How to calculate daily Implied Volatility?

[longthewings]

I need help calculating the implied volatility of a stock/index for a single day. I referenced the site: https://www.optionsanimal.com/using-implied-volatility-determine-expected-range-stock/
Formula: (Stock price) x (Annualized Implied Volatility) x (Square Root of [days to expiration / 365]) = 1 standard deviation.

Here's my attempt, I didn't want to use the IV of a option set to expire a year out because I wanted to be as accurate as possible. So I chose the IV of a weekly option, in this case 6 days left to expiration with the IV of 15.51%.

Current S&P 500 Index: ~2005
Implied Volatility: 15.51%

To calculate the daily IV is the formula:

Standard Deviation = 2005 x 0.1551 x sqrt(1/252)
Daily IV = .97% = Standard Deviation/Current Price?


or

Standard Deviation = 2005 x 0.1551 x sqrt(1/365)
Daily IV = .81% = Standard Deviation/Current Price?


some places are saying there's 252 trading days, but I thought the IV was based on every day of the year including non-trading days.

So confused here...

My 2 cents.....

The basic Black-Scholes vol parameter input uses a 365-day year. So for pricing and hedging purposes, your implied vol parameter needs to be scaled to 365 days. Many traders will track realized vol using the 252-day convention, but if you want to use that number in the model it must be adjusted to a 365-day year first. Consistency is key, as others have stated.

The CBOE VIX also uses a 365-day year. The key is actually the variance, not the vol. The vol is a quoting convention. It's a way of showing a bucket of "implied variance" over the next 30 calendar days. If you want to bridge VIX from it's quoted 365-day value to a 252-day value -- say, to compare to a realized vol time series computed on 252 days -- then do this:

Closing VIX (365-day convention): 20.70%
Expected Variance over next 30 calendar days: 20.70% * 20.70% / 365 * 30 = 0.00352
Expected Daily Variance over next 20 trading days: 0.00352 / 20 = 0.000176
Computed 1-day Standard Deviation (252-day convention): sqrt(0.000176) = 1.33%
Closing "VIX" (252-day convention): 1.33% * sqrt(252) = 21.11%

So be careful and be consistent. Depending on what you're doing, days matter. But again, for pricing and hedging, the vol parameter for Black-Scholes is scaled to 365 days.

 

[Jones75]

My 2 cents.....

The basic Black-Scholes vol parameter input uses a 365-day year. So for pricing and hedging purposes, your implied vol parameter needs to be scaled to 365 days. Many traders will track realized vol using the 252-day convention, but if you want to use that number in the model it must be adjusted to a 365-day year first. Consistency is key, as others have stated.

The CBOE VIX also uses a 365-day year. The key is actually the variance, not the vol. The vol is a quoting convention. It's a way of showing a bucket of "implied variance" over the next 30 calendar days. If you want to bridge VIX from it's quoted 365-day value to a 252-day value -- say, to compare to a realized vol time series computed on 252 days -- then do this:

Closing VIX (365-day convention): 20.70%
Expected Variance over next 30 calendar days: 20.70% * 20.70% / 365 * 30 = 0.00352
Expected Daily Variance over next 20 trading days: 0.00352 / 20 = 0.000176
Computed 1-day Standard Deviation (252-day convention): sqrt(0.000176) = 1.33%
Closing "VIX" (252-day convention): 1.33% * sqrt(252) = 21.11%

So be careful and be consistent. Depending on what you're doing, days matter. But again, for pricing and hedging, the vol parameter for Black-Scholes is scaled to 365 days.

Great info. I used to use the 365 days, but for no explicable reason went to 252. It's like trying to fool yourself into thinking vol expectation is higher when in fact it is not.

 

[longthewings]

In most cases, it really doesn't make that much of a difference. But it is confusing. There was a point in time where I was very frustrated because I could never find a solid discussion of this topic in the literature. Seems most authors and practitioners pick one school of thought (either 365 or 252) and go with it.

It basically taught me that if I'm going to trade or analyze a product (i.e. VIX), it is vitally important to at least read or skim through the technical white paper on the nuts and bolts of how it is calculated, what it truly means, etc.

If I know that, it becomes far more clear what a number is and isn't, and whether it is useful in its current form, or needs to be adjusted in order for it to be useful or more accurate for my application.

 

[longthewings]

Also, this naturally begs the follow up question(s) to "How to calculate daily implied volatility?" of....

Ok, we can calculate daily IV....but is this number even useful? If so, how would you use it?

I think that there are a lot of academics out there that would say IV is essentially meaningless...it is the "wrong number you need to plug into the wrong model to return the correct market price of an option."

That's the short answer though. Maybe these "wrong numbers" still contain useful information that can be utilized in pricing and hedging. There is some good literature out there on this (Rebonato comes to mind).

