# Option gamma: identifying levels of pain

#### Published in Automated Trader Magazine Issue 39 Q2 2016

## The increasing popularity of shorter expiry S&P 500 options has the potential to significantly impact the underlying market. Analyzing the hedging pressure of market makers can help to identify critical points.

Ever since Black and Scholes introduced their famous option pricing formula, traders have been sensible about hedging their exposure to the various risk measures (price of the underlying, time till expiration, volatility, interest rates, dividends). Arguably, the most important exposure an option has is to the price movements of the underlying instrument. This sensitivity is labelled delta. Delta-hedging is a term familiar to every option trader. The importance of options with relatively short maturities - as defined by the trading volume and levels of open interest - has increased. Therefore, the risk in option trading nowadays is not just defined by the option delta. The importance of the option gamma, which measures the sensitivity of the option delta to changes of the underlying price, can create large delta exposures even on smaller moves in the price of the underlying and thus creates volatility spikes. This article shows how to use option gamma to identify 'pain levels' of the market.

### Recap of option delta and gamma

The concept of option delta (Δ) is quite simple. Ex post, one would compare the change of the price of an option (V) with the price change of the underlying instrument (S). So how many dollars does the price of the option change if the price of the underlying changed by some dollar amount:

One important aspect is the t-1 index of delta. Since the prices at time t are not available at time t-1, the delta cannot be calculated exactly at time t-1 and one has to approximate. The Black-Scholes approach comes into play, as delta is simply the first derivative of the price of the option with respect to the price of the underlying:

One should keep in mind that in contrast to equation (1), which
takes the realized option prices of t and t-1 (being subject to
different lifetimes, implied volatilities, dividends and rates),
the Black-Scholes approach in equation (2) keeps everything
except the underlying price equal. Δ _{re,t-1}
therefore is not really a pure measure of changing underlying
prices.

Figure 01 shows the delta of put options calculated with the Black-Scholes approach for different times to expiration. It is easily observable that the delta of an option is very dependent on the option's time to expiration.

#### Figure 01: Black-Scholes Option Delta for Different Expirations

The problem with the delta of an option is that it is not a fixed number, but is itself sensitive to changes in the price of the underlying. Delta grows the further the option goes in the money (below the strike price for puts, above for calls). The rate of change of delta is measured by the gamma (Γ) of the option. In Figure 01 we see that this rate of change also depends heavily on the remaining time to expiration of the option.

To calculate the option gamma, one would compare the dollar change of the option delta with the dollar change of the underlying price:

If we switch to the Black-Scholes approximation, gamma is the second derivative of the option price with respect to the price of the underlying:

Figure 02 shows the Black-Scholes gamma of options with different times to expiration. As expected from the previous chart, the gamma of short dated options is significantly different to the gamma of longer dated options, resembling a normal distribution with lower variance than the longer dated options, peaking when the price of the underlying is at the strike.

#### Figure 02: Black-Scholes Option Gamma for Different Expirations

Finally, the sensitivity of the gamma of an option to changes in the underlying price (sometimes referred to as speed) is the third derivative of the option price with respect to the price of the underlying:

The key points to keep in mind are that the distribution of gamma is much narrower for short dated products. More simply put, this means a much higher sensitivity of the option delta to changes in the underlying instruments price for short dated options.

### Motivation

Short dated products with a volatility component in their pricing structure have been on the rise over the last years. Products like weekly equity index options have shifted the balance of open interest in favor of instruments with a lifetime until expiration shorter than 31 days.

#### Figure 03: Open Interest for Different Expirations

Nowadays, options expiring within the next 7 days account for 15% of the total open interest in S&P 500 options as illustrated in Figure 03.

While these short dated products add a lot of trading opportunities to market participants, they also have an impact on the market structure of the underlying. The combination of large open interest and high gamma (and speed) can lead to increased volatility-of-volatility (sudden spikes and plunges in implied volatility) and in our view was the cause for the sharp declines and recoveries over the recent years (October 2014, August 2015).

