Philip M Halperin  Scribblings
A Couple of Calculations Nobody Ever Makes
How do we close the
Butterfly? Page 1 This part will be fairly trivial for anyone who has traded options or even learned about them academically, but please bear with me, because I am trying to build things up intuitively. notes Imagine for a moment a simple vertical ratio spread:
The profit/loss payoff of this position is well known. Ignoring premia at expiration, we have: So the directional risk in this position begins at strike X and increases without bound with the underlying. So far, so trivial. Now, our geometric intuition tells us that X = 120. After this point the position begins to lose money. So the way to negate the unbounded nature of the risk is to buy one 120 Call: Note that at 120, the long call profits negate the ratio spread position losses. The two positions sum to zero p/l above 120, leaving us with the wellknown long call butterfly:
Here, again ignoring premia, we have a crack at making money anywhere between 100 and 120 and no losses that arise anywhere else. Period. So, the vertical ratio spread can be closed in two ways (actually there are many other ways, but enumerating any of the others gets in the way of discussion at hand):
So much for the introductory insight. We'll return to this point, but for the moment it is sufficient to note the following:
