Question

# A trader creates a long butterfly spread from call options with strike prices of \$160, \$170,...

1. A trader creates a long butterfly spread from call options with strike prices of \$160, \$170, and \$180 per share by trading a total of 40 option contracts (10 contracts at \$160, 20 contracts at \$170 and 10 contracts at \$180). Each contract is written on 100 shares of stock. The options are worth \$20, \$24, and \$30 per share of stock.

1. What is the value of the butterfly spread at maturity as a function of the then stock price?

2. What is the profit of the butterfly spread at maturity as a function of the then stock price?

At maturity, let stock price be S

Value of long call with X = 160

max(0, (10*160 - 10 *S)) - 20*10

Value of short call with X = 170

-(2 *(170*10 - 10*S)) + 20*24

Value of long call with X = 180

max(0,(10*180 - 10*S)) - 30 *10

The total value of the strategy ( for 100 stocks )

= (max(0, (10*S - 10 *160)) - 20*10 -(2 *(170*10 - 10*S)) + 20*24 + max(0,(10*S - 10*180)) - 30 *10) * 100

Profit for the strategy =

Case 1

If S < 160

= (0 - 20*10 - 0 + 20*24 + 0 - 30 *10) * 100

Case 2

If 160 < S < 170

= ((10*S - 10 *160) - 20*10 + 20*24 + 0 - 30 *10) * 100

Case 3

If 170 < S < 180

= ((10*S - 10 *160) - 20*10 -(2 *(170*10 - 10*S)) + 20*24 + 0 - 30 *10) * 100

Case 4

If 180 < S

= ((10*S - 10 *160) - 20*10 -(2 *(170*10 - 10*S)) + 20*24 + (10*S - 10*180) - 30 *10) * 100

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