale of futures), but generally would be quite
upset to sustain large losses on the upside. Using puts, of course, leaves room for
upside profit potential. Third, there is risk of early assignment with short index calls,
although that is of minor significance in this case since the portfolio of stocks would
have been long in any case. Other calls could be written immediately on the day after
the assignment. The only real drawback to using the puts is that premium dollars are
Cbapter 30: Stock Index Hedging Strategies 547
paid out and, if the market stabilizes, the time value decay will cause a loss on the
puts. If one actually suspects that such a stabilization might occur, he should use
futures against his position instead of puts or calls.
INDEX ARBITRAGE
As previously stated, index arbitrage consists of buying virtually all of the stocks in an
index and selling futures against them, or vice versa. Whenever the futures on an
index are mispriced, as determined by comparing their actual value with their fair
value, there may be opportunities for arbitrage if the mispricing is large enough.
When futures are extremely overpriced: buy stocks, sell futures; or when futures are
underpriced: sell stocks, buy futures. In either case, the arbitrageur is attempting to
capture the differential between the fair value price of the futures contract and the
price at which he actually buys or sells the index. First, we will examine fully hedged
situations - ones in which the entire index is bought or sold. After that, we will exam
ine smaller sets of stocks that are designed to simulate the performance of the entire
index.
Hedging indices which contain fewer stocks is easier than hedging larger
indices. Hedging a price-weighted index is probably the simplest type of hedge. As
examples, the same sample indices that were constructed in the previous chapter will
be used.
Whenever futures or index options trade on an index, it is possible to set up
market baskets for arbitrage. The trader should determine, in advance, how many
shares of each stock he will buy or sell in order to duplicate the index. In a price
weighted index, of course, he will buy the same number of shares of each stock. In a
capitalization-weighted index, he will be buying different numbers of shares of each
stock. Let us first look at how the number of shares to buy is determined. Then we
will discuss some of the nuances, such as monitoring bids and offers of the indices,
order execution, and others.
HOW MANY SHARES TO BUY
In advance of actually trading the stocks and futures or options, one should deter
mine exactly how many shares of each stock he will be buying in each index he plans
to arbitrage. Normally, one would decide in advance how many futures contracts or
option contracts he will trade at one time. Then the number of shares of stock to be
bought as a hedge can be determined as well. Essentially, one is going to hedge equal
dollar amounts - that is, he will buy enough stocks to offset the total dollar amount
represented by the index.
548 Part V: Index Options and Futures
Example: Suppose that one decides he will set up his market baskets by using 50
ZYX futures at a time. How much stock should he buy against these 50 contracts?
The futures contract has a trading unit of $500 per point. Assume the ZYX Index is
trading at 168.89. Then the total dollar amount represented by 50 contracts is 50 x
$500 x 168.89 = $422,225. The hedger would buy this much stock to hedge 50
futures contracts sold.
Again note that the index price, not the futures price, is used in order to deter
mine how many futures to sell.
In a price-weighted index, one determines the number of shares t


per point. Assume the ZYX Index is
trading at 168.89. Then the total dollar amount represented by 50 contracts is 50 x
$500 x 168.89 = $422,225. The hedger would buy this much stock to hedge 50
futures contracts sold.
Again note that the index price, not the futures price, is used in order to deter
mine how many futures to sell.
In a price-weighted index, one determines the number of shares to buy by
determining the total dollar value of the index he plans to trade and then dividing
that dollar amount by the divisor of the index. The resulting number is how many
shares of each stock to buy in order to duplicate the price-weighted index.
Example: Suppose that we have a price-weighted index composed of three stocks, A,
B, and C. The following data describe the index:
Stock Price
A 30
B 90
C 50
Price total: 170
Divisor: 1.65843
Index value: 102.51
The number of shares of each stock that is in the index is 1 divided by the divisor, or
1/1.65843 = 0.60298 shares. Thus, if we were to buy .60298 shares of each of the
three stocks, we would have created the index.
Suppose that futures exist on this index and that the trading unit in these futures
is $250 per point. That is, the futures represent a total dollar value of the index times
250. With this information, it is easy to determine the number of shares of stock to
buy to hedge one futures contract: 250 times the number of shares of each stock,
.60298, for a total of 150.745 shares of each stock.
Normally, one would not merely sell one futures contract and hedge it with
stock. Rather, he would employ larger quantities. Say that he decided to trade in lots
of 100 futures contracts versus the stocks. In that case, he would buy the following
number of shares of each stock:
Number of shares = .60298 x $250/point x 100 futures contacts
= 1507 4.5 shares
Chapter 30: Stock Index Hedging Strategies S49
Actually he would probably buy 15,100 shares of each stock against the index,
and on every fourth "round" (100 futures vs. stock) would buy 15,000 shares. This
would be a very close approximation without dealing in odd lots.
The trader might also use index options as his hedge instead of futures. The
striking price of the options does not come into play in this situation. Typically, one
would fully hedge his position with the index options - that is, if he bought stock, he
would then sell calls and buy puts against that stock. Both the puts and the calls
would have the same strike and expiration month. This creates a riskless position.
This position is a conversion.
