If you annualize volatility with 252 days can you use that number in a 365-day option model?

A Moontower reader asked a question that gets into one of the most confusing topics for option traders:

If I’m pricing options using calendar days(365), then I should even annualize realised volatility by multiplying 18.8(√256) instead of 16 (√256, approx trading days). In order to compare the VRP ratio on same scale, am I right?

I know firsthand from watching people wrestle with option models that this topic has put many brains in a blender. It’s worth a blog-post sized answer. My hope is that you will not only walk away clear-headed but bursting with ideas to explore.

A typical starting point

You compute close-to-close realized daily volatility for the past 252 trading days. Those days comprise the past year. You get an average daily vol of 1.875%. You annualize it by multiplying 16 to get 30% volatility.

observations:

• Before the addition of Juneteenth, a non-leap-year had 252 trading days.
• After Juneteenth was added to the holiday calendar, there are 251 trading days.
• A leap year will typically have 1 more trading day than a non-leap year.
• 2024 is a leap year that has 366 days and 252 trading days which is what we expect for a leap year.
• If the daily standard deviation is 1.875% you should annualize by multiplying by √252 but traders will typically just estimate by multiplying by 16 (ie √256). If you are building a model, don’t use the estimate, but a lot of trader workflow involves quick assessments so it’s worth noting where the mental math shortcuts are.

The central question is:

Can you put 30% annual volatility into a typical 365-day option model or should you have annualized by √365?

The answer is a satisfying mix of reasoning and arithmetic.

What’s even better is you will be able to appreciate a range of answers across a spectrum of complexity and be relieved that for 99% of you, the additional complexity is not worth the brain damage. But the insights can still lead to a flood of additional inspiration for anyone interested in volatility!

The key to the question: the meaning of 252 trading days

Straight to the heart of it:

Recognize that when you sampled 252 days of trading data you did in fact sample the volatility that transpired over a 365 day year.

Why?

Because the daily volatility that transpires from close-to-close is not just the volatility from the open to the close!

Close-to-close volatility = close-to-open volatility + open-to-close volatility

[Note: I’m using the word “volatility” in place of the technically correct term “variance”. Variance is volatility squared. Variance is additive across time so it’s the units you use to do the underlying math but the more colloquial “volatility” is  reader-friendly. You probably encounter the word “volatility” an order of magnitude or more than the term “variance” (does Zipf’s law apply to financial glossaries?) so why raise the cognitive load for the typical reader when the advanced reader’s burden of translation is quite low by virtue of them being, well, advanced.]

When you computed the realized vol from 252 days you included the volatility that occurs overnight and over the weekends/holidays. Although you only have 252 samples, it includes information about 365 days.

The core of the issue isn’t that you are missing information, it’s that you haven’t allocated it to the correct containers (ie time periods). You bluntly assigned it to trading days creating the illusion that the information only applies to 252 intervals whose boundaries are restricted to 6.5 market hours.

This is more clear if you compute your own ratios of close-to-open volatility divided by close-to-close volatility. You can start to answer questions like:

• What percentage of an asset’s volatility accumulates overnight?
• Do you think it would be higher for global assets like oil and gold or GME?
• How about weekends…how much of Friday close to Monday close volatility is captured from Friday close to Monday’s open?

All of this reduces to a comforting answer:

If you put your 252-day annualized realized volatility in a 365-day model it will generate a well-priced option assuming next year’s realized volatility is similar.

[Similarly had you annualized by √365 or ~ 19 you will overestimate the volatility and therefore the option price].

Where the brain damage begins: interpreting implied vols

The harder problem is interpreting what an IV means in the first place.

Calendar day (ie 365 day) option models

If we believe every calendar day is an equal contributor to that 30% vol then we are saying that volatility accumulates uniformly across every day, weekend or weekday. This will overstate weekend volatility and understate weekday volatility. In terms of options pricing, the straddle would experience the full theta for every hour from Friday’s close to Monday’s open. But I assure you it doesn’t (and if it looked like it did then IV actually fell — this will be more clear soon).

Business day (ie 252 day) option models

If we use a business day model we are saying no volatility transpires over the weekend. If that were true then the straddle wouldn’t decay at all over weekend.

