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I have read in Bennett - Trading Volatility the following quote.

As shown above, a long gamma (long volatility) position has to buy shares if they fall, and sell them if they rise. Buying low and selling high earns the investor a profit. Additionally, as a gamma scalper can enter bids and offers away from the current spot, there is no need to cross the spread (as a long gamma position can be delta hedged by sitting on the bid and offer).

However, I am not sure why this is the case. After each move in the underlying, the long gamma investor needs to either buy or sell additional shares to maintain his delta hedge. I.e. if the stock drops he needs to buy more and sell more if the stock rises. Thus, he either needs to sell at the bid or buy at the ask. Why would he be able to sit on both sides?

Furthermore, why would this not be available to a short gamma investor? For me it is simply the same situation, but reversed

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Let's assume the underlying trades at spot 100, with bid at 99 and offer at 101. Trader that is long gamma can put an offer in the queue at 101 and bid in the queue at 99, and his offer or bid will get executed automatically when the underlying stock goes up or down (if the trader is long gamma and he doesn't decide to place hedging bids and offers in advance, he'd indeed need to cross the spread if he only decides to hedge once the underlying moves, but that would be shortsighted and inefficient).

If you are short gamma, you are short the options. So if you want to delta-hedge and the underlying goes down, you need to sell, and you need to buy if the underlying goes up. If the underlying goes to 99, it means that someone sold into someone's bid at 99: you can't place an offer at 99 when the underlying is at 100. So you can't pre-hedge like you can when you are long gamma. If the underlying drops to 99, and you need to sell, you need to hit someone's bid at 98 (assuming there's a bid there).

Same if the underlying goes up to 101 and you need to buy: you'll need to hit someone's offer at 102. Again, you can't pre-hedge this, because if the underlying trades at 100 first and you place a bid at 101, it would get executed immediately (if there is an offer waiting in the queue at 101) and the price would move against you.

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