A butterfly spread is a neutral strategy that is a combination of a bull spread and a bear spread using three strike prices. It is typically implemented using call or put options with the same expiration date. In this example, let’s use call options to construct a butterfly spread.
To set up a butterfly spread, we need to perform the following three transactions:
1. Buy one call option with a lower strike price (K1)
2. Sell two call options with a middle strike price (K2)
3. Buy one call option with a higher strike price (K3)
Note that K2 > K1, and K3 > K2. We also assume that the options have the same expiration date.
Let’s denote the option premiums as follows:
- Option with strike price K1: c1
- Option with strike price K2: c2
- Option with strike price K3: c3
The net cost (price) of setting up the butterfly spread is as follows:
Net Cost = c1 - 2 * c2 + c3
Now let’s determine the payoff for this strategy. There are four possible cases based on the stock price (S) at option expiration:
1. S < K1: No options are exercised, and the payoff is equal to the net cost.
Payoff = - Net Cost
2. K1 ≤ S < K2: Only the first call option is exercised, and the payoff is:
Payoff = (S - K1) - Net Cost
3. K2 ≤ S < K3: The first and second call options are exercised, and the payoff is:
Payoff = (S - K1) - 2 * (S - K2) - Net Cost
4. S ≥ K3: The first and third call options are exercised, and the second call option is exercised twice, and the payoff is:
Payoff = (S - K1) - 2 * (S - K2) + (S - K3) - Net Cost = (K3 - K2) * 2 - (K2 - K1) - Net Cost
Now that we have the payoff in different scenarios,
1.S < K1
Payoff = − NetCost = − (c1 − 2c2 + c3)
2.K1leqS < K2
Payoff = (S − K1) − (c1 − 2c2 + c3)
3.K2leqS < K3
Payoff = (S − K1) − 2(S − K2) − (c1 − 2c2 + c3)
4.SgeqK3
Payoff = ((K3 − K2) − (K2 − K1)) − (c1 − 2c2 + c3)
To make it more concrete, let’s use a hypothetical example.
Suppose we have the following option prices:
- K1 = $40, c1 = $5
- K2 = $50, c2 = $3
- K3 = $60, c3 = $1The butterfly spread is set up as follows:
1. Buy one $40 call option for $5.
2. Sell two $50 call options for $6 ($3 each).
3. Buy one $60 call option for $1.
The net cost for the butterfly spread is given by:
Net Cost = c1 - 2 * c2 + c3 = $5 - 2 * $3 + $1 = $0
Finally, let’s examine the payoffs:
1. S < $40
Payoff = - Net Cost = - $0 = $0
2. $40 S < $50
Payoff = (S - $40) - $0
3. $50 ≤ S < $60
Payoff = (S - $40) - 2 * (S - $50) - $0 = (S - $40) - 2 * (S - $50)
4. S $60
Payoff = $10 * 2 - $10 - $0 = $10
In summary, the butterfly spread is a neutral strategy that profits when the stock price remains close to the middle strike price (K2). In this example, the butterfly spread has limited risk since we’ve made it with a net cost of $0. However, in practice, there might be a net cost to set up the butterfly spread which will also limit the maximum loss to that net cost.