20.1 -Considering the  Volatility Smile

In the previous chapter, we briefly examined inter-Greek interactions and how they are manifested on the options premium. This area is important as it will allow us to choose the best strikes to trade. Before we get to that, we'll touch on two topics about volatility: 'Volatility Smile’ and 'Volatility Cone’.

Volatility Smile is a fascinating concept that I consider to be 'good-to-know'. This is why I will only briefly touch on it and not go into detail.

Theoretically, all options with the same underlying expiring on the exact same expiry date should have similar 'Implied Volatilities (IV). However.in reality this is not the case.

Take a look at the image (image 1)
This is the SBI option chain as of September 2015 . SBI trades at 225. The 225 strike is now 'At-the-money'. A blue band highlights the same. These two green bands indicate the implied volatilities for all other strikes. This is what you will notice: as you move away from the ATM option for both Calls or Puts, the implied volatilities rise. In fact, the further you move from ATM the higher the IV. This pattern is evident across all stocks and indices. You will also notice that the implied volatility for the ATM option is the lowest. You can plot the graph below showing all the option strikes and their implied volatility.
(image 2
 

The graph looks like a smiling face; hence, the name "Volatility Smile"

20.2 - Volatility cone

We haven't yet touched on the option strategy called "Bull Call Spread", but I'll assume that you're familiar with it for this discussion.

The implied volatility of options can have a significant impact on the profitability of options traders. Let's say you are bullish about a stock and wish to start an option strategy, such as a Bull Call Spread. You will need to pay high upfront costs and have lower potential profit if you trade options with high implied volatility. If the implied volatility of the options is low, however, you will be able to trade at lower costs and make a higher profit.
 

Nifty trades at 7789 as of today. If the implied volatility of options positions is currently 20%, then a spread of 7800 CE or 8000 CE bull calls would cost 72 and yield a potential profit 128. The implied volatility of option positions is now 35%, instead of 20%. This position would be 82 and have a potential profit of 118. A bull call spread with higher volatility not only has a higher cost but also reduces profitability.

Option traders need to be aware of the volatility level in order to correctly time trades. Option traders also have to consider the selection of strike and underlying (especially if you use volatility-based strategies).

Consider Nifty ATM options, for example, have an IV currently of 25%, while SBI ATM options have an IV at 52%. Should you trade Nifty options if your IV is low, or should you opt to go with SBI options instead?

The Volatility cone is a useful tool that addresses this question for Option traders. The Volatility Cone allows traders to assess the cost of options. Identify options that are trading expensive/cheap. You can trade across different strikes of the same security, as well as across other securities.

Let's find out how to use Volatility Cone.

Below is the Nifty chart over the past 15 months. The expiry dates for the derivative contracts are marked by the vertical lines. The boxes preceding the vertical lines indicate the price movement of Nifty in the 10 days before expiry.
(image 3)
The following table will be generated if you multiply the Nifty's realized volatilty in each box.
(CHART)

The above table shows that Nifty's realized volatilty ranges from a maximum 56% (Feb 2015), to a minimum 13% (April 2015.).

As shown below, we can also calculate the mean and variance of realized volatility.
(Chart
We could repeat the exercise for 10, 20, 30, 45 and 60 day windows. Then we'd get the following table.
(CHART)
The graphic representation of the table would look like a cone, as shown below. This is why the name "Volatility Cone" was given to it.
(image 4)
 

To read the graph, first identify the "Number days to expire" and then examine all data points above it.If the number days remaining before expiry is 30, then you can see the data points directly above that. These are the data points representing realized volatilty. This will let you determine the Minimum-2SD, Minimum-1SD, Average implied volatility, etc. Remember that the Volatility Cone' graphically represents the 'historical realized volatility'.

We can now plot the implied volatility for the current day on the volatility cone that we have constructed. Below is an illustration of Nifty's near-month (September 15th) and next month (October 15th), implied volatility using the Volatility cone .

Each dot indicates the implied volatility for an options contract. Blue are for call options, and black for puts options.

