January 26, 2023
I’d argue the two Greeks at the forefront of the mind of a market taker are most likely delta and theta. How much money am I going to make or lose if the underlying moves a dollar, and how much money am I going to lose a day by holding this contract? When should I take profit, how much leverage am I getting by purchasing X, Y, or Z strike, what’s the probability my contract expires in the money, etc.?
For option buyers crossing the spread on listed markets, it makes sense to think specifically about first-derivative Greeks as they have some of the most direct impacts on the underlying contract prices. But I also think it’s important to discuss and understand the second-order derivative effect of Gamma.
Why do those OTM options give greater leverage opportunities, why are you able to purchase those tail-like scenarios at certain prices, what impact Gamma has directly on the underlying market due to the risk management tactics of market makers, and more.
As some readers may not be familiar with some of the concepts covered, I’ll first quickly review the simplicities of Gamma specifically and some of the trends to notice. Looking at the graph of Gamma exposure can be insightful on its own but thinking about how shifts in spot and volatility will change the magnitude of this Greek is highly important. Gamma shifts due to changes in the market, whether that change be a move in spot, a move in implied volatility, or a shift in time to expiry.
A surface-level definition of Gamma from any financial textbook, or in this case Google, states that: Gamma represents the rate of change between an option's Delta and the underlying asset's price. A higher Gamma value indicates that the Delta could change dramatically with even very small price changes in the underlying.
Gamma typically is the largest when the option is ATM. It decays in a similar fashion to that of a bell curve graph as we drift further away from the strike price, whether OTM or ITM.
Due to the nature of how Delta can change quite quickly around the ATM strike as Gamma is the highest, someone who is long an ATM option can see large absolute swings in their contract value as the underlying price swings closely around the ATM strike.
It’s myopic to only consider the underlying price movement when thinking of Gamma, although this is an important trait. Another trait to consider, however, is how Gamma reacts to implied volatility. Option traders across the board are always concerned with purchasing “cheap” or “expensive” options and thus having the ability to understand how gamma exposure shifts with a move in implied volatility is important.
In a low volatility environment, gamma is concentrated around the ATM strike, with the ATM strike typically being the peak. In higher volatility environments, gamma tends to flatten/widen out across the different strikes although the peak still is close to the ATM strike (not exactly, but we will just assume so to keep it simple).
This means for an ATM option with the ATM strike of $100, if we look at the distribution of Gamma at 100% IV (implied volatility) compared to 50%, we should see the absolute value of Gamma is greater for the ATM option with IV at 50%. For that same option, if we moved spot to $150 or $50, in a high-volatility environment the Gamma of that option would be greater than that of the same option in a low-volatility environment.
Gamma for OTM and ITM options increases as volatility rises. Gamma for ATM options falls. The following image shows this principle.
The final important principle to understand is that shorter-dated options, all else equal, have greater Gamma than longer-dated options. The impact a shorter time to expiry has on Gamma is similar to a low volatility environment, previously described.
Gamma for ATM options with a shorter time to expiry is greater than the same option with a longer time to expiry. If time to expiry is longer, due to the increased amount of time for daily moves to be made, these ITM and OTM options have a higher probability of flipping from OTM to ITM and vice versa which makes sense as to why longer dated OTM options have a higher Gamma than shorter dated OTM options - they have more time to potentially become ITM.
Gamma & Delta
Market participants are long Gamma when they are long options, and short Gamma when they are short options. Given the previous definition of Gamma’s impact on Delta and how Gamma shifts with differing underlying environments, one can take note quickly that when you are long Gamma you get longer Delta as the market goes up and shorter as the market goes down.
For a quick mental exercise on why this is important, say you are SHORT an ATM call option (you want the market to go down) with 0.50 Delta and 0.05 Gamma. Your net position is -50 deltas. Let’s say you go and buy 50 deltas initially to offset your exposure to the underlying price movement and just want to sit and collect theta. If the market moves up by a dollar your ATM option is now $1 ITM, and your delta has gone from -0.50 to -0.55. Your new delta position on that move upwards (given your previous hedge exists) is now -5 deltas. To keep the same unbiased view on the market you purchase 5 more deltas to return back to delta-neutral.