Example: Suppose that cash-based options trade on this index, and that these
options are worth $100 per point as are normal stock options - that is, an option is
essentially an option on 100 shares of the index. The trader is going to synthetically
short the index by buying 100 June 105 puts and selling 100 June 105 calls. Assume
that the index data is the same as in the previous example, that 0.60298 shares of each
stock comprise the index. How many shares would one hedge these 100 option syn
thetics with?
Number of shares = .60298 x 100 contracts x 100 shares/contract
= 6029.8 shares
Note that in the case of a price-weighted index, neither the current index value
nor the striking price of the options involved (if options are involved) affects the
number of shares of stock to buy. Both of the above examples demonstrate the fact
that the number of shares to buy is strictly a function of the divisor of the price
weighted index and the unit of trading of the option or future.
Hedging a capitalization-weighted index is more complicated, although the
technique revolves around determining the makeup of the index in terms of shares
of stock, just as the price-weighted examples above did. Recall that we could deter
mine the number of shares of stock in a capitalization-weighted index by dividing the
float of each stock by the divisor of the index. The general formula for the number of
shares of each stock to buy is:
Shares of stock N Shares of N F . Futures unit
= . . x utures quantity x . to buy m mdex of trading
We will use the fictional capitalization-weighted index from the previous chap
ter to illustrate these points.
Example: The following table identifies the pertinent facts about the fictional index,
including the important data: number of shares of each stock in the index.
550
Stock
A
B
C
Price Float
40 177,000,000
80 50,000,000
60 l 00,000,000
Total capitalization:
Divisor: 147,500,000
Index value: 115.80
Part V: Index Options and Futures
Capitalization
7,080,000,000
4,000,000,000
6,000,000,000
17,080,000,000
Shares
l.20
0.34
0.68
Thus, if one were to buy 1.20 shares of A, .34 shares of B, and .68 shares of C, he
would duplicate the index. Recall that one determines the number of shares of an
individual stock in a capitalization-weighted index by dividing the float of the stock
by the divisor of the index.
Suppose that a futures contract trades on this index, with one point being worth
$500 in futures profit or loss. Then one would buy an amount of each stock equal to
500 times the number of shares in the index. Further suppose that one decides to
trade 5 futures at a time. Thus, the number of shares of each stock that one would
have to buy to hedge the 5-lot futures position would be:
Shares to buy = Shares in index x 5 futures x $500/future
The following table lists that information, as well as totaling the dollar amount
of stock represented by the total. We will verify that the dollar amount of stock pur
chas!:d is equal to the dollar amount of index represented by the futures.
Shares Shares to Buy to S Amount of
Stock in Index Hedge 5 Futures Price Stock Bought
A 1.20 3,000 40 $120,000
B 0.34 850 80 68,000
C 0.68 1,700 60 102,000
$290,000
Thus, $290,000 worth of stock has been purchased. From an earlier example,
we saw how to compute the total dollar worth of a futures trade. In t


ock pur
chas!:d is equal to the dollar amount of index represented by the futures.
Shares Shares to Buy to S Amount of
Stock in Index Hedge 5 Futures Price Stock Bought
A 1.20 3,000 40 $120,000
B 0.34 850 80 68,000
C 0.68 1,700 60 102,000
$290,000
Thus, $290,000 worth of stock has been purchased. From an earlier example,
we saw how to compute the total dollar worth of a futures trade. In this case, the
index is at 115.80, 5 contracts were sold, and each point is worth $500. Thus, the total
dollar amount represented by the futures sale is 5 x 500 x 115.80 = $289,500. This
verifies that our stock purchases hedge the futures sale adequately. Nate that the
slight difference in the stock purchase amount and the futures sale amount is due to
the fact that the number of shares in the index is carried out to only two decimal
points in this example.
Cl,apter 30: Stock Index Hedging Strategies 551
There is an alternative method to determine how many shares to buy. In this
method, one first determines how much stock he is going to buy in total dollars. For
example, he might decide that he is going to buy $10,000,000 worth of the S&P 100
( 0 EX) Index. Next, one determines what percentage his dollar amount is of the total
capitalization of the index. For example, $10,000,000 might be something like .02%
of the total capitalization of the OEX. One would then buy .02% of the total number
of shares outstanding of each of the stocks in the OEX. After the number of shares
of each stock to buy has been determined, one would have to determine ho'Y many
futures to sell against this stock- he would divide $10,000,000 by the index price and
also divide by the unit of trading for the futures. This procedure is demonstrated in
the following example.
Example: Suppose that one wants to set up an arbitrage against the same index as in
the previous example. For purposes of comparison with that example, we will sup
pose that this hedger wants to buy a total of $290,000 worth of stock. In reality, one
would probably use a round number such as $300,000 or $500,000 worth of stock.
However, by making a direct comparison, we will be able to more easily demonstrate
that these two methods produce the same answer.
First, the hedger must· determine the percent of the total capitalization that he
is going to buy. In this case, he is buying $290,000 worth of stock and the total capi
talization of the index is $17,080,000,000 (refer to the table at the beginning of the
previous example). This means that he is buying .0016979% of the total capitalization
of the index.