Start from the left and look at the first set. There are three blue dots and one black dot. Each dot indicates the implied volatility for an options contract. The first blue dot could represent the implied volatility at 7800 CE. Above that, it could be 8000 CE. Above that, it could be 8100 PE.

(image 5)
 

The first set of dots, which begin from left, represent near-month options (September 2015), and are plotted at 12, on the x-axis. These options expire 12 days after today. The next set is for middle month (October 2015), plotted at 43. These options will expire in 43 days.

Interpretation

The 2nd set of dots is to your left. For the middle month option, we can see a blue dot just above the +2SD (topmost line, colored in marsoon). This dot could be for option 8200 CE. It expires 29-Oct-15. If this is the case, it would indicate that 8200 CE is currently experiencing implied volatility. This is because 8200 CE has experienced volatility in the past 15 months, when there have been "43 days until expiry". (Remember, we only considered data for 15 months]. This option has a high IV. Therefore, premiums will be high. One can also consider trading to shorten the "volatility" with the expectation that volatility will cool down.

A black dot at the -2 SD line of the graph is also for a put option. This indicates that this put option is trading at a low price because it has a low IV and thus carries a low premium. You can design a trade to purchase this put option.

Traders can draw a volatility cone for stocks, and then overlay it with the current IV. The volatility cone is a way to get an idea of the current implied volatility relative to past realized volatility.

Options close to the + 2SD line can be considered expensive to trade, while options closer to the -2 SD line may be considered cheap to trade. Trades can be designed to exploit'mispriced IV. Try to avoid trading expensive options and trade cheap options.

Note: The plot should only be used for liquid options.

We hope that this discussion on Volatility Smile has helped us to understand Volatility better.

20.3 -Time v/s Gamma

Let's now turn our attention to inter-greek interactions in the following two sections.

Let's now concentrate a little bit on greek interactions. We will begin by looking at Gamma's behavior with respect to time. These are some points to refresh your memory about Gamma.

• The rate at which delta changes is Gamma..
• Both Calls and Puts have Gamma as a positive number
•    Large gamma risk can be translated through large Gamma.(As taken directional risk)  
• You are long Gamma when you purchase options (Calls and Puts).
• You are short Gamma when you have options (Calls and Puts).
• Avoid shorting options with a high gamma

Last, avoid shorting options with a high gamma. It's fair enough. However, imagine that you are in a position where you want to short an option with a low gamma. You can short the option with a low gamma value and keep the position until expiry to retain the option premium. But the question is: How can we make sure that the gamma stays low over the course of the trade's life?

This is why understanding Gamma's behavior versus time to maturity/expiry is key. Take a look at this graph.
(image 6)
 

The graph below shows how OTM, ATM and ITM options' gamma changes as the 'time until expiry' decreases. The X axis is time to expiry, while the Y axis shows gamma. Contrary to other graphs, you should not look at X-axis from left or right. Instead, look at X-axis from right and left. The value at the extreme right reads 1, which indicates that there is plenty of time for expiry. The value at its left ends reads 0. This implies there is no expiry date.You can think of the time lapse as any period between 30 and 60 days, or 365 to expire. The behavior of Gamma is the same regardless of expiry time.

These points are illustrated in the graph.

• All three options ITM (ATM), ATM (OTM) have low Gamma values when there is plenty of time for expiry. Gamma for ITM options tends to be lower than ATM and OTM options.
• The gamma values of all three strikes (ATM OTM and ITM) are fairly constant until they reach half-way through the expiry.
• As we near expiry, OTM and ITM options race to zero gamma
• As we get closer to expiry, the gamma value for ATM options goes up dramatically

These points are clear. You don't want to shorten "ATM" options. This is especially true if you get close to expiry. ATM Gamma tends be very high.

It is important to realize that we are actually talking about three variables: Gamma, Time until expiry, Option strike. Visualizing the relationship between one variable and another is easy. Take a look at this image.
(IMAGE 7).
 