### Research question

We aim to increase the understanding of the impact of increased volume in short-dated options on the volatility of the underlying instrument due to hedging requirements by analyzing weekly put options on the S&P 500 Index. We will answer the following questions:

What are the implications of the characteristics of short-dated options?

What was the impact of delta-hedging of short-dated options in August 2015?

How can market 'pain levels' be identified?

For this analysis, option data was analyzed on a daily basis (but the arguments hold for any shorter time period as well), assuming the following:

There are two types of market participants, traders and investors:

Investors have a long position in both the underlying and puts and do not rebalance their hedges.

Traders have a short position in options and hedge by buying/selling the underlying or a related instrument.

Trading in options occurs at end of day, meaning hedging is required for the entire open interest of the previous day.

The market risk is defined as downside risk and we therefore only analyze the open interest and delta/gamma in put options.

### What are the implications of the characteristics of short-dated options?

We assume that market participants who are short options will delta-hedge their market exposure in order to maintain a delta-neutral position. Traders will have to sell the underlying instrument if it declines and buy the underlying instrument if it rises. Figure 01 illustrates the option deltas for different lifetimes of options over their strike prices referenced to the price of the underlying. By definition, the shape of the delta distributions is much narrower and steeper for shorter lifetimes, meaning delta changes more if the price of the underlying changes. This is reflected in the gamma shown in Figure 02, creating a larger re-hedging need if the trader wants to remain delta-neutral.

Assuming a sharp decline in the price of the underlying, in the range of a three-standard deviation move, by how much does the delta change? Depending on the lifetime the change in delta could be significant.

#### Figure 04: Delta Sensitivity for Different Expirations

Figure 04 shows the impact of a 3% increase/decline in the underlying price on the delta of options (assuming an implied volatility of 15%, a risk-free rate and dividend yield of zero). The top chart shows options with 8 days to expiration, the bottom chart shows options with 365 days to maturity. Assuming a 2% out-of-the-money put option, there are three important differences between the lifetimes, as the shortest-dated option has:

1.a lower initial delta

2.more skewness in its market exposure

3.a larger gamma

### Lower initial delta exposure

The lower initial delta value potentially leads traders to increase their leverage as the initial market exposure remains small. This behavior becomes evident when Figure 03 is extended to include the distance to the current spot. The average strike distance (weighted by open interest) to the price of the underlying is much smaller for short dated options as illustrated in Figure 05. Roughly 45% of all open interest in options expiring within one week are between -10% and 0% vs. the current spot price.

#### Figure 05: Open Interest Distributions over Expirations and Strikes

There is an economic explanation for this: there is little need to buy/sell a significantly out-of-the-money put on a major equity index with one week until expiration in most circumstances.

### Stronger skewness in market exposure

The risk/reward ratio is heavily tilted for the shorter dated option and the non-linear payout structure of options plays a more important role. In the mentioned example, a 3% change in the underlying price causes a delta change of +0.17 (upside) vs. -0.49 (downside) for the weekly option. The yearly option experiences a delta change of +0.08 (upside) and -0.09 (downside).

### Larger gamma

Unfortunately, the Black-Scholes delta and gamma hide the true risks of this position. The real gamma of short dated options is huge. This does not have to be reflected in the gamma obtained by the Black-Scholes formula since it estimates the change in delta for a very small change in the underlying price only. As we all know there can be large moves in the underlying and even nowadays it is impossible to hedge on a continuous basis (mostly due to trading costs).

'Large' is a fuzzy description of a move, but with a 15% annualized volatility, one should expect daily moves between roughly -1% and +1% on a daily basis, which in turn will lead to a very quickly growing delta on any even expected down-move of the underlying price for short dated options as described above.

In short, in the example above, the delta of the short dated option will increase six times as fast compared to the delta of the longer dated option for the 3% move mentioned.

### EVENTS in August 2015

The theoretical process described can also be applied to real live environments and can explain some of the developments around days of major market turmoil, such as in August 2015.

