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Calculating the VIX—The Easy Part

 
Sunday, November 8th, 2015 | Vance Harwood
 

The movements of the CBOE’s VIX® are often confusing.  It usually moves the opposite direction of the S&P 500 but not always.  On Fridays the VIX tends to sag and on Mondays it often climbs because S&P 500 (SPX) option traders are adjusting prices to mitigate value distortions caused by the weekend.

In addition to these market driven eccentricities the actual calculation of the VIX has some quirks too.  The VIX is calculated using SPX options that have a “use by” date.   Every week a series of SPX options expire.  This schedule of expirations forces a weekly shift in the VIX calculation to longer dated options.  For many years the CBOE’s VIX calculations only used monthly SPX options, but starting October 6th, 2014 it switched to using SPX weekly options when appropriate.  See “Why the Switch” section towards the bottom of this post for more information.

The VIX provides a 30 day expectation of volatility, but the volatility estimate from SPX options changes in duration every day.  For example, on October 13, 2014 the SPX options expiring on the 7th of November provide a 25 day estimate of volatility, while the November 14th options provide a 32 day estimate.  In this case to get a 30 day expectation the VIX calculation uses a weighted average of the volatility estimates from these two sets of November options.

The newly updated S&P 500 VIX calculation is documented in this white paper.  It computes a composite volatility of each series of SPX options by combining the prices of a large number of puts and calls.  The CBOE updates these intermediate calculations using the ticker VIN for the nearer month of SPX options and VIF for the further away options.  The “N” in VIN stands for “Near” and the “F” in VIF stands for “Far”.  These indexes are available online under the following tickers:

  • Yahoo Finance as ^VIN, ^VIF
  • Schwab $VIN, $VIF; historical data available
  • Google Finance INDEXCBOE:VIN, INDEXCBOE:VIN; historical data available
  • Fidelity:  .VIN, .VIF;  limited historical data

The final VIX value is determined using the VIN and VIF values in a 30 day weighted average calculation.  Graphically this calculation looks like the chart below most of the time:

VIX-calc-typ
As shown above the VIX value for October 13th is determined by averaging between the November 7th SPX options (VIN) and the November 14th SPX options (VIF) to give the projected 30 day value.  If you look closely you can see that the interpolation algorithm used between VIN and VIF does not give a straight line result; I provide calculation details later in “The Weighted Average Calculation” section

The chart below shows the special case when the VIX is very close, or identical to the VIF value.

VIX-calc-Wed
Wednesdays are important days for the VIX calculation:

  • The VIX calculation is dominated by the VIF values.
  • The SPX options used switch such that the old VIF becomes VIN and the options with 36 days to expiration become VIF.
  • Once a month on a Wednesday VIX futures and options expire (expiration calendar).  Soon after market open a special opening quotation of VIX called SOQ is generated.  Its ticker is VRO and it’s used as the settlement value for the futures and options.  Unlike the VIX’s normal calculation, the SOQ uses actual trade values of the underlying SPX options not the mid-price between the bid and ask.  Only one series of options, the ones with exactly 30 days to expiration are used.

Although SPX weekly options are available for 5 weeks in the future, the VIX calculation uses the SPX monthly options (expiring the 3rd Friday of the month) instead of the weeklies when they fit into the 24 to 36 day window used by the calculation.   The SPX weeklies expire at market close on Friday but the monthly options expire at market open on Friday.  By using these monthly options the CBOE keeps the VIX futures / options settlement process identical with the previous month based VIX calculation.

Why the Switch?

The chart below illustrates how the CBOE changed the VIX calculation methodology.

VIX-new-VS-old


This particular snapshot  shows the old VIX calculation (ticker: VIXMO) doing an extrapolation using SPX monthly options expiring November 22nd and December 20th (11 and 39 days away from the 30 day target)—a hefty distance.  If you would like more details about the old VIX calculation see “Computing the VIXMO—the easy part“.  The new VIX calculation on the other hand always does an interpolation over a much shorter period of timenever using options with expirations more than +-7 days from the 30 day target.  This CBOE article gives a good overview of the advantages of the new approach.

If you look closely at the chart, you can see that in this case the VIX calculation using the two methods arrives at slightly different answers (black line).  The new method gives a result of 21.16, 1.5% higher than the old method’s 20.85.  While I’m confident that the new calculation will be better in the long run because of the tighter VIN / VIF brackets I do have some concerns about the current volumes and low open interest in the SPX weekly options that are 4 to 5 weeks out.   I have seen the VIX / VIXMO differ by up to 5%—so for the time being I’m keeping both indexes on my watch lists. 

The Weighted Average Calculation

If you want to compute the VIX yourself using the VIN and VIF values you can’t just do a linear interpolation / extrapolation because volatility does not vary linearly with time.  Instead you have to convert the volatility into variance, which does scale linearly with time, do the linear estimation, and then convert back to volatility.  The equation below accomplishes this process.

