What are Options Greeks and their definition? What is Gamma in investing and how is it used? How to take advantage of Gamma Squeezes? A squeeze in finance is a rapid and sudden change in the prices of an underlying asset that obligates investors to force their positions. When this effect occurs, there will be investors ‘squeezed’ who have to take action to prevent being ‘choked’ in the market.

Let’s start at the beginning

First of all, a brief summary of what options contracts are.

Financial options are contracts that grant their buyer the right, but not the obligation, to buy or sell a certain amount of the underlying asset, at a given price called the strike price, within a stipulated period of time or maturity.

Call options are a type of financial contract that provide the buyer with the right to purchase a particular stock, asset, or instrument at a predetermined price during a specified time frame. This predetermined price is referred to as the strike price and applies to the underlying security in question. In the case of put options, the buyer has the right to sell the underlying security at the strike price.

What is Delta?

The Delta (\(\Delta\)) represents the change in the price of an option contract when the price of the underlying stock changes by one unit.

Delta can be a negative or positive value, depending on whether it is a put or a call option. A call option will have a price that moves in correlation with the price of its underlying asset, giving a positive Delta. A put option will have a price that moves inversely to the price of its underlying asset, resulting in a negative Delta.

It can also be thought of as the probability that the option will be exercised at maturity, i.e. the probability of receiving profits at the end of the contract.

Options Risk Metrics: The Greeks

Black-Scholes equation has been used for calculating option prices for the past 50 years but we are going to focus on what they represent conceptually.

The Delta described above is itself a type of Greek. Greeks are used to study risk in the options market. To give some context, we define the rest of the Greeks.

• Gamma (\(\Gamma\) ). Measures the rate of change in the delta of an option for each unit movement in the price of the underlying asset. It is the first derivate of Delta.
• Theta (\(\Theta\)). Measures the rate of change in the price of an option caused from the time between now and the expiry date.
• Vega (\(\nu\)). Defines the sensitivity of an option price to any change in the volatility of the underlying asset.
• Rho (\(\rho\)). Defines the susceptibility of an option price to any change in interest rates.

Relation between Delta and Option “moneyness“

For call options \(\Delta > 0\) and for put options \(\Delta < 0\).

• If the market price at the expiration of the option contract is higher than strike price, a call option is said to be “in the money” (ITM) . If the market price price is lower, a call option is said to be “out of the money” (OTM).
• If the market price at the expiration of the option contract is lower than strike price, a put option is said to be ITM and if the market price price is higher, a put option is said to be OTM.
• If \(\Delta = 0.5\) the option is said to be “At the money” (ATM).

If the option is ITM, its \(\Delta\) will be high since, as we said, it represents the probability that at the end of the exercise you will make a profit, i.e. at the end of the contract date your option will be ITM. As a consequence of this high probability of profit, the premiums will be higher.

On the other hand, if we buy very OTM options, their \(\Delta\) will be small, and therefore we can buy more options at a lower price because their premium will be lower.

We can now give a more detailed definition of Gamma and Delta.

$$
\begin{align}
\Delta = \frac{\partial C}{\partial S} > 0\\
\Delta = \frac{\partial P}{\partial S} < 0
\end{align}
$$

$$
\begin{align}
\Gamma= \frac{\partial²C}{\partial S^2} = \frac{\partial \Delta C}{\partial S} \\
\Gamma= \frac{\partial²P}{\partial S^2} = \frac{\partial \Delta P}{\partial S}
\end{align}
$$

\( S\) represents de stock price. \( C \equiv C(S,t)\) and \( P \equiv P(S,t)\) represent the price of a call and put option, respectively. Note that both variables depend on time.

It is important to note that \(\Gamma >0 \) in a buy situation and \(\Gamma <0 \) in a sell situation.

Let’s look at the following graph. Let the green line be a typical \(\Delta\) and the purple ones the Gamma at each point of Delta.

• \(\Gamma\) takes maximal values when the option is ATM.
• \(\Gamma\) takes minimum values when it is very OTM or very ITM.

