P1.T4.9. Gamma-neutral option positions

[RaDi7]

Hi David,

could you please explain the answer in Problem 18.12:




Will the trader gain from large movements only with this portfolio (long calls + long puts) or is it a general statement? As I think if I had only long calls position (=positive gamma) I would gain from a large increase in the stock price but never from a large decrease.

Thank you very much and have a nice weekend!

 

[David Harper CFA FRM]

Hi @RaDi7

This statement would still be true if the portfolio was only long call options given that there is a dynamic delta-hedge in place (dynamic is implied). This is a situation similar to Hull's delta hedging (19.4 or 18.4). While Hull's 19.4 delta hedge consists of written (short) call options + dynamically purchasing shares, in this question the implied delta hedge consists of a net long option position hedged by dynamically shorting shares. It is key to understand here that the implication is that the company will be buying/selling shares to maintain a neutral position delta. This question is cool because it tests an understanding of gamma exposure.

The company has a portfolio of options and they are delta hedged, such that implied is long options plus short shares. Say the call options have percentage delta = N(d1) = 0.60 and the put options have delta = N(d1) - 1 = -0.40, such that 100 calls and 100 puts implies 60 - 40 = +20 position delta. So the company shorts 20 shares in order to implement a delta hedge; ie, position delta = + 20 due to options - 20 due to short shares. Imagine the underlying stock price jumps up dramatically. What happens immediately to the position delta? Both the call and put percentage deltas increase such that position delta increases! What is the delta hedge trade (to maintain neutral position delta)? The company shorts additional shares at this higher price. Then imagine the price drops dramatically (ie, we are simulating "wild movements" or oscillations in the price). The portfolio position delta consequently drops such that the company, to maintain neutral delta, purchases shares at the lower price. In this way, maintaining delta neutrality against a scenario of oscillating prices implies "short high and buy low" for profits! But this is just operationalizing positive gamma exposure. I hope that helps, thanks!

 

[RaDi7]

Thank you David! It helps!