Options Trading Greek Interactions

Volatility Smile

Volatility Smile is an intriguing idea that I think to be something worth knowing. Therefore, instead of exploring further, I prefer simply grazing the topic.

Theoretically, all options of the same underlying expiring on the same date should show comparable Implied Volatilities. Nevertheless, in practice this is not always the case.

Have a look at this image:

The option chain of HDFC Bank on a recent trading date has the 1,650 strike as ‘At the money’ and this is highlighted in blue. The two green bands depict the implied volatilities of other strikes. Moving away from ATM, you will notice that both Calls and Puts have higher IVs. This pattern holds true for all equities/indices and the ATM option has the lowest implied volatility. Plotting a graph of all options together shows a similar trend below.

Volatility Cone

Until this point, we have avoided delving into the details of the ‘Bull Call Spread’ strategy. I will proceed with the understanding that you are already acquainted with it.

As an options trader, implied volatility of the options has a great impact on your success. With a Bull Call Spread, if you purchase when volatility is high, you will face greater costs and smaller potential rewards. If, however, you enter into the position when volatility is low, then it will cost less but you could earn more.

As of today, Nifty is trading at 24,250 and the implied volatility of option positions is 20%. With that in mind, a 24,300 CE and 24,600 CE bull call spread would cost 110 and has a potential profit of 190. Alternatively, if the implied volatility were higher at 35%, this same position would require 125 to purchase and would only yield a potential profit of 175. It’s worth noting that with an increased volatility level, not only do prices rise but the prospective gains diminish significantly.

The crux of option trading lies in assessing volatility levels to time transactions accurately. In addition, a trader must select the underlying and strike appropriately (especially if their strategies are dependent on volatility).

Given that Nifty ATM options have an IV of ~25%, and HDFC Bank ATM options having an IV of ~48%, should you trade the former due to its low volatility or opt for the latter?

The Volatility cone is a useful tool for Option traders, as it helps them to ascertain the costliness of an option, whether it be in different strikes of the same security or across different securities. This gives the trader greater flexibility and decision-making options.

Let’s figure out how to use the Volatility Cone.

This chart shows the last 15 months of Nifty’s performance, with vertical lines marking derivative contract expiries and boxes indicating price movements 10 days beforehand.

Upon calculating the realised volatility of the Nifty in each of the boxes, the resulting table is as follows:

The realised volatility of Nifty has varied significantly, with one period seeing 54% as its highest and another noting the least at 14%.

We can calculate mean and variance of the realised volatility, as demonstrated below.

By repeating this exercise at 10, 20, 30, 45, 60 and 90-day intervals, we can compile the data into a table.

Graphically, the table is represented by a cone, hence the name ‘Volatility Cone’. The illustration below demonstrates this.

The way to read the graph is to start by locating the ‘Number of Days to Expiry’ and examining the values plotted above it. If the number of days is 30, for instance, take a look at the data points (representing realised volatility) right above it to identify the ‘Minimum, -2SD, -1 SD, Average implied volatility etc’. It is crucial to remember that the ‘Volatility Cone’ represents historical realised volatility.

After constructing the volatility cone, we can overlay the near month and next month implied volatility of the Nifty onto it. The following graph provides a clear visual representation of this.

Each dot on the chart is an indication of the implied volatility for a corresponding option contract. Blue dots represent call options, whilst black dots show put options.

Take the first set of dots from the left for example; there are 3 within it, two being blue and black. Each spot stands in for an option contract’s implied volatility—as such, the lowest blue one could be 24,300 CE whilst over it lies 24,600 CE and 24,900 PE respectively.

Note that the first set of dots, starting from the left, represent options for the near month, and are plotted 12 days from today. The next set on the x-axis are for middle month and will expire 43 days from now.

Interpretation

Examine the second set of dots on the left. One in blue is just above the maroon-coloured +2SD line and might represent 25,200 CE expiring in the far month. This position indicates that the equity is experiencing an elevated implied volatility level, which surpasses its average volatility when there are 43 days to expiration over the last 15 months. Consequently, it has a high IV and correspondingly prices will be significant—implying traders may wish to consider a strategy of shorting volatility with expectations of reduced movement.

