Question

A trader creates a long butterfly spread from options with strike prices X, Y, and Z, where X < Y < Z, and Y is exactly midway between X and Z. A total of 400 options are traded. The difference between X and Y is $12. The difference in the prices of the options with strike prices of Z and Y is $5.88. The difference in the prices of the options with strike prices of Y and X is $6.32. What is the maximum net gain (after the cost of the options is taken into account)? The answer is 1156. How do you solve this?

Answer 1

A long butterfly is created by :

1 Long 1 call option for strike X ( highest premium say C)

2. Short 2 call option for Strike Y (premium =C-6.32)

3. Long 1 call option for strike Z (premium = C-6.32-5.88 = C-12.20)

where X<Y<Z

Here, Y-X=Z-Y=$12

If the price at maturity is below X , none of the call options are exercised

So payoff= 2*(C-6.32) -C-(C-12.20) = - 12.64+12.20 = -0.44

If the price at maturity is between X and Y , call with strike X is exerised but others are not

So, payoff = -0.44 +(P-X) where P is the Price at maturity

the maximum payoff occurs when P=Y

So, maximum payoff = -0.44+ (Y-X) = -0.44+12 = $11.56

If the price at maturity is between Y and Z , call with strike X and Y are exerised but others are not

So, payoff = -0.44 +(P-X) -2* (P-Y) = -0.44+(2Y-X-P) which decreases as P increases

the minimum payoff occurs when P=Z

So, maximum payoff = -0.44+ (Z-X) - 2*(Z-Y) = -0.44+24- 2*12 = -$0.44

If the price at maturity is more than Z , all calls are exerised

So, payoff = -0.44 +(P-X) -2* (P-Y) + (P-Z) = -0.44+(2Y-X-Z) =-0.44+(Y-X -(Z-Y)) =-0.44+(12-12)

= -$0.44 p

So, the maximum payoff occurs when P=Y

and the payoff is $11.56 when there are total 4 options traded

So, maximum gain net of option premium = 11.56*400/4 = $1156 for 400 options traded