Volatility is an important input of the formula that is used for option pricing. It shows the magnitude of variance that exists in the price changes of an underlying security over a period. By using historical data or other sophisticated methods, volatility calculations aim to forecast the degree of fluctuations in an asset’s price. But how accurately can we predict future volatility based on the historical data? How accurately can you predict the future based on the market conditions today? Future is always an uncertainty as the analysts and traders race to predict it relatively better.
Given its unknowns, volatility is a very interesting field in the options world. Let’s take a closer look to understand some of the theoretical and practical concepts that help finance professionals enable the use of this powerful estimation tool.
Most basic approach to calculate the volatility is by calculating the variance of a dataset and taking its square root which gives us the standard deviation.
So, to simplify, in this context volatility is the standard deviation and standard deviation is the square root of variance. And variance can be found with a statistical formula you’re about to be introduced.
If we are looking to calculate the variance for a period of 15 days, we need to calculate the mean deviation of each day’s prices. 2nd step is to square those deviations and divide their sum by the number of observations (in this case: 15) This simple step will give us the variation.
Although it is simple math it can be confusing to someone new to the field and calculations for each price can be tedious. Thankfully in today’s technology spreadsheets help a great deal with such calculations.
The standard deviation number that can be derived from these simple mean and deviation calculations is the volatility input of the Black and Scholes formula itself. With this much knowledge you could create your own volatility calculations just by using the historical data of the underlying prices.
When volatility is calculated by this standard deviation method it is called historical volatility. Let’s check out the example:
Steps for standard deviation calculation:
1st step: Gather data
2nd step: Calculate the mean
3rd step: Calculate mean deviation for each data point (value – mean)
4th step: Square each deviation
5th step: Calculate the mean of squared deviations (variance)
6th step: Calculate the square root of variance (standard deviation)
Let’s see the volatility calculation of Netflix on spreadsheet for a 15 days period:
Average of Prices
Average of Dev. Sq.
Sq. Root of Variance
Data: Yahoo Finance (NFLX)
You can see in the table that mean deviation values are simply the difference between the individual price and average price of the whole dataset.
Averages (mean) in the example above can be calculated with a simple “=Average” formula. The mathematical foundation of this formula is summing each data and dividing them with the n number of observations.
Similarly, you can use Stdev and/or variance formulas to directly calculate the needed values in Excel.
Although it is a very useful concept, historical volatility can be misleading in times of unexpected change which happens quite often and sometimes at most unexpected times in the financial markets.
As an exercise you can consider constructing the same calculation for 30, 60 and 90 day periods. You can simply acquire the price data from Yahoo Finance and use their download data button. There are many other alternative websites and premium data sources for this task.
For instance, traders, market makers and the minds behind the institutions that issue derivative products usually follow volatility values based on different time periods to have a better understanding of market possibilities that may unfold and affect their business.
Historical volatility inherits an assumption that history will always repeat itself which makes it inherently biased. Although it is acceptable to a certain extent there are always price movements and historical events that surprise even the most seasoned professionals. It is wise to be aware of this phenomenon. Many other methods have been introduced to improve the volatility calculations. Historical volatility works well during the trends and small changes until the moment a big crash or direction change happens. When this happens in the markets, volatility jump to a much higher value immediately as there is a big change in the deviation of prices from the mean.
Based on the price calculations, volatility correlates with both call and put option prices positively. In simple terms if volatility increases option prices increase whether they are put or call options.
Another phenomenon that’s even more unexpected than the big price changes and crashes is black swan events. They were successfully theorized by Economist Nassim Taleb and they signify the events that are highly beyond normal expectations of science, politics, technology, history etc. From this definition we understand that if a topic is being widely discussed by many people than it is not a black swan. Black swans catch everyone by surprise and are extremely unexpected in nature. They usually cause people to discuss them for decades in their aftermath.
If the historical data is not always a very reliable metric by itself to estimate future volatility of an asset, index, commodity or another security, what else can be used as to reinforce the future price predictions? How can the shortcomings of historical volatility be overcome? Implied volatility is one of the answers to alternative approaches for volatility calculations. Although it is not perfect it has shed a light to the discussion from a different perspective and has its own advantages.
Black and Scholes and other formulas are used to calculate option prices in the market. What if we use an option price and with reverse engineering the price and formula calculate the volatility that is potentially used for pricing. Strike price, underlying price, interest rates and time to maturity are rather certain values. So, it is not very difficult to use several iterations to extract the volatility value from an option price. This will give the volatility implied by the market prices, hence the term implied volatility. So we can say that implied volatility is calculated by reverse engineering the options already trading in the market and extracting a volatility value based on those option prices in the market.