The reality is somewhere in between. Volatility time doesn’t pass linearly. It passes slower over the weekend (so we experience some decay but not what the full theta predicts) and faster during the week. In other words that 30% vol gets a different weight depending on the day.

The difficulty in interpreting what an implied volatility in an option model is the flipside of the time vs volatility coin — different models disagree on how much time remains in the life of an option where the time remaining is measured as fraction of a model year.

To demonstrate this, imagine it’s the night of December 31st and you are looking at an option that expires on the evening of the following December 31. An option with 365-calendar days until expiry. At this moment both models agree that a full year remains until expiration.

• The 365-day model says there is 100% or 365 out of 365 days remaining.
• The 252-day model says there is 100% or 252 out of 252 days remaining.

Ok, January 1st comes and goes. It’s a holiday.

• The 365-day model says there is 99.7% or 364 out of 365 days remaining.
• The 252-day model says there is 100% or 252 out of 252 days remaining.

Let’s say the price of the option is unchanged.

[For some reason you can see the option price but the market’s closed. The fantasy actually doesn’t screw up the point. Also, if you have traded cotton you know the options market can be open while the underlying futures market is closed — this itself is a conclusive thought exercise on the Schrodinger’s question of does volatility time transpire when a market is closed.

What if Elon dropped dead on a Saturday, do you think TSLA’s share price is unchanged on Monday — if not then you have also answered the question does volatility transpire when a market is closed.  The fact that you can only measure its impact on Monday doesn’t mean it hasn’t transpired. Don’t confuse accounting challenges with reality.]

Both models are looking at the same option price, but the 365-day model thinks there is less time til expiry — it will therefore mechanically imply a higher volatility.

January 2nd is a business day. It comes and goes.

• The 365-day model says there is 99.45% or 363 out of 365 days remaining.
• The 252-day model says there is 99.60% or 251 out of 252 days remaining.

The gap in time remaining between the 2 models has narrowed .15% apart versus .30% apart but the 365-day model must still imply a slightly higher vol to account for “less time to expiry” relative to the 252-day model.

Let’s skip ahead a couple business days to the end of January 6th.

• The 365-day model says there is 98.36% or 359 out of 365 days remaining.
• The 252-day model says there is 98.02% or 247 out of 252 days remaining.

Now the 252-day model has less time until expiration. Again both models are fed the same option price but now the business day model implies a higher volatility!

Visual aids

Let’s pretend we are looking at a $100 stock and a call option struck at $100 (an at-the-money option) that expires in 365 days.

Assume the stock price never changes and the option price every day is the price that makes the IV 30% in a 365-day model (these models are the most common and usually the default when you find an online calculator or in your brokerage software).

I populated a table including the 2024 NYSE holiday schedule.

Earlier when we stepped through the first week of the year, you could sense a sawtooth tug-of-war between “DTE % remaining” between the 2 models.

• A business day rolling off impacts 1/252 of the second model but only 1/365 of the default model.
• A holiday or weekend day impacts 0/252 of the second model but still 1/365 of the default model.

Therefore, as the week progresses, more time comes off the business day model and pushes up the IV relative to the default 365-day model. Then on Monday, the business day IV falls because the option prices will have experienced some weekend erosion, but the business day model thinks no time has passed. The opposite happens with the calendar day model — the volatility falls throughout the week, but then pops up on Monday because the weekend doesn’t experience a full dose of decay.

If we use the time remaining in the default 365-day model as a baseline, we can compute the difference from the 252-day model. Likewise we can display the spread of the IVs between the 2 models. As the fraction of year remaining in the 252-day model falls relative to the 365-day model, the IV implied by the 252-day model increases relatively.

This is a plot of the option’s life where the time spread means the difference in time remaining from the 252-day model vs the 365-day model:

Note that the IV difference (orange line) for the first 10 months is less than 1/4 of a vol point. It’s not until you get into the last 60 days, that the IV differences get more significant and themselves volatile. This makes sense…when you have just a few weeks until expiration a business day rolling off has a larger impact on the “business-to-calendar days remaining ratio” (the self-loathing astute reader will have noticed that this ratio is exactly what drives the volatility difference. Technically, it’s the square root of that ratio — again the volatility vs variance thing).

