Question

1. A bank has a short position in 1,000 options on XYZ stock. Each option has delta of 0.45 and a gamma of 5.

(a) What is the delta of the bank position? How many XYZ shares are needed for a delta-neutral position?

(b) A traded option on XYZ stock (Option 1) has a delta of 0.6, and a gamma of 0.5. How can the bank position be made delta and gamma neutral?

Answer 1

A bank has short position for 1,000 options on XYZ stock.

option's delta = 0.45

option's gamma = 5.

a). So portfolio delta = number of option*option's delta = 1000*0.45 = 450

We know that delta of XYZ share is 1. So, bank need to short 450/1 = 450 shares of XYZ to make their portfolio delta-neutral.

b). traded option's delta = 0.6

traded option's gamma = 0.5

to make bank's position delta and gamma neutral, bank will take a position in traded option to make it gamma neutral and then take position in shares to make it delta neutral.

Gamma of the bank = number of option*option's gamma = 1000*5 = 5000

to make it gamma neutral, bank need to short, 5000/0.5 = 10000 traded option on XYZ

so new delta of the bank = 450 - 10000*0.6 = -5550

so to make it delta neutral, bank need to purchase 5550 shares of XYZ stock.

So final position of the bank is

short 10000 traded option on XYZ stock (Option 1) has a delta of 0.6, and a gamma of 0.5 and but 5550 shares of XYZ stock.