Long Call Butterfly | |
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Long 1 XYZ Sep 50 call @ $2.00, Short 2 XYZ Sep 55 calls @ $1.00, Long 1 XYZ Sep 60 call @ $.50 | |

Total Cost | Option premium paid, $50 |

Maximum Loss | Option premium paid, $50 |

Maximum Profit | Dollar value of difference between outside and middle strike prices minus premium paid, $450 |

Short Call Butterfly: | |

Short 1 XYZ Sep 50 call @ $2.00, Long 2 XYZ Sep 55 calls @ $1.00, Short 1 XYZ Sep 60 call @ $.50 | |

Total Credit Received | Net option premium received, $50 |

Maximum Loss | Dollar value of difference between outside and middle strike prices minus credit received, $450 |

Maximum Profit | Net option premium received, $50 |

Long Put Butterfly: | |

Long 1 XYZ Sep 30 put @ $.25, Short 2 XYZ Sep 35 puts @ $.50, Long 1 XYZ Sep 40 put @ $1.00 | |

Total Cost | Option premium paid, $25 |

Maximum Loss | Option premium paid, $25 |

Maximum Profit | Dollar value of difference between outside and middle strike prices minus premium paid, $475 |

Short Put Butterfly | |

Short 1 XYZ Sep 30 put @ $.25, Long 2 XYZ Sep 35 puts @ $.50, Short 1 XYZ Sep 40 put @ $1.00 | |

Total Credit Received | Net option premium received, $25 |

Maximum Loss | Dollar value of difference between outside and middle strike prices minus credit received, $475 |

Maximum Profit | Net option premium received, $25 |

##### Explanation and Application

Butterflies, condors and "wingspreads" are so-called because with sufficient -- no, make that CONSIDERABLE imagination, their expiration date risk profiles look like something that could fly. That, and anything that can add a bit of color to the otherwise dreary world of option trading is welcome. When talking about butterflies et al., you'll hear self-proclaimed experts speak of options as "body" and "wings". The "body" refers to options with strikes in between the two exoskeletal outermost strikes. The "wings" refer to options at the diaphanous outermost strikes. We use the term "wingspreads" to identify option positions such as "condors", "pterodactyls" and "albatrosses", which look like butterflies that have been stretched out. Rather than come up with a myriad of names to identify these spreads, we use "wingspreads" because they all have similar risk/reward characteristics and sensitivities, and those flying creatures are much more threatening than butterflies.

The risks and potential rewards of butterflies and wingspreads are limited. If you buy a butterfly, the most you can lose is the amount you paid for it. The most you can make is the difference between the "body" strike and a "wing" strike minus the amount you paid for it. If you sell a butterfly, the loss and profit are the inverse of buying a butterfly.

Wingspreads' sensitivity to movement in the stock price is related to the time to expiration. For example, the closer a butterfly is to expiration, the more sensitive its price is to a change in the price of the stock. What this means is that butterflies and wingspreads that are far from expiration don't always change in value that much when the stock moves. This means that all wingspreads aren't necessarily the best tool for exploiting changes in the stock price. Wingspreads can be bullish and bearish – but the closer a wingspread is to expiration, the more bullish or bearish it can be.

Wingspreads reach their maximum value when the stock price is at (in the case of butterflies) or between (for other wingspreads) the middle or "body" strike(s) at expiration. They are at their minimum value when the stock price is either above the higher "wing" strike or below the lower "wing" strike at expiration. Therefore, wingspreads can be effective when you believe that a stock's price will land within a specified range and within a specified time frame. When you believe that a stock will stay at a single price, long butterflies might be a good choice. When you think a stock will stay in between two prices, long wingspreads with middle "body" strikes at the low and high prices of the stock's range might work best. In this sense, long butterflies and wingspreads are like short straddles and strangles, but without the unlimited risk. Look at a graph of the value of a long butterfly at expiration, and the middle of the butterfly looks like a short straddle.

Conversely, if you think the price of a stock is going to move a way from a specific point or outside a specific range of prices, short butterflies and wingspreads might be a good choice. They work a bit like long straddles and strangles, but without the unlimited profit potential. They are also generally less expensive. Look at a graph of the value of a short butterfly at expiration, and the middle of the butterfly looks like a long straddle.