Let’s zoom in on the IVs. Remember, we chose an option price that makes the 365-day model always imply 30%. We are seeing how the 252-day model IV bounces around relatively based on that very same option price. (You could have fixed the 252-day model as the default and saw how 365-day IV moves around).

This is the first 9 months. The IVs are fairly close.

The last 3 months:

The same option price is creating a 5 vol point difference in IVs between the 2 models.

[It makes sense. On the last day the default model says there is 1/365 days remaining and the business day model says there is 1/252 remaining. The square root of (252/365) is 83%. The business day model thinks there is much more time remaining than the calendar day model and therefore to generate the same option price it implies 83% of the 30% IV or ~ 25% IV]

Observations

1. When an option has 1 year until expiration, no matter what model you use you will see the same implied volatility.
   
2. The moment, the clock starts ticking, the way the day is categorized will change DTE when measured as a percentage of a year (which is how the t in an option model works — fraction of a year.)A different t yields a different IV for the same option price.
   This is a big reason why all IVs are “wrong”. There is no right IV. They always depend on the ruler we use to measure. When I was on the floor most traders used a 365-day model. When it got to Friday, traders might start “running Sunday’s sheets”…what that means is they push the days ahead in the model to fit the Friday option prices. This is a kluge so they don’t have to lower their model vols only to have to raise them again on Monday when the straddle doesn’t experience its full model theta. The sawtooth incarnate.
   
3. Bonus observation: It gets better. Vol time doesn’t even pass linearly intra-day either. The fist 30 minutes of the trading day is 1/13 of the business day but far more of the vol time has elapsed. Your “fraction of the year remaining” has passed faster than what the clock says.Intraday volatility decay schedules will look similar to the percentages prescribed by a VWAP algo — the first hour of the day might be 25% of the traded volume and 25% of the accumulated variance. Just like we decompose close-to-close volatility into close-to-open plus open-to-close we can decompose the open-to-close period into hours, 15-minute intervals, or even finer.

The endpoint of all this volatility accounting is a granular calendar which specifies weights to various periods. This framework can flex to accomodate earnings, economic releases, corporate events/conferences, rebalance dates, or whatever your creativity can imagine. The goal is minimize noisy changes in IV that are simply artifacts of lumpy, discrete decay schedules.

The practical takeaways

I have good news.

Unless you are in the business of trading for a fraction of a vol point, almost none of this matters. I was implementing volatility cleaning functions to trade cross asset >15 years ago. I used discrete methods like you see in the table above. Today, option firms are doing the same thing continuously. They imply IVs by integrating under the curve of a smooth, “event aware” voltime function.

For some it’s cute to know about this stuff if you want to explore further or add new friends to your idea sex orgy. But more importantly, there’s enough scaffolding here to walk away with actionable heuristics.

1) Your annualized realized volatilities (252 annualization factors) are acceptable to use in option models.

• Implied vols from option models apply that average volatility uniformly to a set of days. This can make them difficult to interpret without grounding it in assumptions of how the volatility is allocated to business days, overnights, and weekends/holidays. But if you are comparing options to one another much of that fog cancels out.
• And if you are hedging or speculating with options that are more than a few weeks out, the minor IV discrepancies between models are irrelevant.

2) Here’s the one that applies to most:

Don’t worry about small differences in absolute IV measures!!

Why?

• Again: the variation in IV between models is negligible with months until expiry!
• The difference in IV is probably swamped by the width of the spread in your long/short rates.
• If you trade 0dtes or weeklies you are probably better served to think in terms of straddles rather than IV anyway. This renders the IV noise due to “how much time does the model think remains” moot.[Although the topic of “how much volatility should be ascribed to the overnight” is definitely an area worth exploring if you trade short-dated options since those overnights are significant percentages of the variance time.]
• The typical option user is not doing vol arb for a few cents across asset classes (if you trade oil options that expire at 1pm on Wed vs USO options that expire at 4pm because they are relatively mispriced than you need to care about this stuff). Again, for must cases, the IV noise cancels out if you are trading listed options of the same asset type against one another.

Learn more:

⏳Understanding Variance Time (Moontower tutorial)