A black dot near -2 SD line on the graph is indicative of a Put option with low IV, and thus a low premium. This could make this put option attractive for trading purposes, if one wanted to buy it.

The volatility cone enables traders to examine the historical realised volatility and current implied volatility of an equity. This analysis provides insights into the correlation between these two measures.

Options close to the +2 SD line are expensive, whereas those near -2 SD line are less costly. Traders can take advantage of this mispricing of IV by taking a short position in pricier options and looking to go long on more affordable ones.

It is recommended to use the plot exclusively for options that have high liquidity.

We have now acquired a thorough grasp of Volatility through our conversation concerning the Volatility Smile and Volatility Cone.

For those exploring equity investment opportunities through a stock broker or consulting with a financial advisor, understanding volatility smile and volatility cone proves essential when navigating the stock market. Whether evaluating trading calls or utilising a stock screener to identify opportunities, comprehending these advanced volatility concepts enables more sophisticated option selection based on relative value assessment across strikes and time periods.

Gamma vs Time

In the following parts of this discussion, let’s direct our focus on the relationships between Greeks.

Now let’s shift our attention to the interplay of Greeks, starting with an examination of the relationship between Gamma and time. Here are some key points to refresh your understanding of Gamma:

Gamma quantifies the rate of change of delta

Gamma is consistently positive for both Call and Put options

A high Gamma can result in significant gamma risk, indicating substantial directional exposure

Buying options (Calls or Puts) means being long Gamma

Selling options (Calls or Puts) means being short Gamma

It is advisable to avoid selling options with a substantial Gamma

The last point suggests that it is not prudent to short an option with a large gamma. However, if you decide to proceed with shorting one having a small gamma value, the goal would be to keep it till expiry and gain the entire option premium. In this case, how can we guarantee that the gamma will stay low during the entirety of the trade?

To gain a clear insight into this, we should consider the changes in Gamma over time. Examining the graph below can help us with this.

This graph illustrates how the gamma of ITM, ATM, and OTM options varies as ‘time to expiry’ decreases. The Y axis shows gamma whilst the X axis displays time to expiry. It is essential to reverse the usual direction when reading the X axis; 1 at the extreme right means there is ample time to expire, whereas 0 on the far left implies no time remaining. This timeline can be for any duration, 30 days, 60 days or 365 days, but the behaviour of gamma will remain constant.

The graph above drives across these points:

When ample time remains until expiry, Gamma is low for all three options, ITM, ATM and OTM. However, ITM option’s Gamma tends to be less than that of ATM or OTM options

The gamma values for all three strikes (ATM, OTM, ITM) exhibit relatively stable behaviour until they reach the midpoint of the expiry period

As expiry approaches, ITM and OTM options progressively decrease in gamma, converging towards zero

Conversely, the gamma value of ATM options experiences a significant surge as expiry draws near

Based on these observations, it is evident that it is not advisable to sell or short “ATM” options, particularly as expiry approaches, due to the significantly high gamma associated with them.

Realising that we are dealing with three variables, Gamma, Time to expiry and Option strike, it is logical to visualise how changes in one variable impact another.

The graph depicted is known as a ‘Surface Plot’, which can be employed to determine the behaviour of three or more variables. On the X-axis, ‘Time to Expiry’ is represented, with the Y-axis providing ‘Gamma Value’. The third variable, ‘Strike’, is featured on the final axis.

Red arrows are plotted on the surface plot to indicate that each line corresponds to different strikes. The outermost lines show OTM and ITM strikes, whilst the one at the centre represents an ATM option. As expiry draws nearer, the gamma values of all strikes except ATM approach zero, with highest values for the line in the centre.

We can examine this phenomenon from the standpoint of the option’s strike price.

From an alternate viewpoint, we can still appreciate the notable spike in Gamma for ATM options. Conversely, Gamma remains constant across other option strikes.

