# What really is Gamma scalping?

How does Gamma scalping really work? It seems there is no true profit scalped. If we look at the simplest scenario, Black-Scholes option price $V(t,S)$ at time $t$ and the underlying stock price at $S$ with no interest, the infinitesimal change of the overall portfolio p&l under delta hedging, assuming we have the model, volatility, etc., correct, is $$0=dV-\frac{\partial V}{\partial S}dS=\big(\Theta+\frac12\sigma^2S^2\Gamma\big)dt.$$ So the Gamma effect is cancelled by the Theta effect. Where does so called Gamma scalping profit come from?

Note: My condition implies that $$P\&L_{[0,T]} = \int_0^T \frac{1}{2} \Gamma(t,S_t,\sigma^2_{t,\text{impl.}})S_t^2( \sigma^2_{t,\text{real.}} - \sigma^2_{t,\text{impl.}})\,dt$$ coming from the misspecification of volatility is $0$.

Assuming all else remains equal (implied vol has not changed and very little time decay has occurred), Gamma scalping can best be explained by Gamma (or realized volatility) enhancing the value of a delta hedged portfolio.

For example: If you are long an at-the-money call option, you are long 0.5 Delta and long Gamma. If you hedge this position, you will short 0.5 units of stock to be Delta neutral.

If the stock moves up:

Long option value will go up by 0.5 times the stock move + Gamma

Short stock hedge will lose 0.5 times the stock move

Net, the portfolio will be up by your Gamma

If the stock moves down:

Long option value will go down by 0.5 times the stock move - Gamma

Short stock hedge will gain 0.5 times the stock move

Net, the portfolio will be up by your Gamma

You will be up by Gamma. Hence the term Gamma Scalping.

Note: This strategy depends on realized volatility being greater than implied volatility (or the theta decay that you are paying for being long the option).

If you repeat this, the portfolio will go up by the Gamma. The strategy makes money because of the convexity of the option vs the linearity of the hedge.

• Only your note is the true mechanism which is precisely expressed by the second equation in my question. That means this name is really a bad name, as it is misleading and confusing. The trading is really just an arbitrage or bet on the volatility, whereas Gamma is just a multiplier. That is not even true since the multiplier has $S^2$ in it as well. At least Theta scalping would have been a better name as Theta absorbs all the multipliers. – Hans Dec 19 '17 at 0:02

As long as you live in a world where implied and realized vol are the same, there is no net profit (or loss) from gamma scalping. However, if they are different, then you make a gain or loss which is not path dependent. This is all still in a hypothetical world of course with continuous trading.

In reality when rehedging less frequently, pnl becomes random and path dependent with at mean centered around Vega times the difference between realized vol and implied vol.

To me the equation you gave is important because:

• it underpins why you can see option trading together with delta hedging as betting on implied volatility
• it shows how your profit accrues (twice as large move, 4 times the pnl)

Might go too far for your question, but see here Delta Hedging with fixed Implied Volatility to get rid of vega? for an explanation of how what volatility you use in your hedging matters, even if you know that there is a difference between the implied vol you bought the option at and the subsequent realizing volatility.

• I just added an emphasizing and clarifying note derived from the premise of my question. My curiosity is why people talk about Gamma scalping as if it is some kind of trading strategy. Is it just some folk lore coming from people's misconception of how options work? If you can provide a link to a similar question, it will be helpful. I could not find one before I posted my question. – Hans Dec 18 '17 at 7:54

Gamma scalping (being long gamma and re-hedging your delta) is inherently profitable because you make 0.5 x Gamma x Move^2 across the move from your option. (You get shorter delta on downmoves, so you buy underlying to hedge, you get longer on upmoves, so you sell on upmoves, etc.) Because it's inherently profitable across any move, you must pay for the privilege to be long gamma. The cost is that you pay out theta.

Theta (all else equal) of an ATM option can be thought of as the market's expectation of gamma-scalping profits for that day. If the stock moves more than implied by the market, you should make money on the gamma-scalp.

When other posters say it's a bet on volatility, they're correct. More specifically, it's a bet on realized volatility. If the stock realizes a higher vol than implied, gamma scalping makes more money than the option decays through theta.

You say that gamma-scalping profits should be cancelled out by theta. This is only the case in a Black Scholes world and in the case that realized vol = implied vol. This is almost never the case in reality.

It is indeed a trading strategy, and also a byproduct of running an options portfolio. Some people trade near-term options with high gamma in order to directly arb near-term realized versus implied. It's not a folk lore. Hope that answers some questions.