If the market moved down a dollar, your net position would start at -0.50, your call would go from ATM to $1 OTM, and your delta would be -0.45. Given your previous hedge, you are now long +5 deltas and need to sell 5 deltas into the market to stay delta neutral.
Dynamic hedgers who are long Gamma are buying into dips and selling into rallies acting as a stabilizing force in the market. Dynamic hedgers who are short Gamma are selling into dips and buying into rallies acting as a force that catalyzes market volatility.
Thinking about Gamma from a Dealer’s Perspective
As dealers are typically considered the type of participant who would try to dynamically hedge their exposure to the markets, it makes sense to try and track their positioning to see when they may be exposed to pockets of short Gamma.
Most market makers are very systematic and have some sort of risk framework or thresholds they follow to make sure they are not taking on excessive amounts of risk at any given time. They are aware that by selling options they are betting against market movement and any volatility, and are happy to collect theta for the right price.
When dealers are caught with short options portfolios and in turn, short Gamma and volatility, and the market happens to move, they are aware they must crystallize losses by hedging their newly acquired or reduced delta positions by buying into rallies and selling into dips or risk their underlying exposure getting too large.
How to use Gamma Exposure to think about market moves
If we know that options market makers are continuously rebalancing their portfolios to reduce exposure to the underlying market, by looking at regions of highly negative Gamma we can determine likely areas where market volatility is likely to be increased. These negative Gamma regions may lead to forced portfolio rebalancing as dealers keep Delta and Gamma within specific thresholds. To maintain these thresholds while short Gamma, dealers end up buying into rallies and selling into dips. Adding forced, organic, buying pressure into a market rally, or the opposite into a market downtrend.
Something unique to the cryptocurrency markets is the prevalence of a variety of venues to hedge perpetual futures on, typically with large amounts of leverage. In crypto, short-squeezes can drive cascading liquidations which can lead to massive price swings in the perpetual futures markets. Opening up arbitrage opportunities between the spot/perp markets to drive price movements. During a large cascading liquidation one can buy spot and sell the perpetual for a relatively quick arbitrage profit if all connectivity is there and one is quick enough.
If you combine forced buying pressure into low liquidity due to dealers short Gamma, momentum-driven investors seeing a quick rally from $17-19K FOMOing into markets with organic buyer demand, and massive short liquidations throughout the week which culminated in a liquidation cascade on Friday, Jan 13th you get a massive looking short-squeeze-esque trade that looked like a potential outcome all throughout the $17-19K range.
Although dealer hedging of short Gamma positions was only a small portion of everything that needed to happen to cause such a massive move on Friday, January 13th, one can see the effect of short-dated, short Gamma positional hedging causes a self-fulfilling prophecy for call buyers (very similar to the Gamestop trade).
Thinking about Gamma in the way described above has its caveats and assumptions being made. The most important assumption and the most difficult one to solve here is: we are assuming every option buyer is paired against a dealer who controls their risk exposure through dynamically delta hedging their positions. The previously described spot/perpetual buying pressure doesn’t exist if the other side of the trade selling you the option does not mind the convexity of being short an option position without dynamically hedging and just sits and accumulates a mark to market profit or loss rather than crystallizing that P/L and rebalancing their portfolio through delta hedging.
The chart by Amberdata shown above does a great job at trying to accurately determine what they believe to be the gamma positioning of dealers.
Gamma is a highly interesting and important Greek to take note of, and even more interesting to analyze when applied to potential dealer positioning. When you start to take note of pockets of negative, short-dated Gamma on top of organic buying demand flowing into the markets due to FOMO, short liquidations at pivotal resistances being tested, and low liquidity conditions into the weekend, you might just realize it seems like the market is playing with a lit match next to a pool of gasoline.
Proactively assessing short liquidations, funding rates, dealer gamma positioning, and potential pockets of illiquidity is something that could help drive potential trade ideas.