##### Greeks

Note that the price of the butterfly can become very sensitive to changes in the stock price with less than two weeks to go. The greeks of the butterfly also can change very dramatically as the stock price moves up and down. The discussion will concentrate on butterflies, but the same principles can be applied to all wingspreads.

The delta of a long butterfly is mildly interesting because the delta is positive when the stock price is below the middle strike of the butterfly, neutral when the stock price is at the middle strike, and negative when it is above the middle strike. The intuition behind this is that the butterfly maximizes its value when the price of the stock is at the middle strike. Therefore, if the price of the stock is below the middle strike, it has to rise for the butterfly to make money – hence the positive deltas. If the price of the stock is above the middle strike, it has to fall for the butterfly to make money – hence the negative deltas.

What's causing the delta to flip from positive to neutral to negative? Your old friend, Captain Gamma. The gamma of a long butterfly runs from positive to negative. At the outer strikes of the butterfly, gamma is positive, indicating that the butterfly would manufacture positive deltas if the stock price rises, and negative deltas if the stock price falls. This corresponds exactly with the way the delta of the long butterfly works as described above. The gamma of the long butterfly is negative when the stock is at the middle strike. This indicates that the butterfly will manufacture negative deltas if the stock price rises, and positive deltas if the stock price falls. This is precisely what you don't want to happen. That's why the long butterfly wants the price of the stock to stay right where it is when it's at the middle strike.

The theta of the long butterfly is the mirror image of the gamma. Along with the negative gamma comes positive theta, and vice versa. Theta is positive when the price of the stock is at the middle strike, indicating that time passing helps the long butterfly reach its maximum value. At the outer strikes theta is negative, indicating that the butterfly is losing value as time passes.

Butterflies can be good choices to exploit changes in implied volatility in a limited risk fashion. Like the other greeks, the vega of a butterfly changes depending on where the price of the stock is relative to the strike prices of the butterfly. When the stock price is at the middle strike, the vega of the long butterfly is negative. That means that any increased implied volatility in the stock will decrease the value of the butterfly. This makes sense, because a butterfly's value depends on the likelihood that the stock price will be at its middle strike at expiration. Higher implied volatility decreases the likelihood that the stock will stay at the middle strike price; therefore the value of the butterfly would decrease with an increase in implied volatility. Vega is positive for the long butterfly at the outer strikes. Therefore, an increase in the implied volatility of the stock increases the value of the butterfly because of the greater likelihood that the stock will move toward the middle strike by expiration.

Butterflies and other wingspreads are interesting positions to test in the analyzer because their greeks, although somewhat complex, make intuitive sense if you know how and when a butterfly increases and decreases in value. The greeks, then, can confirm analytically what you know about butterflies intuitively.

##### Structure

A butterfly is a wingspread that has options at three equidistant strikes; a long butterfly is long 1 low strike option, short 2 middle strike options and long 1 high strike option, and a short butterfly is short 1 low strike option, long 2 middle strike options and short 1 high strike option. A condor or other wingspread has options at four strikes, with the same distance between the each wing strike and the lower or higher of the body strikes.

It is sometimes useful to think of a butterfly (or any wingspread) in terms of two vertical spreads: one bullish, the other bearish. This will enable you to calculate wingspread prices faster. It will also help you understand how to make adjustments to positions like ratio or back spreads to turn them into butterflies.

Combination of Call or Put Bull Spreads and Bear Spreads Create Butterflies

Calls | Puts | |

Bull Spread + Bear Spread = Butterfly | Strike | Butterfly = Bull Spread + Bear Spread |

It makes sense that a combination of the two P&L graphs of a bull spread at one strike and a bear spread at the next higher strike will form a butterfly spread. Remember that a bull spread can be either long a call vertical or short a put vertical. A bear spread can be either short a call vertical or long a put vertical. So, any combination of bull spread at the lower strikes and bear spread at the higher strikes results in a long butterfly. A bear spread at the lower strikes and a bull spread at the higher strikes results in a short butterfly.

