Question

5. Suppose that you sold an American put option P on the underlying stock ABC. You calculated the delta and gamma of the put option and found Ap = -0.4 and Ip= 0.3. Suppose you want a position which is both delta and gamma neutral. Determine the hedge in the following two cases: (a) Use the underlying stock itself and a call option C on the stock (with different strike and maturity as the put), with Ac 0.25 and Io 0.1 (b) Use the underlying stock itself and a call option with the same strike and maturity as the put option. (Answer: Buy one call and short sell a share.)

Answer 1

In order to make the position both delta and gamma neutral, we first need to make the position gamma neutral and then delta neutral.

Given information:

Portfolio is short 1 put option

Delta = -0.4

Gamma = 0.3

Case a:

To neutralize gamma we need to sell calls. To determine how many we need to sell we need to divide the gamma of our current position with the gamma of the call option.

i.e. 0.3 / 0.1 = 3 call options to sell

Now calculate the total delta: we have sold one put option and will sell 3 call options. so the total delta is:

1*-0.4+3*0.25 = 0.35

In order to go delta neutral, we will have to sell 0.35*100 shares i.e. 35 shares of the underlying stock

Case b:

we need to calculate delta for the call option with the same strike and maturity. The delta of call and put option add up to 1 (ignoring the -ve sign for put option). Hence delta for call option = 1 - 0.4 = 0.6

Gamma for a call and put with same strike price and expiration is the same.

Hence to make the position gamma neutral, we need to buy a call option so that the total gamma is zero. 0.3 - 0.3 = 0

Now calculate the total delta: we have sold one put option and have bought 1 call options. so the total delta is -0.4 + 0.6 = 0.2.

So we need to sell 0.2*100 = 20 shares of the underlying stock to make the portfolio delta neutral.