# The differences between calculating realized volatility for FX, Equities, and Cryptocurrencies

### Level 1 - Zero to Options Hero

Now that we have gone over the importance of realized volatility in professional traders’ thought process of whether an option is cheap or expensive, let’s dive into some of the intricacies of market-specific considerations when calculating realized volatility in FX, Equity, and Cryptocurrency markets.

If you recall from the earlier post on the assumptions behind Black-Scholes, the model for pricing and hedging options assumes that the option trader can trade the underlying asset continuously, with zero transaction costs, without interruption. In reality, asset markets are open only at certain times. For example, Foreign Exchange markets are open 24 hours a day from Monday, 7am in New Zealand to Friday, 5pm in New York. The New York Stock Exchange is open from 9:30am to 4pm Monday to Friday, with the exception of Exchange holidays. In fact, the cryptocurrency markets are most likely the market that has the closest structure to what is assumed in Black-Scholes, in that cryptocurrency exchanges are open for trading 24 hours a day, 7 days a week, 365 days a year.

These market mechanics play into two major considerations when it comes to our realized volatility calculation. The first is the annualization factor, while the second has to do with modifications for accounting for the ‘overnight return’. Let’s dive in!

**Annualization Factor**

Recall that the realized volatility is defined as the annualized standard deviation of the log price returns. The reason that the figure is annualized is so that traders can compare realized volatility across different asset classes where markets trade a different number of days per year. Common factors for daily frequency volatility calculations include sqrt(252) for equities, sqrt(260) for FX, and sqrt(365) for cryptocurrencies. When sampling realized vol at higher frequencies, we need to take that into account by multiplying the daily annualization factors by the square root of the number of observations per day.

**Equity Dividend Adjustment**

As you may well know, the price of a stock will gap lower on the ex-dividend date as the fair value adjusts for the fact that the owner of the stock no longer receives the dividend past the ex-dividend date. This gap would cause what seems to be realized volatility but in reality, needs to be removed in order to get an accurate estimate of the true realized volatility.

The recommended way to make this adjustment is to multiply all the stock prices before the ex-dividend date by an adjustment factor defined as:

This will serve to make the price series smoother in terms of removing the price gap seen on the ex-dividend date.

**Overnight Return**

The overnight return is particularly relevant for equity markets and is the change in the price from yesterday’s market close to today’s market open. There are a few major methods that traders use to deal with this issue:

**Ignore the overnight return entirely, and only calculate realized volatility based on the price returns during market hours**. This is similar to ignoring the price jump after an earnings announcement. While it is simple, it would most likely result in a realized volatility calculation that is ‘too low’ as there are real price returns that occur during the overnight period due to market-moving news and pent-up supply/demand that builds up overnight. Therefore, we do not recommend this method!**Treat the overnight return in the same way as the other price returns in the historical time series for the purposes of calculating realized volatility**. For example, if we are sampling the price of a stock on the NYSE every 30 minutes, every day we would have 14 price observations from 9:30am to 4pm, plus an additional price observation at 9:30am the following day. Therefore, we can calculate the realized volatility using 15 price observations or 14 log returns every trading day.Now, you may say that it does not feel particularly natural to compare a price return over a 30-minute period with one that is over a 17.5-hour period — that is, it does not feel like an apples-to-apples comparison. In which case, we can look at the 3rd method!

This is an interesting method that involves some additional calculations and some parameters that can be tweaked for the aspiring options nerds out there.

**The goal is to estimate the percentage of the total price variance which occurs during the trading day.**One popular way to do this is to compare the realized volatility estimate based on a daily close-to-close calculation with the higher frequency intraday (i.e. every 30 minutes) calculation over a given lookback window.**Be sure to use the correct annualization factor in these calculations!**

Once this percentage is estimated, we can simply take the realized volatility estimate we calculated in method 1 and divide it by this percentage in order to adjust the estimate (higher) to incorporate the additional volatility that occurs during the overnight period.

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We will now run through a quick numerical example of both the **equity dividend adjustment** and **overnight return** adjustments for calculating realized volatility on equities using IBM stock (IBM).

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