Long and short butterflies

A butterfly is a vertical spread and its mirror image stacked at a shared middle strike: buy one option at a lower strike, sell two at the middle, buy one at a higher strike. It is the cheapest way to express "the stock will be here on this date," and its payoff is the tent that every other range structure in this course approximates. The price of the precision is that most of the time the stock is not there.

The long call butterfly

XYZ at $50, 45 days to expiration, IV 30%. Buy the $45 call, sell two $50 calls, buy the $55 call.

• Max loss: the debit, $1.65 per share, $165, if XYZ finishes below $45 or above $55.
• Max profit: width of one wing minus the debit = $5.00 − $1.65 = $3.35, $335, if XYZ finishes at exactly $50.
• Breakevens: $45 + $1.65 = $46.65 and $55 − $1.65 = $53.35.

Long call butterfly 45/50/55 for 1.65 debit

  +335 |            /\
       |           /  \
    0  |---------X------X---------   X = 46.65 / 53.35
       |        /        \
  -165 |_______/          \_______
       +---+---+---+---+---+---+---+
       43  45  47  49  51  53  55  57   XYZ at expiry

The put butterfly at the same strikes (buy $45 put, sell two $50 puts, buy $55 put) costs about the same and has the identical payoff, by the same parity argument as Module 2. Use whichever has tighter markets; for a centered butterfly that usually means using the OTM options for the wings, which is one call wing and one put wing, and that structure is the iron butterfly in the next lesson.

What the ratio actually means

Risk $1.65 to make up to $3.35 looks like 2:1 in your favor. Check it against probability. The expected move to expiration is about $5.30, so the breakevens at ±$3.35 sit at about 0.64 standard deviations; the chance of finishing inside them is roughly 47%. But the max profit needs a pin at $50; inside the breakevens the average profit is nearer $1.70 than $3.35. Expectancy at market odds: 0.47 × $170 − 0.53 × $165 ≈ −$7, effectively zero before costs. The butterfly, like every other structure, is fairly priced. What it offers is a specific shape: small fixed loss, large profit concentrated at one price.

The Greeks, and why the tent only appears late

At entry, with the stock at $50 and 45 days left:

At the center the butterfly is a short straddle wearing wings: short gamma, positive theta, short vega. Move the stock to $45 or $55 and the signs flip, because there the long wing dominates. That is why a butterfly's behavior before expiration is nothing like its expiration table. With 45 days left and the stock still at $50, the butterfly is worth only about $1.90 after ten days, not $5.00. Most of the tent's value arrives in the last two weeks, when the two short $50 calls finally lose their extrinsic value. A butterfly bought 45 days out is a slow trade that becomes a fast one at the end.

Practical consequences:

• Butterflies are usually held longer than condors, often into the last week, because that is when they pay. That means accepting the gamma of the last week, which is the trade.
• Take profit when the value reaches a set multiple of the debit, commonly 2× (sell at $3.30 here) rather than waiting for the pin.
• Buy them further out in time only if you want the low cost; the closer to expiration you buy, the more the price already reflects where the stock is.

The short butterfly

Reverse everything: sell the $45 call, buy two $50 calls, sell the $55 call. Credit $1.65, which is the max profit, kept if XYZ finishes below $45 or above $55. Max loss $3.35 at a pin on $50. It is a bet on a move away from the center, like a long straddle with the upside capped and the cost reduced. Traders rarely use it because a long straddle or strangle does the same job without a cap, and the short butterfly's max profit of $1.65 is small for the $3.35 at risk. It appears mostly as the other side of someone else's long butterfly.

Key idea: A long butterfly buys one option below, sells two at the center, buys one above. Max loss is the debit, max profit is one wing width minus the debit at a pin on the center, and the expectancy at market prices is about zero, like everything else. Its Greeks flip sign between the center and the wings, and most of its value arrives in the final two weeks.

Try it: Build the 45/50/55 butterfly in the options profit calculator at 45 days. Record its value with the stock at $50 at 45, 30, 14, 7 and 0 days. Then record its value with the stock at $53 at the same dates. Note the date at which the tent becomes visible.

Recap

• Long butterfly: +1 lower, −2 middle, +1 upper; debit is the max loss, wing width minus debit is the max profit at the center.
• 45/50/55 on a $50 stock: debit $1.65, max profit $3.35, breakevens $46.65 and $53.35; expectancy at market odds near zero.
• At the center it is short gamma, positive theta, short vega; at the wings the signs reverse.
• The tent forms late; take profit at a multiple of the debit rather than waiting for the pin.
• The short butterfly is the mirror image, a capped bet on a move, rarely better than a straddle.