Realizing profit with Gamma Trading doubt

Question

Lets suppose we have a delta-neutral portfolio and that we want to trade the gamma.

If we are long gamma, we can profit from every rebalancing to keep the portfolio delta-neutral.

Lets suppose the following portfolio:

• long 100 x call ATM (∆=50%, constant gamma = 10)

• short 50 x stocks ($100).

1) Underlying's prices goes down by $1:

• Call's delta goes down (∆=40%)

• I need only -40 stocks to remain delta-neutral, so I buy +10 stocks and realized immediate profit.

  Gamma PnL = (1/2) * (10) * ($1)^2= $5 USD;
  
  Or assuming that delta is discrete and goes from 50% to 40% immediately; -10*($99-$100) = $10 USD. 
  

2) Underlying's price goes up by $1:

• Call's delta goes up (∆=60%)

• I need -60 stocks to remain delta-neutral, so I sell more -10.

And we will sell this additional -10 in a price above that the initially sold.

If price goes down further, we will buy it, and again realize immediate profit.

But what if the price just goes up?

If price's only goes up, and we close the position, we will finally have to buy the underlying, buying it for a higher price, losing money on the underlying by itself.

This loss in the underlying may or may not be followed by a profit on the options (call's price can goes up, but we have Vega, Theta, etc, that can make the option's value decrease).

What am I missing about this?

Answer 1

If the price only goes up (quickly), you make more on the option than you lose on the stock. So if it goes to 150 overnight, the delta of the option quickly goes to 1, and you are long 100 calls versus short only 50 stocks.

If the stock goes up only gradually, you could lose more on Theta than you gain on gamma. However, the point is, if you rebalance every day, what matters is if the gamma pnl exceeds the theta. This will happen if the stock moves by more than a certain amount (the breakeven amount) , whether the move is up or down.