Question

Construct an asymmetric butterfly using the 950-, 1020-, and 1050-strike options. What is the minimum number of each option you should buy or sell? Draw a profit diagram for the position and specify the dollar value of the maximum profit and maximum loss. Assume c(950)=$120.405, p(950)=$51.777, c(1020)=p(1020)=$84.470, c(1050)=$71.802, p(1050)=$$101.214.

Answer 1

Range from 950 to 1050 = 100 price points. Peak point of the butterfly will be 1020.

Range from 950 to 1020 covers 70% and from 1020 to 1050 covers 30%.

[Note: we will construct butterfly using call options only]

Thus if we sell 1 c(1020), we will need to buy 0.3 c(950) and buy 0.7 c(1050).

Fractional values may not be practical, hence we can consider selling 10 units of c(1020), then we will need to buy 3 c(950) and buy 7 c(1050) to construct the asymmetric spread. {answer to number of each option we should buy/sell}

Maximum loss would be the cost of building this spread

= Premium paid for 3 c(950) + Premium received for 10 c(1020) + Premium paid for 7 c(1050)

= -3*120.405 +10*84.470 -7*71.802

=-19.129

Maximum profit will occur at peak of the butterflu when S=1020.

Profit will be = [-3 c(950) + 3*70] + [10 c(1020)] + [-7 c(1050)]

=190.871 (for precise calculations refer excel screenshot below at point S=1020)

{c(1050) will be out of money; c(1020) will be at the money -> so we just use the premium values; c(950) is in the money, so we consider initial premium [-3 * c(950)] and a shift (3 shift per point*70 price points)}

Each individual call profit/loss can be tabulated in excel. Summation should give butterfly profit/loss. (refer attached image). And a payoff can be plotted using S values on x-axis & butterfly value on y-axis.