\(\Delta = 0.5 \) means that for every $1 the stock goes up, the option premium goes up $0.50. If the stock price goes down $1, the option premium goes down $0.50.

When we purchase a call option, our goal is for the underlying security to increase in value. With each upward movement in the underlying’s price, we earn $0.50 (with a multiplier of 100), equating to a gain of $50. Conversely, for every point that the underlying decreases, we will experience a loss of $50.

The Market Maker roll

Suppose an investor buys OTM call options. The Market Maker is obliged to sell. The MM receives the money from the premium paid by the investor, not from “winning” the bet to sell calls. To hedge his position, the Market Maker must buy stocks of the underlying asset of the option in order to obtain a Delta Neutral (the overall \(\Delta\) value of a position is 0) .

As the asset price approaches the strike price, \(\Delta \) will increase and consequently \(\Gamma\) will increase as well . This occurs since it is a derivative and so the Market Maker will increasingly need to buy larger and larger quantities of the underlying asset in order to hedge his position.

When the option is very much ITM, a 100% rise in the stock price implies an additional 100% buy by the market makers. This is called a Gamma Squeeze.

This increase in \(\Gamma\) leads to an increase in the price of the asset. If it turns out that market makers had many short positions in the underlying, they will be forced to change their positions and buy them back but now at a much higher price, taking large losses.

What is the difference between a short squeeze and a gamma squeeze?

A Gamma Squeeze is associated with options positioning, while a Short Squeeze is related to stock positioning.

A Short Squeeze happens when the price of a stock that has been heavily shorted increases rapidly. This cause the short sellers have to unwind their positions and leading to further upward momentum in the stock price. This can be triggered by unforeseen developments affecting the stock’s value or a sudden surge in demand for it.

When a large volume of options are purchased for a particular security, it can cause a Gamma Squeeze. In this situation the significant movement in the price of the underlying stock is caused by the Market Maker’s hedging strategy in response to the increase options buying.

What happened to GameStop in 2021?

GameStop (GME) was a company with very low-priced stocks due to the digitization of commerce. As a result of this, there were major hedge funds with bearish positions against it. It got to the point that there were more short positions than Free Float (stockes in circulation).

On the reddit forum, a movement of investors began to agree to strangle the “shorts”, in order to provoke a Short Squeeze massively buying call options.

The calls bought by long investors in GameStop were OTM and therefore with a small \(\Delta\), we assume that delta= 0.25. Suppose the GME had 1000 calls sold at that delta to cover its position. The Market Maker would have to buy 25,000 shares of GME.

These results are derived from the expression of the hedge ratio, which we will see below.

Assuming that the \(\Delta\) of the stocks is always 1.

$$
\begin{align}
HR = \frac{\Delta_{calls}}{\Delta_{stocks}}
\end{align}
$$

$$
\begin{align}
HR = \frac{1000 \cdot 100 \cdot 0.25}{1} = 25000 \ stocks
\end{align}
$$

In sold option positions \(\Gamma < 0\) . That is, if the underlying asset rises, you have to buy more of it at higher prices to remain in Delta Neutral.

The MM had to buy 25,000 stocks that are not in the market because the shorts had more positions than Free Float. The MM was obliged to buy at any price and as there was no liquidity this price was very high.

The high price paid by the MM drove up the stock price. The OTM calls were closer to being ITM so the \(\Delta\) went from 0.25 to 1 because \(\Gamma\) kept increasing. At this point, applying the same hedge ratio formula, for 1000 call options the MM had to buy 100.000 stocks.

Conclusion

The GameStop saga of 2021 was a historic moment in the financial world that saw the convergence of a short squeeze and a Gamma Squeeze, resulting in an unprecedented surge in the company’s stock price. The coordinated buying activity by retail investors, fueled by social media platforms and online forums, created a market frenzy that sent shockwaves throughout Wall Street. However, while some investors made significant gains, others faced heavy losses when the stock price eventually plummeted.