Given three equidistant strikes, a long butterfly can be a call butterfly, a put butterfly, or an iron butterfly. They all have expiration p/l graphs that look the same. No matter what kind of butterfly it is, its maximum value will be reached if the stock price lands at the middle strike price at expiration. At that point, the lower strike bull spread will maximize in price (the difference between the strikes), and the higher strike bear spread will be theoretically worthless.

To prove the basic equivalency of call and put butterflies, let's look at the resulting position if you buy a call butterfly and sell a put butterfly at the same three strikes. If you buy a butterfly and sell its equivalent, you should have a neutral position. Consider a position of long a call butterfly and short a put butterfly in XYZ Apr options, long +1 50 call, short –2 55 calls, long +1 60 call, short –1 50 put, long +2 55 puts, and short –1 60 put. The position will be long +1 XYZ Apr 50/55 box and short –1 XYZ Apr 55/60 box, which is a neutral position. The long box and the short box intersect at the 55 strike.

Other wingspreads can be understood in exactly the same way as butterflies. They are combinations of bull spreads and bear spreads, and a long call condor and a short put condor at the same strikes equals two boxes. But wingspreads can also be seen as being made up of a row of butterflies. The more butterflies that one has in a row, the wider the wingspread is and the more the profit zone is stretched out.

A note about "iron butterflies" and "iron condors": despite the name, they're nothing special. Iron butterflies, iron condors and iron wingspreads are a seeming contradiction: you receive a credit to be long them. A long iron butterfly has the same risk/reward profile as a long call or long put butterfly, both of which cost money to be long. How is the credit for the long iron butterfly achieved? An iron butterfly is simply a short box added to a long butterfly. An iron butterfly can be either a long call butterfly plus a short box whose strikes are at the lowest and middle strikes of the butterfly, or a long put butterfly plus a short box whose strikes are at the middle and highest strikes of the butterfly.

An iron butterfly is therefore a regular butterfly plus a short market-neutral box spread. Adding a short box to a butterfly doesn't necessarily change its risk/reward profile, it simply adds a cash loan from the market (which is what a short box is). The credit of the iron butterfly or iron wingspread is nothing magic and confers no real advantage.

An iron butterfly can also be seen as a short straddle at the middle or body strike and a long strangle at the wing strikes. This corresponds exactly with the middle part of a price graph of a long butterfly at expiration looking like a short straddle.

Iron wingspreads work in the same way. They are long wingspreads plus a short box, or they are short a strangle at the middle two body strikes, and long a strangle at the outer wing strikes.

##### Pricing

Contrary to popular opinion, butterflies are NOT free. Prior to expiration, wingspread values depend largely on the likelihood of the stock being at a certain price (in the case of butterflies) or between two prices (in the case of other wingspreads) at expiration. The more time there is to expiration, the less certain you can be of where the stock price will be at expiration. The less time there is to expiration, the more certain you can be of where the stock price will be at expiration. This means that if you were to graph the prices of all the consecutive butterflies in each expiration month, the graph would look rather flat in the months far from expiration because of the uncertainty about which butterfly will have the maximum value at expiration. (Remember: a butterfly reaches maximum value when the stock price is at the middle or "body" strike of the butterfly at expiration.) Therefore, butterflies far from expiration have roughly the same value, because only one or two of them can be of any value at expiration, and which one or two will have value are uncertain.

When there is less time to expiration, there is somewhat more certainty where the price of the stock is going to be at expiration. As a butterfly comes closer to expiration, the graph would begin to develop a "hump" at or near the current stock price. In fact, at the same time, the graph of the prices of all the consecutive butterflies begins to look like the p/l graph of a butterfly itself!! The "hump" on the graph is the price of the most expensive butterfly. It's the most expensive because it's the at-the-money butterfly, which has a middle strike that is the most likely price of the stock at expiration.

Generally, wingspreads will be more expensive than butterflies because they have a much larger profit range. That is, a butterfly maximizes its value if the stock price is exactly at the middle strike price of the butterfly. But a wingspread maximizes its value over a range of stock prices. Because a wingspread has a greater likelihood of maximizing its value, it has a larger price than a butterfly.

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