What is Gamma Exposure? A simple Model

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Market Gamma Exposure

The market isn’t theoretical like many economists and finance PhDs like to believe. The market is a bunch of real people buying and selling. The structure of these people affects how the market behaves.

Most people buy stocks.

By most, we mean anything from retail traders to the biggest actors in the market - institutions, endowments, pension funds, hedge funds, family offices, etc.

Very few make a living from shorting stocks. Most buy stocks.

This simple fact creates a structural condition with large consequences in market activity.

To picture this let’s understand the two types of actors in the market:

Investors and Dealers

Investors

Investors is a very general term we will loosely use to describe all who attempt to make profits in the market by predicting price direction.

Warren Buffet is an investor - he buys Apple because he thinks its price will go up.

Johny Table in his Robinhood account is also an investor when he buys Tesla stock because he believes it will go up.

Investors - in this sense - doesn’t just mean buying stocks. You can also short stocks if you believe the price will go down. Bill Ackman is an investor when he shorts Herbalife because he believes its price would go down.

We make no distinction between investors and speculators here. Simply - if you bet on price direction you are an investor.

Most agents in the market are investors. Warren Buffet, pension funds, university endowments, Johny Table - they are all investors and it happens to be the case most of them buy stocks as opposed to shorting stocks.

Dealers

Dealers are around so that investors can buy (or short). A dealer is an agent who does not attempt to make profits from predicting price direction. A dealer attempts to make profits from dealing to those who do predict direction.

Dealers exist in the market to provide liquidity, so that when Warren Buffet or Johny Table come to buy stock - someone is there to sell it to them.

Dealers charge for this activity of course. Their profit sometimes comes in the form of rebates from the exchange but more fundamentally in the Bid-Ask Spread.

A Dealer says - “You interested in Apple? I’ll buy it from you at 329 or sell it to you at 331”. A Dealer puts the bid - 329 - and the ask - 330 - so that when Johny Table or Harvard Endowment come through they can buy Apple or sell it.

What the dealer does which is putting the bid and ask - or bid-ask spread - is called ‘making a market’. Another term for a dealer is a ‘market maker’.

A Dealer’s business is ‘market-making’ and in principle, they do not attempt to make money from price direction. In fact - the art of a successful market maker is often to be able to never be exposed to price direction. This is an art for the following reason:

A Dealer puts a bid-ask for Apple at 329/331. Someone comes and sells 100 shares of Apple. Cool, the dealer buys it from them for 329. Immediately someone comes to buy 100 shares of Apple. Cool, the dealer sells it for 331. Bam, the dealer just made 2 dollars for each of the hundred shares - a nice $200 profit.

What if this second person that came to buy Apple didn’t come ‘immediately’. What if the dealer bought those 100 shares at 329 from the first person and then Apple stock falls to 320. The dealer just lost 9$ on each of the hundred shares - a harsh $900 loss.

Hedging

The dealer of course wants to avoid this. One of the main ways to avoid this is by ‘hedging’.

Hedging means ‘not putting all your eggs in one basket’.

The dealer just bought 100 shares of Apple at 329 - this means all his eggs are on this Apple basket. If the basket breaks he loses. To ‘hedge’ would be to put eggs on a different basket - precisely on a basket that does the exact opposite to what the Apple basket is doing.

This way, if Apple price falls 1$, the other basket makes 1$ and the net result for the dealer is 0. If the price falls 9$ to 320, the other basket gains 9$ - net result for the dealer is 0. This way the dealer didn’t lose anything when Apple fell to 320 because his other basket gained the same exact amount.

Then - finally - the second person shows up to buy Apple. Now the bid-ask the dealer places is at 319/321. The dealer sells Apple at 321. At the end of the day what happened is the following:

• Dealer buys 100 shares of Apple at 329 = -32,900
• Apple shares fall $9 to 320
• Hedge basket gains exactly $9, 9*100 = +900
• Dealer sells Apple at 321 = 32,100

-32900 + 900 +32100 = +100

The dealer made a +100 profit. Notice that had the dealer not ‘hedged’ his Apple position the net result would be an 800 loss.

This is all to show that ‘hedging’ is the essence of market-making.

Now, back to market structure.

Investor and Dealer Motivations From the above, we can see that investors bet on price directions and dealers hedge. That is that.

Enter Options

All the above was talk about dealers in shares. Options are a whole other universe.

As we said - most people buy stocks.

