P M Halperin - Scribblings - Closing the Butterfly

Philip M Halperin - Scribblings

A Couple of Calculations Nobody Ever Makes

How do we close the Butterfly? Page 3
After all the foregoing graphs and examples, we are now in position to get calculate the distance from the opening long strike to the closing long strike for our generalised butterfly. Before we jot down a formula, let us express it in simple language (always a good thing to do to see if you got the concept right):

The distance from the closing strike to the opening strike of the generalised butterfly is going to be the sum of the distances between the middle short strikes and the long strike, stretched or scrunched by the number of options we need to close the butterfly.

And now, because we understand it geometrically, we can finally write this down in algebra:

Kc - Kl = S (Kl - Ks) / (L - S)

Where

Kc = the strike needed to close the fly

Kl = the initial long strike (in our foregoing examples, this was 100)

Ks = the middle short strike (in our foregoing examples, this was 110, but wait)

L = the number of strikes we are long, and

S = the number of strikes we are short.

So, starting from the inside of the parentheses outwards, we can read the generalised butterfly distance formula:

(Kl -Ks) = the distance between the long strike and the short strike

(L - S) = the difference between the number of longs and the number of shorts (ie, the number of options we need to buy to close the fly)

and finally,

Kc - Kl means the distance between the wings of the butterfly.

So now we have closed the generalised butterfly (but only for the one middle strike case)(at some point soon, we will extend this to other varmints, like condors). Equally importantly, we now have a new tool in our belt for closing ratio spreads by turning them into butterflies

The implications of closing the butterfly as a strategy for trading unhedged short volatility:

Now the implications of viewing a ratio spread as simply an incomplete butterfly are fairly obvious: When the time comes to unwind a ratio spread, you now have the choice of simply closing out the spread, or completing the butterfly. Accordingly, the short call risk of the ratio spreads we have examined becomes essentially identical to the characteristic of the options we need to purchase to complete the fly. For example, in the simple case of the 1x3 100x110 call spread, we need to buy two 115 calls. So we can view the short directional risk of this 1x3 as being essentially equal to that of two short 115 calls.

Moreover, for the short option spec trader (the market makers do not have such precise control over the geometry of their positions to make this study more than an academic exercise, in most cases), the process of buying tails to capture butterflies at a credit becomes an integral part of options trading strategy. Conceptually, the trader can look at initiating vertical ratio spreads simply as the first leg of a dynamic strategy of "capturing" underlying ranges at a credit. It was in this manner that I traded from the short side through the EMS crisis of 1992 in Swissy (USD-CHF), selling volatility at 17% and buying back at 22% or more, and made money in the process, without spot hedging. If you continuously follow a process of acquiring long butterflies at a credit, you will make money. In general, you will make much of the credit from the short options spread, and every now and then you will make money from the long inside options as well.

Now one indicator of the relative worth of the strategy will be the price of the closing call versus the cost of taking the spread off. In almost all cases (anomalies do occur), it will cost more to complete the long butterfly than to reverse out the ratio spread. Again, let us look at a 1x2 100x110 call spread. Purchasing a 120 call should nearly always cost more than selling 1x100 and buying 2x110 calls. (This is partially because, in the real world, the tails will always be bid to correct for the illness-of-fit between the accepted model and the observed distribution of market prices). This merely reflects the fact that the long butterfly is an asset; if you could acquire it for the same price as you can acquire a null position (no position), an arbitrage opportunity would exist (and we all know that that possibility has been postulated out of existence by the theoreticians).

So, when it comes time to close the spread, the thoughtful trader will carefully consider the alternatives on a cost/benefit basis, especially if he put the spread on with a view to closing the butterfly. In my experience, having a policy of closing the butterfly more or less consistently will dominate in the long run.

Coming soon:

Applications with generalised ratios: The FX and other OTC Markets
The generalised generalised butterfly: Condors and other Critters

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