Next, he uses this percentage and multiplies it by the float of each stock. That
is, he is going to buy .0016979% of the total number of shares outstanding of each
stock in the index. This results in purchases as shown in the following table:
Stock
A
B
C
Float
177,000,000
50,000,000
100,000,000
Shores to Buy
3,005
849
1,698
Compare these share purchases with the previous example. The number of
shares to buy is the same, allowing for rounding off in the previous example. Thus,
these two methods of determining how many shares to buy are equivalent.
Before leaving this section, it should be pointed out that arbitrageurs can also
establish an arbitrage when futures are underpriced. They can sell stocks short and
buy the underpriced futures. This is a more difficult type of arbitrage t~,establish
552 Part V: Index Options and Futures
because short sales must be made on plus ticks. However, when futures are under
priced for an extensive period of time - perhaps during extreme pessimism on the
part of speculators - it is possible to set up the arbitrage from this viewpoint.
PROFITABILITY OF THE ARBITRAGE
The key for many arbitrageurs and institutional investors is whether, after costs, there
is enough of a return in this stock versus futures strategy. The method in which we
previously computed the fair value of the futures will be used in determining the
overall incremental return of doing the arbitrage.
The major cost in executing the arbitrage is the cost of commissions. Since there
are large quantities of stocks being bought or sold when an entire index is traded, the
commission rate is generally quite low. For example, an institutional investor might
pay 3 cents per share or even less. This still could be a substantial cost, especially
when a large index such as the S&P 500 Index is being purchased. Even profession
al arbitrageurs may have to pay commission costs if they are using the services of a
computer firm to buy stocks. These methods of trading stocks are described in the
next section.
Once one's rate of commission charges is known, he can convert that into a
number that represents a portion of the index price. He does so by multiplying his
per-share commission rate by the current index value and then dividing that result by
the average share price of the index. The following example describes that method of
conversion.
Example: Suppose that one is going to buy the entire ZY:X Index at a commission
rate of 3 cents per share. The index is trading at 185.00. Furthermore, assume that
the average price of a share in the index is 45 dollars per share. With this informa
tion, one can determine how much he is paying in commissions, in terms of the index
itself.
Commission rate
Commission in terms = -~p_e_r_s_h_ar_e __ x_I_n_d_e_x_v_al_u_e_
of index Average price
per share
.03 X 185.00 =-----
45
= .123
Thus, a commission rate of 3 cents per share translates into 12.3 cents of index value.
Chapter 30: Stack Index Hedging Strategies 553
The most difficult factor to determine in the above equation is the average price
per share for a capitalization-weighted index. There is a shortcut that can be used. It
is easy to determine the average price per share for a price-weighted index, such as
XMI. The average price per share for large-capitalization indices such as the OEX
and S&P 500


hapter 30: Stack Index Hedging Strategies 553
The most difficult factor to determine in the above equation is the average price
per share for a capitalization-weighted index. There is a shortcut that can be used. It
is easy to determine the average price per share for a price-weighted index, such as
XMI. The average price per share for large-capitalization indices such as the OEX
and S&P 500 is about 80% of that of the XML
Example: It is a simple matter to compute the average price per share. for a price
weighted index. Merely divide the index value by the number of stocks comprising
the index and then multiply that result by the divisor of the index. Suppose the XMI
is trading at 352 and the divisor of the XMI is 4.4. Then, since there are 20 stocks in
the XMI, we can quickly determine the average price per share of stocks in the XMI:
Average price = Index price x Divisor
per share Number of stocks
in index
352 X 4.4 =----
20
= 77.44
Given this information, we can estimate that the average price per share for stocks in
a broader-based index would be about 80% of that figure, or something like 62 or 63
dollars per share.
Now that the commission rate has been converted into an index value, one can
determine his net profit from trading the exact index against the futures. One must
figure in his futures commission costs as well. The following example demonstrates
the net profit from executing the arbitrage, including all costs. Once the net profit
has been calculated, a rate of return can be computed.
Example: Suppose that the ZYX Index is trading at 185.00 and the futures, which
expire in two months, have a fair value premium of 2.00 points, but are trading at
188.50, a premium of 3.50. The futures are worth $500 per point. Thus, the futures
are expensive and one might attempt to buy stocks and sell the futures. His net prof
it consists of the premium over fair value less all costs of entering and exiting the
position.
As we saw in the previous example, at 3 cents per share stock commission, we
pay an index value of .123 to enter the position. Similarly, we would pay .123 in index
value to exit the position at a later date. Thus, the net round-tum stock commission
is approximately 25 cents of index value.
554 Part V: Index Options and Futures
Commissions on futures are generally charged only when the position is closed
out. Generally, a futures commission on an S&P 500 contract might be reduced to
something like $10 per contract for this type of hedging. Since 185.00, the index
value, represents 11500th of the value of the futures contract, we can reduce the
futures commission to an index-related number by dividing the actual dollar com
mission by 500. Thus, the futures commission is, in index terms, 10/500, or .02. The
total commission for entering and exiting the position is thus 0.266 of index value,
0.123 each for the purchase and sale of the stocks and .02 for the futures.
