Lesson 19 – Implied Volatility vs. Historical Volatility

Lesson Objective: Understanding the main differences between Implied Volatility (“IV”) and Historical Volatility (“HV”) and how to use them effectively in your trading of options.

Historical Volatility

First, let’s start with some basics.

Take an imaginary stock XYZ with a current price of $100 at the opening session. Let’s fast forward 10 (business) days and see its distribution of daily returns based on the closing prices over the past 10 trading days, and then let’s calculate the average rate of daily return over the 10-day period:

The average rate of daily return is +1.37% but the stock has been moving widely from -14% on Day 8 to +16.3% on Day 9. It appears that the stock is quite volatile. Let’s see how volatile.

Statistically, standard deviation (σ) is a fair proxy for volatility¹. Standard deviation is a mathematical term that quantifies the amount of variability or dispersion around an average.

It is calculated as the square root of the sum of the squared variances between the value (daily return) at each observation less the average value for all observations divided by the number of observations:

Capital sigma means “sum of”, x bar is the average value for all observations, and n is the number of observations (10 in our example).

So, let’s determine σ:

So, the standard deviation σ (“volatility”) of the distribution of XYZ prices over these past 10 days is about 9%.

In mathematical terms (assuming that the daily returns of XYZ follows a normal distribution – see footnote 1), it means that there is about a 68% probability that the daily returns of XYZ will fall within one standard deviation i.e., between -9% and +9% over the 10-day period. In our example, it was indeed in-range 7 times out of 10, so 70% of the time.

This would be called the 10-day Historical Volatility (“HV”) – sometimes referred to as Statistical Volatility (“SV”), or Realized Volatility (“RV”) by some brokers. HV is backward looking and can be extended to any period of time: 1 month, 2 months, 360 days, and so on. HV is typically given for a 52-week period by brokers.

HV is useful on a standalone basis as it provides a sense of the inherent volatility of an underlying over some period of time in the past. However, you should also analyze it within the context of its historical range.

For instance, 9% appears to be very high, but what if the 10-day HV had been between 8% and 13% for XYZ? Then 9% would be high in absolute terms but low in relative terms: in fact, 9% would be at the 20% range of the 10-day HV’s historical range [20% = (9%-8%) / (13%-8%)].

Usually your broker will have this “range” information available online. You should use it to understand if the current HV is an outlier or not before you rely on it.

The higher the historical volatility value, the riskier the security, but also the faster and more likely it can move to your target (strike) price, which is what traders hope for when buying options.

Compare the price fluctuations for the two stocks A (in red) and B (in green) in the graph below:

It is pretty clear that stock A is more volatile than stock B and HV for stock A has been higher (wider fluctuations) than for stock B, so one would conclude that A is riskier than B since it could normally lose more in one single move. On the other hand, because of stock A’s more significant swings, it could reach a higher target price than B could in that time window. As you can see, the risk/reward profile of an underlying is intimately linked to its volatility.

Although historical volatility is a good indicator of the past, it says nothing of the market’s views of volatility going forward. That’s where implied volatility – which focuses on the future – comes in. Traders are usually more interested in implied volatility than historical volatility as they constantly try to gauge future market movements.

Implied Volatility

Like Historical Volatility, Implied Volatility (“IV”) is always used in reference to a specific underlying. However, contrary to Historical Volatility, IV is a forward-looking indicator that reflects the collective view of market participants in terms of the expected move of the underlying over a specific time frame. Note that IV is only concerned with the extent of the move, not with its direction (up or down).

As you may recall from Lesson 10, IV is not directly observable as opposed to the other five inputs which are used in option pricing models. IV is generally derived from an option pricing model as the “plug” that will make the theoretical price of the option (per the model) match the option’s price as provided by the market forces of supply and demand².

Market participants generally use at-the-money (ATM) options to determine IV because most option trading volume typically occurs in ATM options.

Mathematically speaking, IV represents a one standard deviation range for the underlying and can be calculated for any period of time (e.g., 10 days, 1 month) although most of the time it is annualized. As such, IV can be used to determine the probability of an underlying reaching a specific price by a certain time. Going back to the definition of standard deviation above, IV provides a confidence level on the underlying’s expected range (the high and low of the underlying) by expiration. So, if you compare the range suggested by IV with the strike price(s) at which you trade your options, this will (1) give you an idea of whether the market agrees with your outlook, and (2) help you decide on your positioning in terms of risk/reward profile.

Assume for instance that the 30-day IV is at 10% while the underlying (XYZ) is at $100. This means that there is a 68% probability that XYZ will be between $90 and $110 in a month from now. If you decide to sell a one-month put option with a $89 strike price, you should be “ok” according to the view of market participants because it is theoretically less probable that the option be exercised at the $89 level (i.e., it is more likely than not that you will not be assigned the underlying, in which case you will be allowed to keep the full premium and will receive the maximum profit).