For a couple of decades now - there has been a very common practice of squeezing out more profits from a portfolio of stocks. This is done with options. Here is how it works.

You have ZM stock at 200. You believe ZM is going up (which is why you bought it in the first place) but you don’t think it will go up to 250 this month. Regardless, if it goes to 250 this month you feel you might even want to sell some to secure some profits - but primarily you don’t think ZM will rise to 250 this month.

With this in mind, you sell some 250 Calls that expire this month. The premium you keep enhances your portfolio. Because unless ZM rises to 250 this month, the return on your portfolio will be enhanced by the premium of selling these calls.

Niceeee, enhancing that stock portfolio.

That’s not the only thing though in your mind. You are also worried that ZM will crash and your position will lose a significant amount of value. You want some protection on the downside. So you purchase some Puts.

Ideally, you try to make it so that the sale of the Calls finance the purchase of the Puts, in such a way that you have obtained insurance ‘for free’.

Now, this isn’t only Johny Table we are talking about.

Imagine billions of dollars of pension funds, endowments, family offices, institutions, etc., all of whom are long stocks - doing the exact same thing.

As we said - most buy stocks. Most of these billions of dollars are long on stocks. And they all want downside protection - so they buy Puts. And they all want to finance that downside protection or enhance returns - so they sell Calls.

Billions of dollars in selling calls and buying puts.

Who is on the other side of this trade?

Dealers.

Dealers are buying those calls and selling these puts - that is their business after all. (This behavior is related to the implied volatility skew - which is a topic for another day.)

And just like in the Apple stock example above - a dealer does not want to be exposed to price direction. A dealer profits precisely by NOT being exposed to price direction. And what does a dealer do to not be exposed to price direction? They hedge! More specifically - they Delta Hedge.

Delta Hedging

You are Johny Table and you have 100 shares of Apple. You want to buy a Put to insure yourself against a 10% downturn in Apple for the next month. With Apple trading at 333.46 you are looking at the Puts with strike 300.

These puts are trading for 1.74. A single Put contract covers the right to sell 100 shares of the underlying. So you only need 1 put.

Who sells this Put to you? The dealer.

This Put has a Delta of -11.

This means that for every move in Apple, the Put’s value will change by negative 11 times that move. That is for the owner of the Put of course. For the seller of the Put its the opposite, positive 11

As we mentioned - the dealer is not interested in being exposed to price direction that way. In other to hedge away the price exposure the dealer must do something about this 11 exposure to Apple’s price direction. The dealer must sell 11 shares of Apple.

Why 11? Each share of Apple has a Delta of 1 - this means that if the price of Apple goes up by 1, well, the price of Apple goes up by 1.

11 shares of Apple have a Delta of 11 - so if you have 11 shares of Apple in your portfolio and thus a Delta of 11, when Apple price moves by 1, your portfolio will move by 1*11. And selling 11 shares of Apple gives you the opposite price exposure, negative 11.

Ok, but why 11? Because the dealer sold a Put which has a Delta of -11. So his position changes by 11 (the opposite of -11, since he is the seller) for every price move in Apple. This exposure of 11 combined with the exposure of -11 from selling the 11 shares gets him to 0! The Dealer now has Delta 0 = exposure to price direction of 0!

Being Delta Neutral means having 0 exposure to changes in the underlying’s price movement

Only for small price movements though.

What?

Being Delta Neutral means constantly rebalancing the portfolio subject to changes in the price of the underlying - this process is known as Dynamic Hedging

Yes, the dealer is hedged only for small price movements in Apple. Unlike Apple stock that will always have a Delta of 1 at all times - an option contract, in this case a Put, does not have a constant delta value. The delta of an option changes when the price of the underlying changes.

The delta of an option is related to the probability that the option will be worth something at expiration.

The value of a Put Strike 300 with 30 days to expiration on Apple is worth something insofar as Apple has a probability of falling below 300 in the next 30 days. Actually, the Delta is this probability (not really but sort of - more on this another day, let’s just say this loosely for now). The probability assigned by the market of Apple falling below 300 in the next 30 days is - you guessed it, the Delta, 11%.

Now it’s clear to see that if tomorrow Apple falls from 333 to 320, then the probability of it falling below 300 is higher. This means the Put with strike 300 has a higher chance of being worth something at expiration, it is more sensitive to changes in the price of Apple, its Delta is higher.

Its Delta might now be -20. This means that for the dealer it has Delta of positive 20.

