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Solution to Butterfly Problem[?]

Here is a spreadsheet model of the problem:

[pic]

The only assumption cell is A5, which is the index price in one month.

There are four forecast cells: F2 is the profit if all of the money were invested in the index, F3 is the profit on the basket of call options, F4 is the profit on the “butterfly strategy” (buying the options and investing all remaining money in the index), and F6 is the difference between the butterfly strategy’s profit and the profit if all money were invested only in the index.

Notes on Butterfly Spreads

“At expiration a butterfly will always have a value somewhere between zero and the amount between the exercise prices. It will be worth zero if the underlying contract is below the lowest exercise price or above the highest exercise price, and it will be worth its maximum if the underlying contract is right at the inside exercise price.

Since a butterfly has a value between zero and the amount between exercise prices (5 points in our example), a trader should be willing to pay some amount between zero and 5 for the position. The exact amount depends on the likelihood of the underlying contract finishing right at or close to the inside price at expiration. If there is a high probability of this occurring, a trader might be willing to pay as much as 4.25 or 4.5 for the butterfly, since it might very well expand to its full value of 5 points.”[?]

“The way butterflies respond to price swings is unique. With options that have a great deal of time left, a butterfly does not react dramatically to moderate price moves. Only large moves produce a significant response in the spread price. So sensitivity to price movement is reduced to a minimum if expiration exceeds 60 days.

Volatility affects butterflies, but, again, their sensitivity is minimal as long as there is a lot of time remaining. Time decay does not affect a butterfly, primarily because the out-of-the-money options become completely worthless, but most decay comes in the last few weeks before expiration.”[?]

Given that the pricing of butterflies is resistant to volatility, underlying price movement, and time decay, they often serve as a trader’s inventory. In a butterfly, a trader has a combination of 4 different options that acts as a relative store of value while remaining malleable enough to facilitate future trades. A trader can collect these semi-complicated spreads and use the pieces to create other spreads when market forces make it financially more attractive to do so.[?]

Here are Crystal Ball histograms of the index-only profit and the butterfly profit:

|[pic] |[pic] |

|Forecast Profit: Total Profit - Index Only |Forecast: Total Profit – Butterfly Strategy |

|Mean = $1 |Mean = $16 |

|Standard Deviation = $4,957 |Standard Deviation = $4,957 |

|Mean Std. Error = $70.10 |Mean Std. Error = $70.10 |

The butterfly strategy in this problem turns out not to have much impact on either the expected return or the risk of the portfolio. It does, however, shift the distribution of the profit such that the expected profit is lower over most of the range of possible one-month index values but significantly higher within a narrow range around the middle of the butterfly. In other words, the investor who uses this strategy is placing a bet that the index will end up somewhere between the $600 and $610 values (the strike prices of the two “outer” call options in the butterfly).[?]

The implications of the butterfly are evident in this graph, which did not require the use of Crystal Ball (it was drawn using the DataTable Excel feature). We see that over most of the range of possibilities the butterfly has a lower expected return, but that it has a higher expected return in the range close to the expected value of the index.

[pic]

[1] By Eric Schwesinger (MBA ’02) and David Juran.

[i] Natenberg. Option Volatility and Pricing: Advanced Trading Strategies and Techniques. Page 147

[ii] The Options Institute: The Educational Division of the Chicago Board of Options Exchange. Options: Essential Concepts and Trading Strategies. Page 285.

[iii] EHS’s normative 2 cents.

[iv] Alex Johnson (MBA ’03) reports the following informative website that graphically displays many hedging strategies (straddles, strangles, straps, spreads, strips, etc.) that are beyond the scope of this class:



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