VIX-VIN-VIF-eq

The Myth of Option Weekend Decay

 
Friday, January 13th, 2017 | Vance Harwood
 

While doing simulations on volatility and the square root of time I started thinking about how options experience time—is it calendar time, market time, or something in-between?  The CBOE’s VIX® calculations use calendar time, a 365 day year, but most option gurus recommend using a 252 day year for volatility calculations—the typical number of trading days per year in the USA markets.

When it comes to option decay most people, including the gurus, believe that option values decay when the markets are closed—a position I believe conflicts with the 252 day approach to annualizing volatility.

The experimental discovery that led to the current theory of option decay occurred in 1825 when the botanist Robert Brown looked through his microscope at pollen grains suspended in water and noticed they were moving in an irregular pattern.  He couldn’t explain the motion but later physicists including Albert Einstein showed it was the result of water molecules randomly colliding with the pollen. This effect was named “Brownian Motion” in honor of Mr. Brown.

If you effectively stop time in Mr. Brown’s experiment (e.g., freeze the sample), the pollen will stop moving.  Or if you close a casino for a day (probably a better model for the market) the net worth of the associated gamblers stops dropping.

Defenders of the calendar time approach point out there are many activities / events with broadband impact that can move the value of the underliers while the market is closed.  Things like extended trading hours, activity in foreign markets, corporate announcements, geopolitical events, and natural disasters.

However it occurs to me that most noteworthy events that happen outside of market hours tend to be bad news.  For example, I’m not expecting to see headlines any time soon stating, “ISIS disbands, ‘We realized it was all a terrible misunderstanding’”, or “Harmless landslide reveals huge cache of gold”.  This tendency towards negative moves is reflected in the average annual growth rate of off market hours for the last 20 years, -0.37% vs +9.59% for market hours.   And bad news tends to make option prices go up…

If option time is still running when the markets are closed I would expect the market’s opening value to be different from the closing value.  Below is a quick look at the last 20 years of data:

S&P 500 Returns 1-Jan-1994 through 22-Aug-2014 (5197 market days)

Market Time: Open to Close (occurrences) Market Time: Close to Open (occurrences)
No change 0.1%  (5) 58% (3046)
Change less than 0.05% 5.2%  (270) 81% (4249)
Changes >= 1% 27% (1396) 0.04%  (3)


I was surprised how often the market opened at no-change from the previous close (3046 times) and how seldom it has gapped overnight more than +-1% (3 times).

So what?

So far my arm-waving arguments give the edge to market time over calendar time, but really, so what?

Practically there are two things where this makes a difference: the dynamics of option decay and the accuracy of implied volatility calculations on soon to expire options.

Option Decay

Novice options traders are usually disappointed if they try to profit from Theta decay over the weekend.  If the underlying doesn’t move, options prices typically open on Monday unchanged from the Friday close.  Commentators explain this phenomena noting that market makers, not wanting to be stuck with Theta losses over the weekend, discount prices, overriding their models before the weekend to move their inventory—just like a fruit vendor would.

I think the market makers are right for the wrong reason.  Their computer models are (or at least were) based on calendar day assumptions—which assume option decay during the weekend.   By overriding their models they are pricing according to what really happens—no decay when the market is closed.

Annualizing factors  

For longer term expectations of volatility it doesn’t matter much which approach you use.  For options expiring a month from now the differences in implied volatility are only a few percent between the 365 vs 252-day models.  However, for shorter expirations the differences can be dramatic.

The chart below compares per minute values between the two annualizing approaches and shows the percentage difference.  The calendar based approach is the black line and the green line is the market time.  Notice how the difference peaks at Monday open and drops to near agreement at Friday close.

CalvsMrkt-ann

This “weekend” effect is sometimes visible in the CBOE’s VIX index and is pretty dramatic with their shorter term VXSTSM index—not surprising since this index is based on S&P 500 (SPX) option prices with at most 9 days until expiration.

There are good reasons to use a calendar day approach to annualization.  It isn’t sensitive to holidays, unexpected market stoppages, or differences in trading calendars between countries.  I expect that’s why it became a de facto standard in the implied volatility world.  But the rise of shorter term volatility products like weekly options has shifted the volatility landscape enough that I think we need to at least know what is technically correct.

 An analytic approach to a solution

Normally we take a shorter term (e.g., daily) volatility and multiply it by the appropriate annualizing factor to get the annualized volatility.  Since the annualizing factor is the thing in question I decided to take the historical annual volatility for the last 64 years of the S&P 500 and divide it by the daily volatility to solve for the actual historical annualizing factor.

