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By Lawrence G. McMillan

This article was originally published in The Option Strategist Newsletter Volume 10, No. 23 on December 13, 2001. 

This strategy was mentioned in the “Striking Price” column in Barron’s last Sunday, and we have received several questions from subscribers asking about the strategy. The strategy has been around for a long time – since the inception of index options, actually – but it is something of a professional strategy, so it’s not widely know. However, it is gaining more popularity lately, so it is the subject of this week’s feature article.

In essence, the strategy is fairly simple to describe. One sells (overpriced) index options and buys (cheaper) options on the individual components of the index. This “arbitrage-like” spread should make money by expiration, if not sooner, because the overpriced/underpriced nature of things must disappear by then. In reality, the mechanics of running a dispersion strategy can get quite involved.

By the way, the term “dispersion strategy” is a relatively new one. I’m not sure who coined that term, but when I was running an arbitrage desk on Wall Street in the ‘80's, we used the strategy but called it “thedic” (rhymes with medic) – named after someone on our trading desk.

The strategy is most often applied by using index put options, since they are so expensive – especially the out-of-the-money puts. If you look at the table of index option implied volatility (page 7, box on the right), you’ll see that out-of-the-money $OEX puts are trading with an implied volatility of about 26%. while the at-the-moneyputs have an implied volatility of 23%. Moreover, the actual (statistical) volatility of $OEX is about 18% right now. So, if one were to sell those puts with an implied volatility of 26%, he would be selling something that is “overpriced” with respect to the volatility of the actual underlying index and its at-the-money options.

Out-of-the-money index puts have been consistently overpriced since the Crash of ‘87. As a result, the strategy of selling out-of-the-money index puts and buying puts on individual stocks in the same index has been fairly popular among professionals and well-heeled investors ever since.

In theory, trying to take advantage of a mispricing of index options as compared to options on individual stocks in the index could be done with calls, and could sometimes involve buying the index options and selling the stock options – if things lined up properly from a theoretical perspective. In reality, it’s usually done with puts because that’s where the greatest pricing discrepancies lie.

A Sample Index
In order to demonstrate how the strategy works, let’s assume that there is a theoretical price-weighted index composed of three stocks of varying volatility:

Stock    Price    Volatility
GOGO     60       60%
OIL      40       20%
CARS     55       40%

Assume that the index in this case is the sum of the three stock prices, multiplied by 3. All price-weighted indices are computed in a similar manner: sum the prices of the individual stocks in the index, and then multiply by an arbitrary constant (usually, it’s referred to as a divisor, but you get the idea).

Even though the individual stock volatilities are shown in the above table, the volatility of the index will not merely be the average of those three volatilities. In reality, an index has a lower volatility than the average of its component’s volatilities because all stocks don’t move up and down in unison. On a given trading day, some stocks are going to be up and others are going to be down. The net result affects the index movement, and it is therefore less volatile than the average of its components. This is true for any index – whether capitalization- or priceweighted.

So, even though the average of the three volatilities in the above table is 40%, the index itself might have a volatility of 30%, say. As a real-life example, the average of the volatilities of the 100 stocks that comprise $OEX was 34.9% (as of 12/6/01), but the 20-day historical volatility of the $OEX index itself was only 18%.

To see how the strategy might work in practice, let’s make one more theoretical assumption: let’s assume the each of the stocks and the index itself have options that expire in 3 months and that each has a striking price that is 10% out-of-the-money for a put option. Finally assume that the out-of-the-money index puts are trading with an inflated volatility of 45% – above the historical volatility of 30%. If so, the following theoretical values would exist:

Underlying   Volatility    3-Month Put
                         10% Out-of-Money
GOGO         60%             4.20
OIL          20%             0.25
CARS         40%             1.95

Index       45%*             20.33
      *:out-of-money puts have inflated volatility

The three individual stock puts above cost 6.40 in total. Since the multiplier of the index is 3, it would be necessary to buy 3 of each of the individual stock puts in order to hedge the sale of 1 index put. Hence 3 times 6.40, or 19.20, would be spent on index puts. Meanwhile, 20.33 credit is received from the sale of the index puts.

Hence the position is completely hedged and is established for a credit. If prices rise and all the puts expire worthless, the credit is then the profit. If prices fall, the position should also make money because it is hedged. No matter where the stocks are at expiration, their value will equal the index value, since the stocks are owned in the proper ratio to the index (3-to-1).

Practical Problems

In real life, things don’t always work out as easily as the above theoretical example, but the concept is the same. First of all, there won’t be a striking price exactly 10% out of the money for each stock and the index, nor will all the options have the same expiration date. Furthermore, the implied volatility of the out-of-the-money puts might not be large enough to allow establishment of the position for a credit.

The professional trader usually wants to establish the position for a credit. So, many such traders use the strategy in a delta neutral manner. They buy at-the-money puts in the individual stocks, computing the appropriate delta in index terms. Then they sell the expensive out-ofthe- money index options in a delta neutral ratio. In essence they establish the equivalent of a put ratio spread wherein the long side consists of at-the-money stock put options, and the short side consists of a greater quantity of out-ofthe- money index puts. This approach does have some downside risk, though, if the index should fall too far while the position is in place.

Another approach that is sometimes taken with a capitalization-weighted index (such as $OEX, $SPX, and many other indices besides the Dow), is to buy puts on only the larger-capitalized stocks and then sell index options. Again, a credit would be received when the position is established. However, there may be downside risk in this approach as well.

Furthermore, the larger-capitalized stocks will not perform exactly like the index. That difference in performance introduces “tracking error,” which merely means that the index might drop faster than the particular combination of the stocks that were utilized. If that happens, downside risk magnifies.

You might think that the put spread could be established for a debit as a way to alleviate some of the downside risk. That is true, but if it’s established for a debit and the market rallies, then the position will lose money unless the trader sells out some of his longs as the market rallies – a tactic which would eventually place him in a position similar to the strategies in the above two paragraphs where downside risk is a possibility.

Margin can be another problem. The exchanges don’t allow for cross-margining of this type of position, even though one might have a perfect index hedge (as in the above theoretical example). So one must pay for the long options in full and then must margin the short index puts as naked options. This can require a good deal of margin .


Overall, this strategy is attractive to highlycapitalized professionals because one is able to sell relatively expensive options and simultaneously buy relatively cheap options on the same entity – the index. Since a pure arbitrage is not usually available, the trader will undertake a certain amount of downside risk in order to establish the position at a credit (which means there is no upside risk). The “rub” occurs when adverse moves occur on the downside, but usually the position can be closed or downside defensive action can be taken to keep any such losses in check.


I have occasionally seen reference to the reverse of the strategy described above. It is generally false that one can buy the index put options and sell the stock put options profitably – at least not unless the volatility skew that has lasted since 1987 disappears.

The false argument goes like this: since stock options trade for higher implied volatilities than index options, one can sell the stock options and buy the index options (in the proper ratio) and profit. Do you see the fallacy in this argument? It misses the point that the combination of stock options behaves just like the index. That is, even though the stock options individually have higher implied volatilities, the combined effect of selling all of them reduces the net position to the equivalent of the index position. Hence there is no “edge” unless the index options happen to be cheaper – an unlikely occurrence as far as puts are concerned.

On a rare occasion when the volatility skew in index options is especially cheap, it might be the case that out-of-the-money calls are cheaper than their stock option counterparts and therefore the index calls could be bought and the equity calls sold. Even so, the reason such a trade is possibly feasible is not that the stock options individually have higher implied volatilities than the index options do.

This article was originally published in The Option Strategist Newsletter Volume 10, No. 23 on December 13, 2001.  

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