Question

When is it appropriate for an investor to purchase a butterfly spread? Suppose three put options on a stock have the same expiration date and strike prices of $65, $70, and $75. The market prices are $3.50, $6, and $7.50, respectively. Explain how a butterfly spread can be created. Construct a table showing the profit from the strategy. For what range of stock prices would the butterfly spread lead to a loss?

Answer 1

It is appropriate to purchase a butterfly spread when:

• The stock price is expected to remain neutral
• The stock price is expected to remain by and large same
• The stock price is expected to move to s very small extent in either direction.

Thus a butterfly spread is a neutral price strategy that works the best if stock price doesn't change at all. It works fine if stock price changes marginally in either direction. It results in losses if stock price change substantially in either direction.

The losses are limited and so are the profits.

Let K denote the strike price and P denote the Put Premium. All financials below are in $.

K1 = 65, K2 = 70, K3 = 75, P1 = 3.50, P2 = 6.00; P3 = 7.50

If S is the stock price on expiration, then

Payoff from long (buy) put option = max (K - S, 0)

Payoff from short (sell) put option = - max (K - S, 0)

Let's create the butterfly spread as follows: All financials below are in $.

Sl. No.	Position	Immediate cash flow (A)	Payoff at expiration (B)	Gain / (Loss) from the position (A) + (B)
1.	Buy (Long) 1 no. of Put with least strike price	- P1 = - 3.50	max (K1 - S, 0) = max (65 - S, 0)	max (65 - S, 0) - 3.50
2.	Sell (Short) 2 nos. of Put with mid strike price	2 x P2 = 2 x 6 = 12	- max (K2 - S, 0) x 2 = - max (70 - S, 0) x 2	12 - 2 x max (70 - S, 0)
3.	Buy (Long) 1 no. of Put with highest strike price	- P3 = - 7.50	max (K3 - S, 0) = max (75 - S, 0)	max (75 - S, 0) - 7.50
	Portfolio (Total)	12 - 3.5 - 7.5 = 1		max (65 - S, 0) + max (75 - S, 0) - 2 x max (70 - S, 0) + 1

Total Gain / (Loss) matrix:

S	Gain / (Loss)
	max (65 - S, 0) + max (75 - S, 0) - 2 x max (70 - S, 0) + 1
50.00	1.00
55.00	1.00
60.00	1.00
65.00	1.00
70.00	6.00
75.00	1.00
80.00	1.00
85.00	1.00
90.00	1.00
95.00	1.00

And the Gain / (Loss) diagram is:

Gain/ (Loss) from Straddle 7.00 6.00 6.00 5.00 4.00 3.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 70.00 75.00 50.00 55.00 60.00 65.00 80.00 85.00 90.00 95.00 Stock Price, S (S)

This spread is always making profits. Hence, there is no range of share price when this strategy will make loss.

Note:

There appears to be some issue with the way the prices of the put options are given. Ideally a butterfly spread should require investment at t = 0. Here, in this case, the spread is leading to an inflow at t = 0 and no losses in future. Thus it's an arbitrage case, you get a riskless profit to begin with and then some more profit or no profit later on.