I honestly don't know how useful the calculation of "daily IV" really is, or if you can use it to make money more effectively. For me, I came to the conclusion that predicting volatility is hard and I will likely get it wrong, so I better learn how to statically hedge options with other options (easier said than done) so that I can avoid going down this path to begin with.

 

[Jones75]

Also, this naturally begs the follow up question(s) to "How to calculate daily implied volatility?" of....

Ok, we can calculate daily IV....but is this number even useful? If so, how would you use it?

I think that there are a lot of academics out there that would say IV is essentially meaningless...it is the "wrong number you need to plug into the wrong model to return the correct market price of an option."

That's the short answer though. Maybe these "wrong numbers" still contain useful information that can be utilized in pricing and hedging. There is some good literature out there on this (Rebonato comes to mind).

I honestly don't know how useful the calculation of "daily IV" really is, or if you can use it to make money more effectively. For me, I came to the conclusion that predicting volatility is hard and I will likely get it wrong, so I better learn how to statically hedge options with other options (easier said than done) so that I can avoid going down this path to begin with.

Sometimes you can find a mis-pricing, as in getting a bargain or hosed, by checking the one day vol.

I just recently switched over to synthetic long put, delta neutral, straddles for earnings play. In one day, out the next. Though it won't dictate whether it's a buy or pass, I'll check the one day vol to see the expected move. Depending on the price of the underlying, it can be an eye opener, either way.

 

[Brighton]

My 2 cents.....

If you want to bridge VIX from it's quoted 365-day value to a 252-day value -- say, to compare to a realized vol time series computed on 252 days -- then do this:

Closing VIX (365-day convention): 20.70%
Expected Variance over next 30 calendar days: 20.70% * 20.70% / 365 * 30 = 0.00352
Expected Daily Variance over next 20 trading days: 0.00352 / 20 = 0.000176
Computed 1-day Standard Deviation (252-day convention): sqrt(0.000176) = 1.33%
Closing "VIX" (252-day convention): 1.33% * sqrt(252) = 21.11%

Should .00352 be divided by 21 trading days for a more "precise" trading month? Precise is in quotes because it's just an average of the 252 convention/12 months. I realize for 99% of people the difference wouldn't matter (me included), I just want to make sure I understand what you're explaining. Thx.

 

[Bond James]

>> longthewings: The basic Black-Scholes vol parameter input uses a 365-day year.

I second what longthewings says. Implied volatility as used by Black-Scholes formula conventionally means 365 days per year.

The 252 days convention is used when computing historical realized volatility from close prices of the stock. In this case you got approximately 252 observations in a year so you're not referring to calendar days but to trading days. There's also a spooky effect of the weekends: from friday close to monday close there is a span of three days, 3x more than the span from close-to-close during the week. By a constant and normal volatility assumption (the Black-Scholes model), the variance (average absolute movement) of the stock prices should be higher over weekends than over one trading day. The stock definitely moves more over a span of one year than over one hour (an exaggerated difference to help understand why variance depends on time). However, taking historical prices and doing the computations one can observe that the weekend movements from friday close to monday open are actually of significant lower variance. And from friday close to monday close they are approximately similar to the close-to-close returns during the week.

In conclusion it makes sense to ignore the weekends and only consider trading days, or close-of-previous-trading day to close-of-next-trading day prices. This gives about 252 days in a year and thus the convention for realized volatility.

 

[panzerman]

If you are at all worried about the precision of the calculation using 252 vs 365 days, you should instead use the number of hours until expiration. That will give you much more precise number. For retail, the difference between 252 and 365 isn't all that important. I use 252 days, but most professional shops will use hours.

 

[newwurldmn]

If you are at all worried about the precision of the calculation using 252 vs 365 days, you should instead use the number of hours until expiration. That will give you much more precise number. For retail, the difference between 252 and 365 isn't all that important. I use 252 days, but most professional shops will use hours.

The difference between 252 and 365 is huge.

Hours only matter when you start getting to a few days away from expiration.

Most market makers will use hours and will even tweak their implied levels for weekend and holiday volatility. They also study close to open expected volatility through a variety of measures.

Unless you are running a highly leveraged book (like a MM or a vol stat arb book), I think it's overkill. Personally, I use 256 days/year because it's a reasonable enough approximation and I can use 16 as my benchmark.

 

[longthewings]

Should .00352 be divided by 21 trading days for a more "precise" trading month? Precise is in quotes because it's just an average of the 252 convention/12 months. I realize for 99% of people the difference wouldn't matter (me included), I just want to make sure I understand what you're explaining. Thx.

Sure that's fine. My point was that the VIX quoting convention is on a calendar year. To rescale to 252, you have to first "unwrap" the implied variance ( VIX^2 / 365 * 30).