August 24 ^{th} will be remembered as one of the larger
intraday moves in the modern history of the US stock markets. The
S&P 500 Index initially dropped 5.3% while S&P futures
dropped 7.1% shortly after market open (both compared to the
previous day's close). A quick recovery of most of this initial
decline took place within the next hour.

For the calculation we use:

date t = 2015-08-24

open interest (OI) of t-1 (2015-08-21)

every strike (k) of all puts (100 to 3500)

end of day level at time t-1 of the S&P 500 Index (S) of 1970.89

In a first step, the open delta position (OD) is calculated at t-1 and level S and multiplied by 100 to account for the options multiplier:

Then the price level (L) is set to 1950, 1925, 1900, 1875 and
1850 and 1893.21 (the closing price of the S&P 500 Index on
August 24 ^{th} ) to calculate the delta for every strike
and different levels (L) of the S&P 500 Index:

Using these deltas we repeat the process described in equation (6) for every level L.

For a real world analysis, gamma (Γ) is replaced by ΔDelta which is a combination of the methodology presented in equation (1) and equation (2), namely:

Doing so allows us to easily estimate the total gamma for discrete price levels.

### Total option impact

The top chart of Figure 06 shows the open delta exposure for all
price levels. The light blue line indicates the open delta
position on August 21 ^{st} 2015 (t-1), the darker blue
lines the simulated open delta position for simulated price
levels. The red line shows the open delta position at end of day
for August 24 ^{th} 2015.

#### Figure 06: Hedging Requirements for Option Exposure on August 24th 2015

As we can see, roughly half of the total open put delta originates from out-of-the-money strikes which will experience increasing delta and gamma if the market declines. In-the-money-puts will experience increasing delta but decreasing gamma. Increasing gamma will lead to a non-linear increasing hedging requirement.

The bottom chart in Figure 06 shows the total change in the open
delta on August 24 ^{th} versus the previous day's open
delta. From the bottom chart, the convexity caused by the gamma
is clearly visible as the slope of the hedging requirement gets
steeper for lower price levels due to the increased gamma.

If we keep in mind that the delta of one E-Mini S&P 500 future (the most liquid hedging instrument) is 50 only (the futures contract size), a ΔDelta of ~140 million (S&P 500 at 1850) means that 2.8 million futures contracts need to be sold to delta-neutralize the option position.

The total traded volume of the front month contract of the E-Mini
S&P 500 future on August 24 ^{th} 2015 was 5.4
million only.

Obviously, E-Mini S&P 500 futures are not the only way to hedge option positions and it is unlikely that all investors had naked short positions in puts. On the other hand, there are multiple other products with identical risk profiles (weekly S&P 500 futures options, weekly ETF options and others) and a multitude of products with related and thus similar risk exposures.

### Short dated option impact

The crucial point in this analysis is that a large fraction of
the increased delta risk comes from the very short-dated part of
the option chain. If we single out options with expiration within
the next two weeks (that is 13% of all available puts), these
carry ~25% of the initial delta risk (OD _{t-1} ) but
account for ~34% of the increase in risk (ΔDelta) as shown
in Figure 07 below.

#### Figure 07: Hedging Requirements for Option Exposure with Expiration less than or equal to 15 Days on August 24th 2015

We see that the initial delta risk of puts with a lifetime of 15
days can rise significantly during a sharp decline in the price
of the underlying (OD _{t-1, 1971} ≈ 0.7
ΔDelta _{t, 1850} ). Recalling the growing
importance of the short dated products illustrated in Figure 03,
investors should be alert as it is likely that an event as in
August 2015 will occur again.

### Liquidity comparison

As mentioned, the maximum requirement for hedging was around 3 million E-Mini S&P 500 futures contracts. While the total number of contracts traded that day was in excess of 5 million, at time of the market open the total traded volume so far was below 1.5 million and the increase in hedging requirements in the first minute of trading alone was roughly 750,000 contracts.

#### Figure 08: Hedging Requirements for Option Positions

The top chart in Figure 08 shows the intraday price (in CDT) of the E-Mini S&P 500 future with expiration September 2015. The lower chart shows the total futures volume traded and, in comparison, the number of futures contracts required to create a delta neutral position for all puts (offsetting the increase in delta). As we can see, the number of contracts required for delta hedging spiked instantaneously on market open.