Net profit = Futures
price
Futures fair Commission
value costs
= 188.50 - 187.00 - 0.27
= 1.23
This absolute net profit number can be converted into a rate of return by annualiz
ing the profit and dividing by the current index price. Suppose that there are
two months exactly remaining until expiration. Then the rate of return is computed
as follows:
Incremental = Net profit x ( 1/Time remaining)
rate of return Index price
1.23 X ( 12/2)
185.00
= 3.99%
For the two-month time period, his return is about 2h of one percent.
At first glance, a rate of return of almost 4% does not seem like much. But what
we have computed here is an incremental rate of return. That is, this return is over
and above whatever rate we used in determining the fair value of the futures. Thus,
if an institution were going to invest its cash at the prevailing short-term rate, and that
rate were used to determine the futures fair value in the above example, then the
institution could earn an additional 4%, annualized, if it arbitraged the futures rather
than put its money in the short-term money market.
TRADE EXECUTION
Most customers are not concerned with how the trades are executed, for they give
the order to their broker and let him work out the details. However, for those who
are interested in the actual trade execution, a short section dealing with that topic is
in order.
Chapter 30: Stock Index Hedging Strategies 555
Ideally, one should be able to monitor the progress of his index in terms of bid
prices, offer prices, and last sales. There are several modern quote services that allow
such monitoring. It is important to know the bids and offers because, when one actu
ally executes the trades, he generally will be trading on the bid and offer, not the last
sale.
Example: Suppose the fair value of the futures contract is represented by a premi
um of 1.25 points, but that the actual future is trading with a premium of 2.00 points:
The index is at 165.75 and the futures are 167.75 (last sale). This might seem like
enough "room" to execute a profitable arbitrage - buy the stocks in .the index and sell
the futures. However, the index value of 165. 75 is the composite of the last sales of
each of the individual stocks in the index. If one were to look at the offering prices
of each stock and then recompute the index, he might end up with an index value
that was 50 cents higher. This, then, would mean that he would be doing the arbi
trage for 25 cents less costs, which is not enough of a margin to work with.
Similarly, when one is looking to sell out the stocks he has bought and simulta
neously buy back the futures, he needs to know the bid value of the index in order to
see what kind of premium he is paying to take his position off.
One normally executes the hedge by giving a series of stock orders to brokers
on the floor of the exchange and simultaneously giving a futures order to the futures
exchange. The actual trader controlling the order generally lets the stock


simulta
neously buy back the futures, he needs to know the bid value of the index in order to
see what kind of premium he is paying to take his position off.
One normally executes the hedge by giving a series of stock orders to brokers
on the floor of the exchange and simultaneously giving a futures order to the futures
exchange. The actual trader controlling the order generally lets the stocks be execut
ed at prevailing market prices, but might try to control the futures execution more
closely.
There are two basic ways in which the actual stock order entry takes place. One
is the conventional method of giving orders to brokers on the floor of the New York
Stock Exchange. This method can be somewhat computerized by having a computer
at the broker's main location send a series of orders to various locations of that bro
kerage firm on the floor of the exchange. The orders are then given to several bro
kers who quickly execute them. The speed with which the executions must take place
demonstrates why the price paid is usually the offering price and not the last sale:
There simply isn't enough time for the broker to try to save an eighth of a point on
one stock, since he has several other orders to execute.
The other method of order entry is completely computerized. The computer
knows the quantity of each stock to buy and, when prompted, sends those buy orders
via telecommunications lines to one of the automatic order execution systems on the
exchange floor. The most common automatic system is the POT system on the
NYSE. The system attempts to guarantee the offering price for large quantities of
stock. In this highly sophisticated method of order entry, the entire execution proce-
556 Part V: Index Options and Futures
dure may take place in about one minute for the entire index. This method of order
entry is so quick and accurate that some brokerage firms with this capability offer it
for a commission fee to other brokerage firms that do not have the capability.
INSTITUTIONAL STRATEGIES
Holders of large portfolios of stocks can use futures and/or market basket strategies
to their advantage. There are two basic strategies that can be easily used by these
large traders. One is to buy futures instead of buying stocks, and the other is to sell
futures instead of selling stocks. Both of these strategies will be examined in more
detail.
When one of these large institutions has money to invest in buying stocks, it
might make more sense to buy Treasury bills and futures instead of buying stocks. Of
course, this alternative strategy only makes sense for the institution if the stock pur
chase were going to be broad-based - something akin to duplicating the S&P 500
performance. The institution does not necessarily have to be intent on purchasing an
exact index, but if the purchase were going to be diversified, the purchase of index
futures might help accomplish an equivalent result. If, however, the purchase were
going to be quite specific, then this strategy would probably not apply.
This strategy works best when futures are underpriced. If the equivalent dollar
amount of underpriced futures can be purchased instead of buying stocks, the entire
amount intended for stock purchase can be put in Treasury bills instead. Recall that
cash will have to be put into the futures account if the futures mark at a loss (main
tenance margin). Even so, there can be substantial savings to the institution if the
futures are truly underpriced.
