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# Volatility Depends On The Resolution

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Realized volatility for the same lookback window depends on how frequency you sample it. In this document, you will:
• learn to compute realized volatility
• appreciate the difference between the sampling window and lookback
• learn to annualize realized volatility
• examine market data to see how realized volatility varies with the sampling period

## Introduction

These short posts explain the relevant computations. If you have ever computed a standard deviation, you are 99% of the way there.

Computing Realized Volatility

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Annualizing Realized Volatility

Now we compute realized volatilities on actual historical prices to see what we can learn.

## Data Exploration

#### Setup

Since we want to compare the effect of sampling period on realized volatility we will keep the lookback period constant.
As a reminder, if we computed daily volatility for the past 6 months:
• the sampling period is daily or 1 day
• the lookback is 6 months
We call this “6-month realized sampled daily”
Since volatility tends to “cluster” (low vol periods follow low vol periods and high vol periods tend to follow high vol periods), option markets for short or even medium terms maturities will give more weight to recent realized volatility. If you were pricing a 3-year option then you’d be more inclined to examine longer a longer lookback which smooths out the sharper peaks and valleys.
This idea is echoed in the concept of a “vol cone”

For our data exploration, we will
1. use constant lookback of 40 business days which is close to 2 months worth of trading days. We call this 40-day vol .
1. sample the vol using daily, 2 day, 5d , 10d, and 20d windows.
• This maps to 40-day vol being computed from samples of the following quantities respectively:
• daily (n = 40)
• 2 day (n = 20)
• 5d (n = 8)
• 10d (n=4)
• 20d (n =2)
1. compute 40-day vol for each business day from 1/2/2013 until 8/16/2024 (>11.5 years)

#### Example using AAPL

Let’s step through an example with AAPL.
1. We compute the realized vol for the 40 day lookback at with carious sampling windows.
• In the snippet below, we also compute the ratio vols from each sampling period to the realized vol computed from daily samples

1. Scatterplot the ratio of 40-day realized vol sampled at different frequencies vs 40-day realized vol sampled daily
1. A point on the black dotted line means the 40-day vol sampled daily was the same as the 40-day vol at the respective frequency
2. If a point is above the line, the vol sampled at a slower frequency is higher than the vol sampled daily (you might interpret this as a market that was trending. For example if a market went up 1% per day, then the daily vol would be about 16% but the 5d vol would be approximately 5% x √52 or 36%!
3. The regressions show that vol sampled at slower frequencies is lower than vol sampled at a higher frequency.
1. Compute the aggregate stats for over 11.5 years There are 2 observations worth noting:
1. The longer the sampling period, the lower the vol for the same 40-day lookback
2. The standard deviation of that ratio grows with the sampling period. If sampling periods for the same lookback differ greatly there will be more variation in the measured realized vols relative to each other! The variation in the ratio by sampling period is easily seen with box-and-whisker charts

## Concluding thoughts

The most general results applied to every symbol:
• However, in every case, the slower the sampling frequency (ie every 4 weeks instead of every 1 week), the more volatile the realized vol itself is compared to the daily sampled vol.

These results reinforce an intuitive idea: the longer your sampling period, the smaller the sample size for a given lookback, therefore:
The more frequently you sample vol the faster you converge on a better estimate of the volatility

If you only looked at annual returns it would take many years to get a sense of how volatile an asset or strategy is. This is also why evaluating investment managers on monthly returns is dangerous. It hides the risk.

Assorted observations:
• Periods when less frequently sampled vol give us a higher realized vol indicate trending (long option player will have wish they hedged less frequently or on a lower delta and vice versa!)
• Option traders think in straddles for shorter dated options but in vol for longer dated vols. Vol-thinking prompts you to ask “will this thing move X% per day” for the next year vs can this stock move 5% in a month. There’s a puzzle embedded in this revelation — the long gamma trader will conclude they should hedge daily to “capture” the higher vol, but a non-option trader may conclude that this is an argument for mean reversion on shorter time scales.
• An interesting scenario to filter for would a situation where realized volatility sampled slowly (ie weekly) was higher than higher frequency sampled vol for the same look back.
• Like the earlier example of a stock that goes up 1% a day (16% realized vol sampled daily) vs >30% vol sampled weekly.
• The reasoning here is: if the vol market is more focused on higher frequency sampled vol to price options it might underprice implied volatility.
• I used the moontower.ai backend to filter for scenarios in a few names where monthly realized vol sampled every 3 days was higher than monthly realized vol sampled daily. I called the ratio “Trend Premium” so I filtered for situations where it was > 1.
• I then filtered again where the options were cheap — a VRP < 1 (the RV in VRP coming from daily samples, making this analysis conservative)
• This was an example from QQQ over the last 2 years. The left chart highlights VRP < 1 AND “Trend premium” > 1

The right chart shows how the lagged VRP a month later (so what is the ratio of IV today divided by the realized vol that was eventually realized)

The realized vol underperformed the IV more than it usually does. So much for that. (I actually toggled through a bunch of names in this way but just eyeballing I couldn’t see a bias one way or the other. A sample of 2 years plus using overlapping data as it’s rolling windows is not exactly a proper study on this idea. The overlapping windows is not an issue for the general study earlier in the post which really just tries to understand the relationship of RV sampled at different frequencies for the same lookback. Overlapping windows does shrink your sample size if you are mining for a signal as I was flailing with this “Trend Premium” filter)

• Finally, here’s a summary table of the 40-day realized vol sampled daily for the past 11.5 years from our studies: