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3. Ratios and butterflies: when the position stops being symmetrical

Derivatives 8: Combined positions

We carry on with the index at 100, the implied at 20% and the expiry at 30 days. At those prices, the calls we are going to use are worth the following:

Take a good look at that last figure before going on, because the whole module hangs on it.

From two legs to four: the butterfly

A long butterfly with calls consists of buying a low-strike call, selling two of the middle strike and buying one of the high strike, with the three strikes equally spaced. With our numbers: you buy the 100 call, sell two 105 calls and buy the 110 call.

It is a position capped on both sides, cheap, with a ratio between what it can make and what it can lose of a little over three to one. On paper it looks very attractive, and that is precisely why it deserves a close look.

The ratio: the same position with a wing missing

A 1x2 long ratio with calls consists of buying a 100 call and selling two 105 calls. Nothing else. It is the previous butterfly without the 110 call.

Compared with the butterfly, the ratio costs 12 euros less and pays 10 euros more in its best scenario. It looks strictly superior. And below 105 it is. The trouble starts on the other side.

Above 105 the ratio has one sold call too many, with nothing covering it. From 108.9 it starts losing, and it never stops:

Twelve euros separate the capped from the open-ended

Go back over the two lists above. The only difference between the two structures is a 110 call costing 12 euros. With it, your worst conceivable case is 120 euros. Without it, your worst case has no number.

That 12-euro call is the one most often left out, and always for the same reason: it looks like it is worth nothing, like throwing money away, like "it will never get there". It is exactly the reasoning of level 6 applied to a position that looked harmless. The difference between paying 110 and paying 122 is 11%; the difference between the two tails is infinite.

And there is a practical consequence that gets forgotten: the complete butterfly is fully paid for and demands no margin beyond the debit, whereas the ratio leaves an option sold naked and generates a margin requirement that grows with the underlying and with the implied, by the mechanism level 6 already described. It is not only the worst case that changes: what also changes is who decides when you are taken out of the position.

How you end up with an open tail without meaning to

Almost nobody decides to sell a tail. You get there by accretion, and there are three common routes:

In all three cases the payoff profile stops being symmetrical and the trader carries on describing the position by its old name. Sinclair singles this out as one of the costliest failures in running an options book: the position you think you hold and the one on the account statement have stopped matching.

Where the maximum profit sits and how likely it is

Back to the complete butterfly, which is the sensible version. Its 380 euros of maximum profit are collected if the index ends exactly at 105, not at 104 and not at 106. A single point.

With the implied at 20% over thirty days, the standard deviation of the move is around 5.7%. Under the usual assumption, the probability of ending within one point of 105 is around 5%, one in twenty. Ending anywhere in the profit zone, between 101.2 and 108.8, has around a 36% probability, but there the average payout is far below the maximum. Collecting at least half the maximum requires ending between 103.1 and 106.9: around 18%.

And these numbers come from a distribution whose tails you already know are too thin, so treat them as orders of magnitude.

Hence the uncomfortable rule: a butterfly's maximum profit sits where the market thinks it unlikely to end up, and that is why the position is cheap. It is not a bargain, it is a price.

Worked scenario: the butterfly that became a ratio

Set out as the worked scenario. You build ten 100/105/110 butterflies for 1,220 euros. A week later the index is at 103, the position is going well and the 110 call is quoted at 0.08 points. You decide to sell the ten 110 calls and pocket 80 euros, on the grounds that "they no longer protect against anything".

Three weeks later a piece of news arrives and the index closes at 130. With the complete butterfly you would have lost 1,220 euros, your declared maximum. With the ratio you lose 21,100 euros. The 80 euros you got back by selling the wings cost almost twenty thousand.

Education, not a recommendation: the purpose of the example is to show where the number sits that does not appear on the payoff profile.

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