The second institutional strategy is applicable when futures are overpriced and
the institution wants to sell stock. In such a case, it makes more sense to sell the
futures than to sell the stock. First, there are large savings in transaction costs ( com
missions). Second, the overpriced nature of the futures actually means that there is
additional profitability in selling them as opposed to selling the stocks. Again, this
strategy only makes sense if one were going to sell a diversified portfolio of stocks,
something that is broad-based like the S&P 500 Index.
Of course, institutions may want to participate in the arbitrage regardless of their
market stance. That is, if a money manager has a certain amount of money that he is
going to put into short-term instruments (perhaps T-bills), he might instead decide to
participate in this arbitrage of stocks versus futures if the incremental return is high
enough. Recall that we saw how to determine the incremental return in a previous
section of this chapter. If he were going to get a 7½% return from T-bills but could
get an 11 ½% return from futures arbitrage, he might opt for the latter.
Chapter 30: Stack Index Hedging Strategies
FOLLOW-UP STRATEGIES
557
Once any hedge has been established, it must be monitored in case an adjustment
needs to be made. The first and simplest type of monitoring is to take care of spin
offs or other adjustments in stocks in the market basket that is owned. Of a more seri
ous nature, in terms of profitability, one also needs to monitor the hedge to see if it
should be removed or if the futures should be rolled forward into a more distant expi
ration month.
Adjusting one's portfolio for stock spinoffs is a simple matter which we will
address briefly. In many cases, one of the stocks in the index will spin off a division
or segment of its business and issue stock to its shareholders. Such a spinoff is gen
erally not included in the price of the index, so that the hedger should sell off such
items as soon as he receives them, for they do not pertain to his hedge.
In a similar vein, in any portfolio certain stocks may occasionally be targets of
tender offers or other reorganizations. If one does nothing in such a situation, he will
not lose any money in terms of his portfolio versus the underlying index. However, it
is generally wise for one to tender his stock in such situations and replace it at a lower
price after the tender. Sometimes, in


s hedge.
In a similar vein, in any portfolio certain stocks may occasionally be targets of
tender offers or other reorganizations. If one does nothing in such a situation, he will
not lose any money in terms of his portfolio versus the underlying index. However, it
is generally wise for one to tender his stock in such situations and replace it at a lower
price after the tender. Sometimes, in fact, such a tender offer will entirely absorb an
index component member. In that case, one must replace the disappearing stock with
whatever stock is announced as the new member of the index.
Technically, in an arbitrage hedge one should adjust his portfolio every time the
divisor of the index changes. Thus, in a capitalization-weighted hedge he would be
adjusting every time one of the components issues new common stock. This is really
not necessary in most cases, because the new issue is so small in comparison to the
current float of the stock. Such a new issue does not include stock splits, for the divi
sor of the index does not change in that case. A more common case is for one of the
stocks in a price-weighted index to split. In this case, one must adjust his portfolio.
An example of such an adjustment was given in Chapter 29. In essence, one must sell
off some of the split stock and buy extra shares of each of the other stocks in the
price-weighted index.
Let us now take a look at follow-up methods of removing or preserving the
hedge.
ROLLING TO ANOTHER MONTH
As expiration nears, the hedger is faced with a decision regarding taking off the mar
ket basket. If the futures premium is below fair value, he would probably unwind the
entire position, selling the stocks and buying back the futures. However, if the futures
remain expensive - especially the next series - then the hedger might roll his futures.
558 Part V: Index Options and Futures
That is, he would buy back the ones he is short and sell the next series of futures. For
S&P 500 futures, this would mean rolling out 3 months, since that index has futures
that expire every 3 months. For the XMI futures and OEX index options, there are
monthly expirations, so one would only have to roll out 1 month if so desired.
It is a simple matter to determine if the roll is feasible: Simply compare the fair
value of the spread between the two futures in question. If the current market is
greater than the theoretical value of the spread, then a roll makes sense if one is long
stocks and short futures. If an arbitrageur had initially established his arbitrage when
futures were underpriced, he would be short stocks and long futures. In that case he
would look to roll forward to another month if the current market were less than the
theoretical value of the spread.
Example: With the S&P 500 Index at 416.50, the hedger is short the March future
that is trading at 417.50. The June future is trading at 421.50. Thus, there is a 4-point
spread between the March and June futures contracts.
Assume that the fair value formula shows that the fair value premium for the
March series is 35 cents and for the June series is 3.25. Thus, the fair value of the
spread is 2.90, the difference in the fair values.
Consequently, with the current market making the spread available at 4.00, one
should consider buying back his March futures and selling the June futures. The
rolling forward action may be accomplished via a spread order in the futures, much
like a spread order in options. This roll would leave the hedge established for anoth
er 3 months at an overpriced level.
Another way to close the position is to hold it to expiration and then sell out the
stocks as the cash-based index products expire. If one were to sell his entire stock
holding at the time the futures expire, he would be getting out of his hedge at exact
ly parity. That is, he sells his stocks at exactly the last sale of the index, and the futures
expire, being marked also to the last sale of the index.
For settlement purposes of index futures and options, the S&P 500 Index and
many other indices calculate the "last sale" from the opening prices of each stock on
the last day of trading. For some other indices, the last sale uses the closing price of
each stock.
Example: In a normal situation, if the S&P 500 index is trading at 415, say, then that
represents the index based on last sales of the stocks in the index. If one were to
attempt to buy all the stocks at their current offering price, however, he would prob
ably be paying approximately another 50 cents, or 415.50, for his market basket.
Similarly, if he were to sell all the stocks at the current bid price, then he would sell
the market basket at the equivalent of approximately 414.50.
Chapter 30: Stock Index Hedging Strategies 559
However, on the last day of trading, the cash-based index product will expire at
the opening price of the index. If one were to sell out his entire market basket of
stocks at the current bid prices at the exact opening of trading on that day, he would
sell his market basket at the calculated last sale of the index. That is, he would actu
ally be creating the last sale price of the index himself, and would thereby be remov
ing his position at parity.
The problem with this is that it is correct theory, but difficult to put into prac
tice. For example, if one has several million dollars of stocks to sell, he cannot expect
the marketplace to absorb them easily when they are all being sold at the last minute
on a Friday afternoon. We will discuss this more fully momentarily, when we look at
the impact of stock index arbitrage on the stock market in general.
There is another interesting facet of the arbitrage strategy that combines the
spread between the near-term future and the next longest one with the idea of exe
cuting the stock portion of the arbitrage at the close of trading on the day the index
products expire. Use of this strategy actually allows one to enter a


we look at
the impact of stock index arbitrage on the stock market in general.
There is another interesting facet of the arbitrage strategy that combines the
spread between the near-term future and the next longest one with the idea of exe
cuting the stock portion of the arbitrage at the close of trading on the day the index
products expire. Use of this strategy actually allows one to enter and exit the hedge
without having to lose the spread between last sale and bid or last sale and offer in
either case. Suppose that one feels that he would set up the arbitrage for 3 months if
he could establish it at a net price of 1.50 over fair value. Furthermore, if the fair
value of the 3-month spread is 2.10, but it is currently trading at 3.60, then that rep
resents 1.50 over fair value. One initiates the position by buying the near-term future
and selling the longer-term future for a net credit of 3.60 points. At expiration of the
near-term future, rather than close out the spread, one buys the stocks that comprise
the index at the last sale of the trading day, thereby establishing his long stock posi
tion at the last sale price of the index at the same moment that his long futures expire.
The resultant position is long stocks and short futures that expire in 3 months at a
premium of 3.60. Since the fair value of such a 3-month future should be 2.10, the
hedge is established at 1.50 over fair value. The position can be removed at expira
tion in the same manner as described in the previous paragraph, again saving the dif
ferential between last sale and the bids of the stocks in the index. Note that this strat
egy creates buying pressure on the stock market at expiration of the near-term side
of the spread, and selling pressure at the latter expiration.
The final way to exit from one's position is to remove it before expiration.
Sometimes, there are opportunities during the last two or three weeks before the
futures expire. If one hedged with long stock and short futures, the opportunities
to remove the hedge arise when the futures trade below fair value - perhaps even
at an actual discount to parity. If the futures never trade below fair value, but
instead continue to remain expensive, then rolling to the next expiration series is
often warranted.
560 Part V: Index Options and Futures
MARKET BASKET RISK
There are some uncertainties in this type of hedging, even though the entire index is
being bought. Since one owns the actual index, there is no risk that the stocks one
owns will fail to hedge the futures price movements properly. However, there are
other risks. One is the risk of execution. That is, it may appear that the futures are
trading at a premium of 1.50 points when one enters the orders. However, if other
hedgers are doing the same thing at the same time, one may pay more for the stocks
than he thought when he entered his order, and he may sell the futures for less as
well. This "execution risk" is generally small, but if one is too slow in getting his stock
orders executed, he may have set up a hedge that was not as attractive as he first
thought.
One major risk is that interest rates might move against the arbitrageur while
the position is in place. If he is long stocks and short the futures, then he would not
want interest rates to rise. In the previous example, the incremental return for the 2-
month time period that the position was going to be held was 2h of one percent. If
short-term rates should rise by more than that, on average, for the 2-month period,
the incremental strategy would be inferior. His carrying costs would have increased
to the point of wiping out the profit from his arbitrage.
For institutional arbitrageurs who don't exactly have cost of carry, this situation
would be viewed in the following manner: If rates increase, he may find that he
would have been better off having his money invested in a money market fund at the
prevailing short-term rate than in the incremental arbitrage strategy. Conversely, if
the arbitrage was originally established with short stocks and long futures, the arbi
trageur would not want rates to drop for similar reasons.
One might leave a cushion against a movement in rates. If rates are currently
8%, then one might decide to use a 10% rate as a cushion. Hedges established that
are profitable at the higher rate level will consequently be able to withstand rates
moving up to 10%.
Example: Suppose that one would normally use a rate of 7½% and would establish
the long stock versus short futures hedge at an incremental rate of return of 1 ½%.
This is a relatively narrow cushion and if the hedge is on for a moderate length of
time, rates could move up to such an extent that they advance to 9½% or higher.
Such a move would make the hedge position unprofitable. Instead, one might calcu
late the fair value of the futures using a rate equal to his current prevailing rate plus
a cushion. That is, if his current rate is 7½%, he might use 8½% and still demand an
incremental return of 1 ½%. Ifhe established the hedge at these levels, he could suf
fer a move of 1 %, the cushion, against him and still earn his incremental rate of 1 ½%.
Cbapter 30: Stock Index Hedging Strategies 561
Another risk that the arbitrageur faces is that of changes in the dividend payout
of the stocks in the index. Suppose that he is long stocks and short futures. If there
are enough cuts in dividend payout, or dividend payments are delayed past the expi
ration date of the futures, then he will lose some of his return. Arbitrageurs who are
short stocks and long futures would have similar problems if dividend payout were
increased especially if a large special dividend were declared by a company that ,is
a major component of the index - or payment dates were accelerated.
If one holds the arbitrage until expiration, he will be able to unwind it at parity.
However, if he decides to remove the arbitrage before expira


ageurs who are
short stocks and long futures would have similar problems if dividend payout were
increased especially if a large special dividend were declared by a company that ,is
a major component of the index - or payment dates were accelerated.
If one holds the arbitrage until expiration, he will be able to unwind it at parity.
However, if he decides to remove the arbitrage before expiration, he might incur
increased costs that would harm his projected return. Instead of selling his stocks at
the last sale of the index, as he is able to do on expiration day, he would have to sell
them on their bids, a fact that could cost him a significant portion of his profit.
In a later section, where we discuss hedging the futures with a market basket of
stocks that does not exactly represent the entire index, we will be concerned with the
greatest risk of all, "tracking error" - the difference between the performance of the
index and the performance of the market basket of stocks being purchased.
IMPACT ON THE STOCK MARKET
The act of establishing and removing these hedge positions affects the stock market
on a short-term basis. It is affected both before expiration and also at expiration of
the index products. We will examine both cases and will also address how the strate
gist can attempt to benefit from his knowledge of this situation.
IMPACT BEFORE EXPIRATION
When bullish speculators drive the price of futures too high, arbitrageurs will attempt
to move in to establish positions by buying stock and selling futures. This action will
cause the stock market to jump higher, especially since positions are normally estab
lished with great speed and stocks are bought at offering prices. Such acceleration on
the upside can move the market up by a great deal in terms of the Dow-Jones
Industrials in a matter of minutes.
Conversely, if futures become cheap there is also the possibility that arbi
trageurs can drive the market downward. If positions are already established from
the long side (long stock, short futures), then arbitrageurs might decide to unwind
their positions if futures become too cheap. They would do this if futures were so
cheap that it becomes more profitable to remove the position, even though stocks
must be sold on their bid, rather than hold it to expiration or roll it to another series.
562 Part V: Index Options and Futures
When these long hedges are unwound in this manner, the stock market will decline
quickly as stocks are sold on bids. In this case, the market can fall a substantial
amount in just a few minutes.
Once long hedges are unwound, however, cheap futures will not cause the mar
ket to decline. If there is no more stock held long in hedges, then the only strategy
that arbitrageurs can employ when futures become cheap is to sell stock short and
buy futures. Since stock must be sold short on upticks, this action may put a "lid" on
the market, but will not cause it to decline quickly.
Having long stock and short futures when these large discounts occur is so valu
able to an arbitrageur that some traders will establish the long stock/short futures
hedge for no profit or even a loss. They hope that subsequently futures will plunge
to a large discount and they can unwind their positions for large profits. If that never
occurs, they only lose a few cents of index value. Assume futures fair value is 3.50
over. Such arbitrageurs might buy stock and sell futures at a net cost of 3.45 over.
That is, if they hold the position until expiration they will lose 5 cents, but if a large
futures discount ever occurs, they will profit.
Regulatory bodies have become increasingly concerned over the years as to the
effect that program trading and index arbitrage have on the stock market. In reality,
when stocks and futures are executed more or less simultaneously, these strategies
should not overly disturb the stock market. However, since they are not executed
simultaneously (there are no rules in the futures markets against frontrunning), the
New York Stock Exchange has imposed an arbitrary limit, called a "circuit breaker,"
on these activities. If the Dow-Jones averages rise or fall more than a specified dis
tance (for example, 200 points) at any time during a trading day, all computer-driven
program order entry is prohibited for the rest of the day, or until the Dow-Jones
moves back to within 25 points of being unchanged on the day. This trading ban
effectively shuts down program trading and index arbitrage, although it is still allow
able to do it by having the trades executed by individual floor brokers. Some of the
larger trading houses, trading for their own accounts, are therefore still able to exe
cute index arbitrage if the discount in the futures is extremely wide, even when the
trading ban is in effect.
The trading ban is really meant to halt declining markets, although in the inter
est of fairness, the ban goes into effect on days when the Dow-Jones rises by the
specified distance (for example, 200 points) as well. Of the various measures that
have been tried in order to stop a declining market from becoming a crash (e.g., cir
cuit breaker trading limits on S&P 500 futures), this seems to be the most effective.
There have been many times when the Dow-Jones has accelerated to the limit, and
then once the trading ban went into effect, the Dow-Jones would slide slowly back
the other way, but at a much more leisurely pace.
