Butterfly spread is constructed by using options at 3 different
expiry. It is used when the trader believes the price of the
underlying asset will not deviate much from the current price.
Hence, it is a neutral strategy.
A butterfly spread is constructed using 4 options, at 3 different
strike prices. the middle strike price is called the body and the 2
strike prices at the end are called as winds.
If the trader believes that the price of the underlying will not
move much, it can use a long butterfly strategy. If the trader
believes that the price of the underlying will move a lot, it can
use a short butterfly spread
The problem states that maximum payoff is attained when the stock
price at expiration is 23 (body) and the payoff is positive between
stock prices 20 and 25 (wings). Hence, the assumption is the trader
wants to construct a long butterfly spread where he does not expect
large price movements.
Long call butterfly-
Buy 1 call each at strike price 20 and 25 and sell 2 calls at
strike price 23
Cash flow from premiums
=-3.58-1.89+2*2.45 | -0.57 |
Stock price at expiry | 23 |
Options payoff at expiry | ||
strike price | State | Payoff |
20 | In the money | 3 |
23 | At the money | 0 |
25 | Worthless | 0 |
Net cash flow | ||
=3-0.47 | 2.53 |
Long put butterfly-
Buy 1 put each at strike price 20 and 25 and sell 2 puts at strike
price 23
Cash flow from premiums | |
=-2.62-5.7+2*4.36 | 0.4 |
Stock price at expiry | 23 |
Options payoff at expiry | ||
strike price | State | Payoff |
20 | Worthless | 0 |
23 | At the money | 0 |
25 | In the money | 2 |
Net cash flow | ||
=2+0.4 | 2.4 |
Payoff table from options (ignoring the premiums)- | ||||
Strike price | Call butterfly | Call butterfly | Put butterfly | Put butterfly |
20 | =0+0+0 | 0 | =0-2*3+5 | -1 |
21 | =1+0+0 | 1 | =0-2*2+4 | 0 |
22 | =2+0+0 | 2 | =0-2*1+3 | 1 |
23 | =3+0+0 | 3 | =0-0+2 | 2 |
24 | =4-1*2+0 | 2 | =0-0+1 | 1 |
25 | =5-2*2+0 | 1 | =0+0+0 | 0 |
Average payoff | 1.5 | 0.5 |
It appears that long call butterfly is more profitable. Hence, we will use - Long call butterfly strategy. Since this is an asymmetric butterfly, the payoffs are different. At spot price of 21 at expiry, the payoff from options is $1. Including the option premiums, the net payoff becomes- 1-0.57= $0.43
6. The following table shows the premiums of European call and put options having the same...
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