On the other hand, if you decide to be more aggressive and sell a one-month put at the $93 or $94 strike, then there is a much higher chance that you will be assigned the underlying. However, you will receive a higher premium as compensation. As you can see, your risk/reward profile is much different than in the first case above.

As previously stated, we have assumed a normal distribution of price movements for simplicity’s sake (see footnote 1). Under this assumption, you could increase the odds in your favor by determining what price range would give you a 95% or even a 99% confidence level that the price of the underlying would stay within such a range at expiration. Statistically, this represents two standard deviations and three standard deviations, respectively.

So, using the same example of a 10% 30-day IV with XYZ at $100 (IV = one standard deviation = 10%):

There is a 68% chance that XYZ will be between $90 and $110 in one month (one standard deviation ⇒ +/- 10%)

There is a 95% chance that XYZ will be between $80 and $120 in one month (two standard deviations ⇒ +/- 20%)

There is a 99% chance that XYZ will be between $70 and $130 in one month (three standard deviations ⇒ +/- 30%)

Graphically, we can see the three confidence levels and their boundaries as follows:

Keep in mind that these numbers all belong to a mathematical world. In reality, there are occasions where a stock moves outside of the ranges set by the second or even third standard deviation: in the 1987 equity crash, the market made a 20 standard deviation move. In theory, the odds of such a move are about once every 1⁸⁶ (1 followed by 86 zeros) years – although virtually impossible, it did actually occur for real.

Reality can be bigger than theory, and the theoretical outcome is only as good as the inputs used in the model. Market participants could be wrong on their IV forecast for instance. In fact, most of the time market participants overestimate IV, as we will discuss in the last section below.

But first, let’s point out that IV at any given time should be compared to its historical range, just like in the case of historical volatility. If IV for stock XYZ is currently at 10% but its range is between 10% and 25%, then it is at the bottom of that range and relatively very low. This is an important data point that you need to be aware of before entering into a trade, the chances are that IV will pick up steam again at some point and will move back up towards its average value (mean reversal), in which case the price of XYZ options will tend to increase (all else being equal).

The Potential Edge of Options Sellers

The other comparison that is worth looking at is current implied volatility vs. historical volatility (this is also readily available on most brokers’ platforms).

In the last two decades of market history, implied volatility has been overstated most of the time when back testing it against historical volatility.

Recall that IV conveys the market participants’ view on volatility in the future. Generally speaking, market participants are fearful of a black swan event and tend to overestimate the potential moves of a stock or ETF in the future. The fear factor is reflected in the pricing of options through the premium demanded by market participants and will inflate IV accordingly.

However, when we look back and analyze how much of the implied volatility has actually been realized, we find that – except for the times of actual market corrections (such as in 1999/2000 and in 2007/2008) – implied volatility has been overpriced in the majority of cases.

This means that options sellers could have an edge as they would be able to capture the extra premium paid while experiencing less realized volatility than expected.

Let’s go back to the example in the first section on historical volatility: XYZ stock’s current price is $100 and you sell an XYZ Put with an expiration date in 10 (business) days while the 10-Day IV is at 11%, which means that the market expects XYZ’s price to be between $89 and $111 at expiration (1 standard deviation). As an option seller, you know that IV is usually overpriced, so you decide to be slightly “aggressive” and sell the put with a strike price of $90 (i.e., right above the market’s expectation of the lower end of the range) for $1.05.

As we now know, XYZ ended up at $110 at expiration and its realized volatility was only 9%, not 11%. Had the market priced a 9% volatility in the option’s premium instead, the premium received would have been lower, say $1.00 instead of $1.05 – so you earned an extra 5% of premium due to the market participants being over-conservative in their forecast of implied volatility.

We like to be net sellers of options overall ourselves. Option buying can still be included – and often is – but as part of an overall strategy rather than on a standalone basis.

¹ All option pricing models assume a “log normal distribution” because a stock can only go down to zero, whereas it is not bound on the upside. However, for the sake of simplicity, this lesson assumes a “normal distribution” whereby the underlying can move equally in either direction. Under this premise, standard deviation (“sigma”) is used as a proxy for volatility and one sigma encompasses 68% of the normal distribution, two sigmas 95% of it, and three sigmas 99% of the normal distribution.

² Note that this approach applies to the vast majority of cases but assumes that there is enough liquidity (i.e., supply and demand) to provide robust market-determined prices for the underlying. If liquidity was insufficient, the market maker could take over and impose the model’s price instead, which would obviously create a different outcome for IV.