This graph is known as a "Surface Plot", and it is useful for observing the behavior of three or more variables. The X-axis includes 'Time to Expiry' while the Y-axis contains the gamma. Another axis contains "Strike".

A few red arrows are plotted on the surface plot. These arrows indicate that each line the arrow points to refers to different strikes. The line at the middle corresponds to ATM option. These lines show that, as we get closer to expiry, the gamma value of all strikes other than ATM tends towards zero. ATM strikes and a few others around ATM have gamma values that are not zero. Gamma is actually highest for the line in the middle - which is ATM option.

It can be viewed from the viewpoint of the strike price.
(image 8).
 

20.4 -Implied volatility v/s Delta

Options traders are in interesting times. Look at this image -
(image 9)
 

This snapshot was taken 11 th September, when Nifty was trading at 7,794. This is the snapshot of 6800 PE, currently trading at Rs.8.3/+.

This means that 6800 is 1100 points away from the current Nifty level at 7794. 6800 PE trading at 8.3 means traders expect the market will move 1100 points lower in 11 trading sessions (note that there are 2 trading holidays starting now).

The odds of Nifty moving 1100 (4% lower than the current level) in 11 trading sessions is low. So why is the 6800 PE trading at 8.3 instead? Are there other factors that are driving options prices higher than expectations? The graph below may have the answer.
(image 10).
 

The graph shows the delta's movement with respect to strike prices. The graph is shown above.

• The delta of a call option is the blue line, which indicates that the implied volatility is 20%.
• The delta of a call option is the red line, which indicates that the implied volatility is at 40%.
• The delta of a Put option is shown in green. Implied volatility is 20%.
• The delta of a Put option is the purple line, which indicates that the implied volatility is at 40%.
• The call option Delta ranges from 0-1
• The put option delta varies between 0 and -1
• Let's say that the stock price at the moment is 175 and you have an ATM option.

Let's now look at the following points.

• Start from the left. Notice the blue line (CE Delta when IV is 20%), considering that 175 is an ATM option and strikes such as 135, 140, 145 etc. are all Deep-ITM options. Evidently, Deep ITM options have an ITM delta of 1.
• The IV drops to 20% and the delta becomes flattened at its ends (deep OTM or ITM options). This means that the rate at Delta moves is low, and further implies that the premium rate moves. Deep ITM options behave almost exactly like a futures contract, with low volatility. OTM option prices will also be close to zero.
• Similar behavior can be observed for put option with low volatility (observe green line).
• The red line (delta CE at 40% volatility) shows that the end (ITM/OTM), is not flattened. In fact, the line seems to be more responsive to price movements. When volatility is high, the rate at the which an option's premium changes relative to the change in the underlying is high. This means that ATM options are highly sensitive to volatility and spot price changes.
• Similar observations can be made regarding the Put options in high volatility (purple line).
• The delta of OTM options drops to almost zero when volatility is low (see the blue and green lines). The delta of OTM options does not decrease when volatility is high. It maintains a small value that is non-zero.

Let's go back to our initial thought: Why is the 6800 PE trading at Rs.8.3/?, 1100 points away?

That's because 6800PE is a deep OTM option. As the delta graph below suggests, deep OTM options can have non zero delta values when volatility is high (see the image below).

I suggest that you pay attention to the Delta-versus-IV graph. In particular, take a look at the Call Option delta (maroon line) when implied volatility is high. The delta doesn't collapse to zero, as can be seen by the blue line - CE Delta when IV is low. This is why the premium does not seem to be very low. The OTM option has a respectable premium due to the time value.

(image 11).
 

keypoints

1. The Volatility Smile helps you to visualize that OTM options often have high IVs
2. A 'Volatility cone' allows you to visualize the implied volatility of today in relation to past realized volatility
3. Gamma is high at ATM options, especially towards the expiry.
4. Gamma for OTM and ITM options decreases to zero as we get closer to expiry
5. When IV is low, Delta can have an impact on ATM options with a lower range of options. Its influence will increase when volatility is high.
6. The far OTM options tend to have a delta value of non zero when volatility is high