Cl,opter 30: Stock Index Hedging Strategies 563
Readers should remember, of course, that the stock market can move inde
pendently of the overpriced or underpriced nature of index products. That is, if
futures are overpriced, the stock market can still decline. Perhaps there is a prepon
derance of natural sellers of stocks. Similarly, if futures are cheap, the stock market
can still go up if enough traders are bullish. Thus, one should be cautious about try
ing to link every movement of the stock market


move inde
pendently of the overpriced or underpriced nature of index products. That is, if
futures are overpriced, the stock market can still decline. Perhaps there is a prepon
derance of natural sellers of stocks. Similarly, if futures are cheap, the stock market
can still go up if enough traders are bullish. Thus, one should be cautious about try
ing to link every movement of the stock market to index products.
PORTFOLIO INSURANCE
Portfolio insurance is the generic name used to describe a strategy in which a port
folio manager uses the index derivatives market to protect his portfolio in case the
market crashes. He could either sell futures or buy puts.
The generic concept was put into effect using futures in the mid 1980s. In the
form of the strategy that was being practiced at the time, the portfolio manager did
not sell futures against his entire portfolio right away. Rather, he sold only a few to
begin with. This allowed him to retain a good deal of upside profit potential for his
portfolio. If the market dropped further, then he would sell more. Eventually, if it
dropped far enough, he would keep selling futures until his entire portfolio was prop
erly hedged. There were computer programs that calculated when to sell the futures
and how many to sell in order for the portfolio manager to eventually end up with the
proper amount of insurance at the right price.
Unfortunately, the concept did not work properly in practice. In fact, it has
often been identified as one of the major factors in the 500-point crash of October
19, 1987. What happened during the days leading up to that date was that the mar
ket was already going down fast. Futures, as a result, began to trade at a discount.
The portfolio insurance strategy assumes futures are sold at fair value, more or less.
Thus, the portfolio insurance managers did not sell their futures when they had orig
inally intended; or they could not sell enough without driving the futures to tremen
dous discounts. In any case, the market kept going further down without any rebound
(essentially from about mid-afternoon on Thursday, October 15, through the close on
Monday, October 19), a total of over 650 points on the Dow-Jones averages." As the
market plunged, the portfolio insurance strategy kept demanding that more futures
be sold, and they were, but often at prices well below where the strategy had origi
nally dictated. This continued selling kept futures at a discount, which triggered even
more selling by other program traders and index arbitrageurs.
As a result, the portfolios were not completely protected - although it should be
noted that they were somewhat protected since they had been selling some futures.
Hence, the portfolio managers were not pleased. Stock market regulators were not
pleased, either, although nothing illegal had been done. The strategy lost most of its
adherents at that time and has not been resurrected in its previous form.
•This represented a decline of over 25% from the relatively low levels (mid-2000s) that the index was trading
at during that time frame.
S64 Part V: Index Options and Futures
However, the concept is still a valid one, and it is now generally being practiced
with the purchase of put options. The futures strategy was, in theory, superior to buy
ing puts because the portfolio manager was supposed to be able to collect the pre
mium from selling the futures. However, its breakdown came during the crash in that
it was impossible to buy the insurance when it was most needed - similar to attempt
ing to buy fire insurance while your house is burning down.
Currently, the portfolio manager buys puts to protect his portfolio. Many of
these puts are bought directly over-the-counter from major banks or brokerage hous
es, for they can be tailored directly to the portfolio manager's liking. This practice
concerns regulators somewhat, because the major banks and brokerage houses that
are selling the puts are taking some risk, of course. They hedge the sales (with futures
or other puts), but regulators are concerned that, if another crash occurred, it would
be the writers of these puts who would be in the market selling futures. in a mad fren
zy to protect their short put positions. Hopefully, the put sellers will be able to hedge
their positions properly without disturbing the stock market to any great degree.
IMPACT AT EXPIRATION - THE RUSH TO EXIT
Some traders persist in attempting to get out of their positions on the last day, at the
last minute. These traders are not normally professional arbitrageurs, but institu
tional clients who are large enough to practice market basket hedging. Moreover,
they have positions in indices whose options expire at the close of trading (OEX, for
example). If these hedgers have stock to sell, what generally happens is that some
traders begin to sell before the close, figuring they will get better prices by beating
the crowd to the exit. Thus, about an hour before the close, the market may begin to
drift down and then accelerate as the closing bell draws nearer. Finally, right on the
bell that announces the end of trading for the day, whatever stock has not yet been
sold will be sold on blocks - normally significantly lower than the previous last sale.
These depressed sales will make the index decline in price dramatically at the last
minute, when there is no longer an opportunity to trade against it.
These blocks are often purchased by large trading houses that advance their
own capital to take the hedgers out of their positions. The hedgers are generally cus
tomers of the block trading houses. Normally, on Monday, the market will rebound
somewhat and these blocks of stock can be sold back into the market at a profit.
Whatever happens on Monday, though, is of little solace to the trader trapped
in the aftermath of the Friday action. For example, if one happened to be short puts
and the index was near the strike as the cl