But wait, the dealer only sold short 11 shares of Apple! He used to be Delta Neutral but now he has 20 Deltas from the Short Put and -11 Deltas from shorting the 11 shares - that is a net +9 Delta Exposure! This is not Delta Neutrality!

Now the dealer must sell another 9 shares of Apple to be back at Delta Neutral.

And that, that is Delta Hedging.

Ok, but what was this entire illustration for?

To reach the gamma.

Gamma

Remember how the Delta of that Put changed from -11 to -20? That’s gamma.

Gamma tells you how much the Delta will change when the price of Apple (the underlying) changes.

In this case, we saw how there was a negative change in price (Apple went from 330 to 320) and there was a negative change in Delta (Delta went from -11 to -20). This means that the Gamma is positive, and in this case close to 1. For every -1 move in Apple you saw almost a -1*1=-1 change in Delta.

Now let’s get back into the Dealer’s shoes.

Maybe this Gamma metric can tell us something about what the Dealer has to do? Maybe even let us predict the Dealer’s behavior?

When the dealer sold the Put they had a Delta of 11. Then Apple fell 10 points and the dealer suddenly had a Delta of 20 on that Put.

Apple price was negative and Delta for the Dealer was positive - this means that their gamma is negative.

Now if we review the mechanics of what the dealer had to do we see the following:

• Dealer is Delta Neutral
• Price of Apple Falls
• Since Dealer has negative Gamma, their Delta became positive
• Dealer must sell Apple shares to become Delta Neutral

Hmmm…

In other words - if we know that the Dealer is delta negative, then we know that if Apple price falls they will have to sell more Apple shares!

Now, the market isn’t just one dealer.

Market Gamma

Imaginary Example:

The Vanguard Group holds 336 million shares in Apple. They want to buy Puts to protect their 112 Billion dollars worth in Apple stock holdings (Apple price 333.46) - they want ‘Protective Puts’.

They want puts for ALL their holdings. A single put represents the right to sell 100 shares of the underlying at the specified strike price. This means they want to purchase 3.36 million Puts. They want to protect themselves against anything larger than a ~10% downturn so they are looking at the Puts with strike 300.

Puts strike 300 are trading for $ 1.74. Each put is for 100 shares and they ware looking to buy 3.36 million puts. That is a total of approximately 585 million dollars.

(Now you learned that today it costs around 585 M to insure 112 B of Apple Stock for more than a 10% downturn. Less than 1% for that insurance doesn’t seem too bad.)

Anyway, we are interested not in The Vanguard Group that is buying this but rather in the Dealer who is selling them 585 M worth of Apple Puts.

If the dealer is short 3.36 million Puts they would have a Delta of around 36 Million. Meaning that for every 1 dollar move in Apple they are exposed to that move times 36 million. The dealer will not expose themselves to that if they want to stay in business - they will Delta Hedge.

To Delta Hedge they must sell short 36 million shares of Apple.

Remember what happened to the dealer when Apple fell to 320? The same thing happens to this one - they must now Delta Hedge again.

Their Puts have a Delta of around 67 Million, they had only shorted 36 million shares which means they need to short around 31 million more shares!

This thing that changed their Delta from 36 Million to 67 million is the gamma.

Just as we spoke about the Gamma for a single dealer - we can speak about the gamma of ALL the dealers.

Market Gamma Exposure

The single dealer case had negative gamma, which meant that when Apple price fell they had to sell Apple to rebalance Deltas. The Vanguard imaginary example the dealer also had negative gamma - millions of negative gamma. So when Apple price fell they had to sell millions of shares to rebalance their Delta.

What we get from this is the following: if the overall market is short gamma, it will exacerbate price directions in a vicious feedback loop.

When the market (Dealers) are short gamma and Apple price falls:

• Apple price falls => dealers must sell Apple to delta hedge
• Dealers sell Apple => Apple price falls more
• Apple price falls more => dealer must sell more Apple to delta hedge
• and so on…

Although we didn’t go through it in detail it works on the upside as well, so when the market is short gamma and Apple price rises:

• Apple price rises => dealers must buy Apple to delta hedge
• Dealers buy Apple => Apple price rises more
• Apple price rises more => dealers must buy more Apple to delta hedge

and so on…

And so we can see that:

When the market is Short Gamma - directional movements exacerbate as price falls call for more price falls, and price rises call for more price rises

The opposite is true when the market is Long Gamma - dealers delta hedging does not exacerbate price movement but rather calm it since when the price rises dealers must sell and if the price falls dealers must buy - thus dampening volatility in the market.