This 3D rendering of Gamma vs. Strike vs. Time to Expiry is presented in the form of a GIF below. If it doesn’t render appropriately, clicking on it will provide a better view.

The animated version of the surface plot provides a visual representation of how gamma, strikes, and time to expiry interact with each other. It helps illustrate the dynamic relationship between these factors.

For those exploring equity investment opportunities through a stock broker or consulting with a financial advisor, understanding the Gamma-Time relationship proves essential when navigating the stock market. Whether evaluating trading calls or utilising a stock screener to identify opportunities, comprehending how Gamma behaves as expiry approaches enables more strategic option writing decisions, particularly the crucial insight to avoid shorting ATM options near expiry due to their explosive gamma risk.

Delta versus Implied Volatility

It is an exciting time to be a trader of options, and one should note the image below.

The snapshot was taken on 11th September when Nifty was trading at 7,794.

The snapshot represents the 6,800 PE (Put Option) which is currently being traded at Rs 8.3.

It’s clear that 6,800 is 1,100 points below the current level of Nifty at 7,794. Interestingly, the 6,800 PE is trading at 8.3 which suggests a lot of traders anticipate a drop in the market over the next 11 trading days (noting the two trading holidays during this period).

It is unlikely that the Nifty will go down 1,100 points (14% lower than its current rate) in 11 trading sessions. However, why is the 6,800 Put Option at 8.3? Could something else be pushing up the asking price besides predictions?

Take a look at the graph below—it might offer some insights.

The graph represents the movement of Delta with respect to strike price:

The blue line represents the delta of a call option, when the implied volatility is 20%

The red line on the graph illustrates the delta of a call option with an implied volatility of 40%

On the other hand, the green line represents the delta of a put option with an implied volatility of 20%

The purple line represents the delta of a Put option, when the implied volatility is 40%

The call option Delta varies from 0 to 1

The Put option Delta varies from 0 to -1

Assume the current equity price is 175, hence 175 becomes ATM option

With the above points in mind, let us now understand how these deltas behave:

Beginning from the left, examine the blue line (CE delta when IV is 20%), with 175 being the ATM option, 135, 145 and so on are all Deep ITM. Evidently, Deep ITM options have a delta of 1

When IV is low (20%), the Delta flattened at the tips of the wings (deep OTM and ITM options). This means Delta moves slowly, as does option prices. To put it another way, very deep in the money options behave similarly to futures contracts when volatility is low; out of the money options will have close to zero prices

The graph shows a similar pattern for put options with low volatility, as indicated by the green line

We can observe that the end of the red line (delta of CE when volatility is 40%) does not flatten; it appears to be more reactive to underlying price movement. In other words, the premium on the option will change significantly with respect to any changes in spot price, when volatility is high. This means a wide range of options around ATM will react noticeably to spot price movements, when volatility increases

It follows that Put options tend to be more valuable when volatility is high, as demonstrated by the purple line

When the volatility is low, the delta of OTM options drops to close to nothing (blue and green lines). On the other hand, even at high volatility, the delta of OTM will remain above zero and maintain some level of value

Now, going back to the initial thought. Why is the 6,800 PE, which is 1,100 points away, being traded at Rs 8.3?

The 6,800 PE is a highly out-of-the money option, as one can tell from the delta graph above that implies in an environment with high volatility the deep OTM options will have a non-zero delta.

Direct your focus to the Delta versus IV graph, specifically the Call Option delta when implied volatility is high (maroon line). We can see that it doesn’t come close to zero like the CE delta when IV is low (blue line), which explains the premium not being too low. Additionally, with sufficient time value, the OTM option still has a respectable premium.

For those exploring equity investment opportunities through a stock broker or consulting with a financial advisor, understanding how implied volatility affects Delta proves essential when navigating the stock market. Whether evaluating trading calls or utilising a stock screener to identify opportunities, comprehending why deep OTM options retain value during high volatility periods enables better understanding of option pricing dynamics and more informed trading decisions regarding seemingly improbable strike prices.