This describes another problem arising due to the nature of this self-fueling process. The further the market has to be sold in order to hedge, the larger the hedging requirement grows until it levels off (if all puts have a delta of -1.0 there is no more additional hedging needed).

### Identifying 'pain levels'

As this 'leveling off' is a function of gamma, it can be calculated. For this analysis, gamma (Γ) was replaced by ΔDelta. This way, one can estimate the 'danger zone' (of increasing gamma). In this range there is additional selling pressure in an already falling market, creating liquidity issues which will likely cause rapid price movements.

The estimation of the 'danger zone' through the market delta here is done by calculating the deltas for every option (equation (7)) for a large range of N underlying prices levels:

L = {0.5 × S _{t-1} , …, 1.5 × S
_{t-1} }

In a second step, the total open delta (equation (6)) is calculated for every price level.

Finally, the difference between the calculated total option delta is the Market Gamma:

Figure 09 illustrates this Market Gamma or 'danger zone' for
August 24 ^{th} 2015 expressed in futures contracts
required for delta-hedging for every point change in the S&P
500 Index. We see that for every point decrease of the S&P
500 Index, the marginal hedging requirement changes. It increases
first (reading from right to left), has its turning point around
an index level of 1946 and then decreases.

#### Figure 09: Identifying Peak Hedging Requirements

One can infer from this chart that there is increased additional hedging requirement (read: selling pressure) until the turning point is reached, most likely leading to reduced available liquidity and thus higher price volatility.

In normal circumstances the turning point is further away from
the current price of the underlying. On August 18 ^{th}
2015, the S&P 500 Index level was 2096.92 (indicated by the
red line), two days later it was at 2035.73 (indicated by the
yellow line). Historically, except for periods of extremely high
overall volatility (2008, 2011) the pressure point had been 5%
below the current market price. In recent years the pressure
point has moved closer to the current market price.

A positive value of the marginal hedging requirement means that the puts are in-the-money on average, meaning that they have a delta in excess of -0.5. This happens after sharp declines and can also lead to quicker recoveries.

The top chart of Figure 10 shows the historic evolution of this pressure point and the most notable feature is the steady upward trend starting in 2010. The closer this pressure point to the current price of the underlying, the greater the hedging need for smaller moves in the underlying market.

Not only was the pressure point closer to the current market price in the recent past (especially in August 2015). In addition, the total hedging requirement at this pressure point had grown substantially.

#### Figure 10: Historical Pressure Point

The yellow line in the bottom graph shows the marginal hedging requirement at the peak Market Gamma point. The red line shows the total hedging requirement, which is the cumulative number of futures contracts that would need to be traded until one got to the peak. Both are measured as a percentage of the total volume traded in E-Mini S&P 500 futures on that day.

We can see a steady upward movement in both measures since 2010. In combination with the smaller distance to the pressure point this has to cause more frequent sell-offs on even minor market shocks.

### Conclusion

The introduction and widespread acceptance of short dated options has changed the behaviour of the underlying. To use a metaphor: it almost seems as if the tail is wagging the dog, even though the analysis includes only listed S&P 500 Index options, which represent a fraction of the available short dated options.

Due to the higher gamma of these short dated options even smaller downward moves in the underlying can cause significant selling pressure on the instruments used to hedge the option exposure. This results in quick and severe volatility spikes, such as the one experienced in August 2015. As the mechanism works both ways, the likelihood of very quick recoveries has also grown.

The liquidity needs for the required hedging have also grown. While the trading volume in the underlying products has likewise increased, the available liquidity in periods of market stress nowadays dries up very quickly as we know from the experiences of the latest market shocks.

The option market pressure can be estimated through the process described and offers an insight into the current market sensitivity to external shocks (even minor ones).

Option sellers are often accused of "picking up pennies in front of a steam roller". The process described above turbo charges that steam roller.