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 $13. the difference in the prices of the options with strike prices of z and y is $5.03. the difference in the prices of the options with strike prices of y and x is $6.67. what is the maximum net gain (after the cost of the options is taken into account)

$1006

Explanation:

A long butterfly is created by following these steps

Here, Y-X = Z-Y =$11.70

i) whenever the price at maturity goes below the price of x ( no call option is executed )

payoff =  2*(C-6.67) -C-(C-11.7) = - 13.34 + 11.70 = - 1.64

ii) when the price at maturity is between X and Y, only call with strike X is executed

hence  payoff = -1.64 +(P-X) where P is the Price at maturity

p - x = y-x = 11.70

hence maximum payoff = - 1.64 +  11.70 = $10.06

iii) When the price is between Y and Z , only call with strike X and Y are executed.

hence, payoff = -1.64 + (P-X)  -2* (P-Y) = -1.64 +( 2Y - X - P) and this value decreases as P increases

the minimum payoff occurs when P=Z

So, maximum payoff = -1.64 + (Z-X) - 2*(Z-Y) = -1.64 + 23.4 - 2*11.7 = -$1.64

iv) When the price at maturity is more than Z , all calls are executed

hence, payoff = -1.64 +(P-X) -2* (P-Y) + (P-Z) = -1.64+(2Y-X-Z)

=  -1.64+(Y-X -(Z-Y)) = -1.64+(11.7 - 11.70)

= -$1.64  

 the maximum payoff occurs when P=Y

considering the  four options  traded the maximum payoff = $10.06

Finally determine the maximum net gain when 400 options are traded

= 10.06 * 400 / 4

= 10.06 * 100 =  $1006

b. 0.67

Explanation:

UCL = 1 + 0.10

        = 1.10 inch

LCL = 1 - 0.10

       = 0.9 inch

standard deviation = 0.005 inch

mean = 1 inch

Cpk

= min[(UCL - mean)/(3*standard deviation) , (mean - LCL)/(3*standard deviation))]

= min[(1.10 - 1)/(3*0.05) , (1 - 0.9)/(3*0.05))]

= min[0.67 , 0.67]

= 0.67

Therefore, Theprocess capability index (Cpk) if the long-run process mean is 1 inch is 0.67