Delta neutral question

[sarita]

HI David,

how are you- this question is from 2003 FRM exam:

A portfolio of stock A and optons on Stock A is currently delta neutral, but has a positive gamma. Which of the following actions will make the portfolio both delta and gamma neutral:
1)buy call options on stock A and sell stock A
2)sell call options on stock A and sell stock A
3)buy pu options on stock A and buy stock A
4)sell put options on stock A and sell stock A

The answer is (4) - sell put and sell stock A.

I understand that since gamma is positive; this is an indicative of a long position so you have to sell and the options and stock? is that right?

why not sell call option and sell stock A? not sure who to choose between choice 2 and 4.

Would appreciate your help in understanding this.

Thanks,
S

 

[David Harper CFA FRM]

Hi saray -

Yea, thanks for sharing a *difficult* question, this is sort of classic FRM in the way the both (2) and (4) are tempting.

In case it is helpful, my favorite "rule" here is:
position Greek = percentage Greek * position quantity, where (+) for long quantity and (-) for short quantity.

Percentage Gamma is always positive, which is meant when you read "gamma is always positive for call or put." But if you are short the option, your position Gamma is negative because = negative for the position * positive percentage Greek; e.g., assume 3,000 options with each percentage gamma of +1.5.
Then we can have either:
Positive position gamma if long 3,000 option = 1.5 * +3,000 = 4,500 position gamma, or
Negative position gamma if short 3,000 options = 1.5 * -3000 = -4, 500 position gamma

So in this case, we start with a positive (position) gamma.
As the stock has zero gamma, this implies we are either net long calls or puts.

First, "fix" the position gamma to zero. Shares cannot change the gamma. We must go short either calls or puts. So, for the first step only, to neutralize delta, either (2) or (4) is correct as both add negative position gamma to the portfolio.

What if we used short call options (per answer 2)?
Position delta = negative quantity * positive (percentage) delta = negative position delta; i.e., as call option deltas are positive, shorting them necessarily adds negative position delta. If that was step 1, then step 2 must be LONG shares to "fix" delta back to zero. But if step 2 were to sell stock (given answer 2), this adds negative position delta (delta of share = 1.0, so -quantity * delta = -position delta) to already negative delta and does not work.

What if we used short put options (per answer 4) in step #1? Negative quantity * negative (%) delta = positive position delta; i.e., shorting put options adds positive position delta. To fix that, we short the shares. Answer (4) is the only one that works.

Hope that helps, David

 

[sarita]

Thanks so much Peter. It does make sense now.

Best,
S