Download s09a-02-butterfly

Solution to Butterfly Problem1
Here is a spreadsheet model of the problem:
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
$
$
$
$
B
C
605 initial price
605
mean
30
stdev
Total
601
D
Cash Out
$ 100,000
$
18
$ 100,000
E
Cash In
$ 99,339
$
100
$ 99,421
F
($661)
$82
($579)
G
H
I
J
= Index Profit
= Options Profit
= Profit with Index + Options
=F4-F2
$82.12
Difference
(positive indicates butterfly strategy is better)
Qty
Qty Bought Sold
Strike
$580
$585
$590
$595
$600
$605
$610
$615
0
0
0
0
100
0
100
0
=100000/A2
index (no calls)
index (with calls)
0
0
0
0
0
200
0
0
Cash Out
Cash In
Option Bid
Option Ask
index
price 1
month
$0.00
$0.00
$0.00
$0.00
$1,589.00
($2,790.00)
$1,219.00
$0.00
$0.00
$0.00
$0.00
$0.00
$100.00
$0.00
$0.00
$0.00
$25.54
$22.84
$20.33
$18.01
$15.79
$13.95
$12.09
$10.60
$25.64
$22.94
$20.43
$18.11
$15.89
$14.05
$12.19
$10.70
$601.00
$601.00
$601.00
$601.00
$601.00
$601.00
$601.00
$601.00
payoff if bought payoff if sold
$21.00
$16.00
$11.00
$6.00
$1.00
$0.00
$0.00
$0.00
($21.00)
($16.00)
($11.00)
($6.00)
($1.00)
$0.00
$0.00
$0.00
=(100000-D3)/A3
165.29
165.26
0
0
=B17*G17-C17*F17
=SUMPRODUCT(B17:C17,I17:J17)
=$A$5
=MAX(0,H17-A17)
=MIN(0,A17-H17)
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
1
By Eric Schwesinger (MBA ’02) and David Juran.
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.”i
“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-ofthe-money options become completely worthless, but most decay comes in the last few
weeks before expiration.”ii
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.iii
Here are Crystal Ball histograms of the index-only profit and the butterfly profit:
Forecast: Index Only Profit
5,000 Trials
Forecast: Total Profit
Frequency Chart
12 Outliers
5,000 Trials
Frequency Chart
12 Outliers
.096
480.9999999
.117
584
.072
360.7
.088
438
.048
240.5
.058
292
.024
120.2
.029
146
0
.000
.000
($15,000)
($7,500)
$0
$7,500
$15,000
0
($15,000)
$
($7,500)
$0
$7,500
$15,000
$
Forecast Profit: Total Profit - Index Only
Mean = $1
Standard Deviation = $4,957
Mean Std. Error = $70.10
Forecast: Total Profit – Butterfly Strategy
Mean = $16
Standard Deviation = $4,957
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).iv
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.
B60.2350
2
Prof. Juran
Benefits from the Butterfly Strategy
$600
Butterfly Benefit
$500
$400
$300
$200
$100
$$590
$595
$600
$605
$610
$615
$620
$(100)
Ending Index Price
Natenberg. Option Volatility and Pricing: Advanced Trading Strategies and Techniques. Page 147
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:
http://www.options-academy.com/
i
ii
B60.2350
3
Prof. Juran