Question

Assume that you wish to form a delta-neutral position by using two options. At the current level of the underlying stock’s price, Option A has a delta of -0.43 and Option B has a delta of 0.33.

a. If you wish to write 800 Option As, determine how many Option Bs should be traded. Also, determine whether the position in Option B should be long or short.

b. Suppose you estimate that if the price of the underlying stock increases by TL1, the delta of Option A becomes -0.42 and the delta of Option B becomes 0.334. In light of this information, estimate the position gamma for the delta-neutral position you formed in part (a).

c. If the underlying stock’s price indeed increases by TL1 and you want to reestablish the delta-neutral position using the underlying stock, determine the number of stocks you would need and the position you would need to take.

Answer 1

Solution

Option A delta = -0.43 (Put option)

Option B delta = +0.33 (call option)

Writing a put option will give positive delta and buying will give negative delta

Buying a call option will give positive delta and selling will give negative delta

Part A ) writing 800 option A will give delta = 800 * 0.43 = +344

So we need -344 delta from option B and since option B has positive delta so we have to short this option by 344/0.33 = 1042.42 = 1042

So total option = 1042 and position is short

Part B )

Option A change in delta = -0.42-0.43 =0.01 Gamma = 0.01

Option B change in delta =0.334-0.33=0.004 gamma= 0.004

Gamma due to 800 ooption A = 800 * 0.01 = 8

Gamma due to 800 ooption B = 1042*0.004 = 4.17

Total gamma = 8 + 4.17 = 12.17

Part C )

Change in delta due to option A = 800* - 0.01 = -8 ( Because .43 becomes .42 and delta has decreased )

Change in delta due to option B = 1042.42 *-0.004=-4.17 ( becuase .33 becomes 0.334 and negative delta increased )

Total change in delta = -8 - 4.17 = - 12. 17

Stock has delta of 1 and buying stock will increase delta and selling will decrease it

So we have to purchase 12 shares (rounded off to whole number )

Number of stock = 12 and position is long

Since,