There are benefits and shortcomings of implied volatility. First of all, by checking the common sensation of the market you are able to tell any volatility measures that are different than the theoretical historical volatility calculations. If there is a prestigious institution, you can judge the volatility by looking at their options. It is likely that they might have a quantitative department that utilizes highly advanced models to estimate the volatility for their option traders. If you saw an option that is way higher in price compared to others you can suppose that the issuer potentially knows something that can pose a risk to the current markets or they are making a mistake. When such mistakes are caught by other finance professionals they may take advantage of it by engaging in a strategy called volatility arbitrage. Although it may sound cool it’s usually not very feasible for individual traders and such mistakes are rather rare.
On the other hand, implied volatility also encourages a herd behaviour. If many people start judging the volatility only by implied volatility this will create a close loop feedback system where everyone relies on the same implied volatility values. For this reason, it is very important for option traders to stay critical of volatility values as they are estimation methods one way or another.
Another important and fundamental point to keep in mind is that, volatility refers to the potential of price movements. However, it doesn’t tell the direction. Generally, in a flat market a move in either direction, up or down, would increase the volatility, in an upward trend a sudden drop or in a downward trend a sudden rise would immediately increase the volatility. You could ponder about the deviation calculations in the historical volatility section to get a deeper understanding of relation between volatility and price changes.
A concept to measure the magnitude of price fluctuations.
Types of Volatility:
Historical volatility, implied volatility, stochastic volatility,
Vol., IV (implied volatility), HV (historical volatility)
S&P 500 Volatility Index (VIX)
S&P 100 Volatility Index (VXO)
Nasdaq 100 Volatility Index (VXN)
Knowledge Tip: What’s the volatility change on a stock that is increasing 2% everyday. (In a 90-day range)
The answer is zero. Since stock maintains the same level of change on a day to day basis the variance and standard deviation would remain zero and the same.
The Cboe Global Markets® (Cboe®) calculates and maintains a variety of volatility indexes that can be interesting to observe. You can also find options and futures that use these indexes as underlying security.
Here are some examples: (Source: CBOE® Date: for January 8, 2020)
How to interpret vega?
Let’s look at an option based on NASDAQ 100 index. Nasdaq 100 is an index that tracks 100 stocks that are listed on the Nasdaq stock exchange. QQQ ticker represents an ETF based on Nasdaq 100, while Nasdaq 100 is just an abstract index calculation, QQQ is a tradeable security that can be bought and sold.
There are also options based on QQQ, let’s analyze a few of them. Let’s say underlying QQQ is trading at $219. Both options are exactly ATM (at the money).
The call option has a Vega of 0.41 and this means 1% change of volatility in the underlying price causes $0.41 price change in the option. If volatility was to increase 10%, call option price in this example would also increase $4.10 in value.
The put option on the other hand also has a Vega of $0.41. As volatility affects option prices in the same direction in both put and call options, a 1% increase in the volatility would also cause the put option price to increase $4.10.
Knowledge Tip: Delta here is the option price sensitivity related to the underlying price change. In the call example, if underlying price was to increase $1, option price would increase $0.37 and for the put option example, if underlying price was to increase $1, option price would decrease $0.36.
You can find more detailed explanations of option greeks in the post dedicated to option greeks.
One criticism with Black & Scholes model is that, it assumes that volatility is a constant value. Although this was still a breakthrough and useful, we know obviously that volatility is not a constant concept. Stochastic Volatility (SV) is one of the answers to the shortcomings of other volatilities.
Usually falling under the practice of quantitative finance, stochastic modeling deals with random distribution model of variance.
Heston model is probably the most utilized SV model. Heston uses square roots of variance in a random differential process. Founded by Mathematician Steve Heston, the model is often formulized as a least squares problem with cost function minimization of Heston model prices and realized prices similar to the regression analysis used in Machine Learning and Artificial Intelligence algorithms.
CEV, SABR (Stochastic Alpha, Beta, Rho) and GARCH (The Generalized Autoregressive Conditional Heteroskedasticity) are some of the other stochastic volatility models that are used by option traders, quants, researchers and scientists.
Although there are other stochastic volatility concepts, they fall outside the scope of this post. Nevertheless, it can be an interesting subject to explore for traders with deeper mathematics and programming knowledge.