First I validated this approach with a Monte Carlo simulation1 that computed the theoretical annualizing factor for a simulated 64 year market period—and then repeated that exercise 10000 times to get the statistics of the calculation.  I then applied the same calculation to the S&P 500’s returns2 over the last 64 years. The result:

Sim-Ann-Factors

The square of the annualizing factor comes is only 0.87% from the theoretical median value3 of 252 and the actual S&P 500 result of 243.5 is only 2.5% from the median value.  The S&P result of 243.5  is almost 3 sigma away from the competing answer of 365.

The S&P 500 data is consistent with a 252 day based annualizing model—which doesn’t support option decay while the market is closed.  The data also indicates that when you see suspiciously high short term volatility numbers at the beginning of the week you should chalk it up to flawed algorithms, not anything real in the market.

 

Notes:

  1. For each day of the simulation, I used the standard deviation of the previous 252 days natural log of daily returns for the short term volatility number.  For the yearly return, I used the simulated market value one year hence divided by the current day’s market value.  Volatility drag is an important second order effect that needs to be included in the calculations.
  2. I offset the actual results by the average annualized growth rate to compensate for the non-zero mean of actual returns over the last 64 years
  3. My simulation results have a median value of 252.2 (0.08% error) if I use a volatility drag coefficient of 0.6 instead of the standard 0.5.  I believe my model slightly under corrects for volatility drag.

Hedging the S&P 500 with Volatility

 
Tuesday, August 9th, 2016 | Vance Harwood
 

It’s expensive to buy securities that track volatility.  Their holding costs are so high that your timing has to be exquisite in order to end up with a profit.  However, if you’re hedging a short volatility position, or poised to jump into the general market at a possible transition point a long volatility position might make sense.

Consider this chart:
spytrend2
Will the S&P 500 bounce off this trend line for the fifth time, or will it go into a correction?

If SPX breaks through the trendline it’s likely volatility will really spike.   Alternately if  the market rallies then volatility will quickly fade, so an asymmetric bet (e.g., call options) is attractive.   If volatility spikes you benefit from the rapid run-up, but if it’s a false alarm your losses are limited.

The next question is to determine what underlying volatility product is best for this hedge and how large a position is needed to balance the risk in your general market position.  Investing in the CBOE’s VIX® would be ideal, but unfortunately there’s no way to directly invest in the VIX, so we’re left with a set of compromised choices—volatility Exchange Traded Products (ETPs) like TVIX, VXX, or VIXM  (see volatility tickers for the complete list), or VIX futures.  Later in this post I’ll analyze how three specific investments would have performed during an actual correction, but first I’ll examine a key issue—how much will the volatility products move up if the market drops.

 

The Choices

The chart below shows how the volatility ETPs have historically reacted during negative S&P 500 (e.g., SPY) market moves.  The data uses simulations of ETP prices from 2004 until their inceptions and actual data after that.

 

The median value of these ratios stays fairly stable over a wide range of percentage moves.  For example the median percentage moves of 1X short term ETPs like VXX will consistently cluster around negative 2.25 times the percentage moves in the S&P.  A daily -1% move in SPY typically results in a VXX positive move of around 2.25%.

These ratios aren’t guaranteed—they’re statistics.  In fact 20% of the time the volatility products move in the same direction as the S&P 500.  Fortunately, when the market is dropping the distribution of ratios tightens up

The chart below shows the historical distribution of VXX percentage moves compared to SPY moves of  > -0.1% and  > -1%.  SPY moves of less than +-0.1% are excluded because they can generate high ratios that aren’t meaningful.

 

When the S&P makes a 1% or larger negative move the median doesn’t shift much, but the number of results on the positive side drops from 21% of the total down to under 5%.

Since these ratios are relatively stable regardless of the size of the market moves we can view these ratios vs. the various ETPs / indexes.

 

Remember these are one day relative % ratio numbers.  While TVIX & UVXY ratios are close to the VIX’s on this metric, the contango losses in holding these ETPs other than during a market downswing are ruinous.   The 1X short term ETPs (e.g., VXX) aren’t much better.

So far I’ve only discussed the CBOE’s indexes and some of the volatility ETPs.  There are also VIX futures that have various sensitivities to the moves of the S&P 500.  These products differ from the indexes and ETPs in that they have expiration dates like options.


As these futures get closer to expiration their sensitivity increases.  Interestingly, a simple natural log relationship (shown on the chart)  gives a good match to the data.

There are also VIX weekly futures based on the CBOE’s 9 day VXST index, but I’ll discuss those in a different post.

 

The Hedge

Circling back to the trend chart at the beginning of this post—what would be a volatility hedge that would protect you if you bet on a 5th upward bounce?

There’re a lot of moving parts here (e.g., security, strike price, expiration date) and a lot of different strategies.  I’ll pick one general approach, and work through the details if the hedge had been applied during the 30-July-2014 through 8-Aug-2014 period.

My assumptions:

  • $100K invested in the SPY (betting that the market will start climbing again)
  • One percent of the market investment ($1K) invested in a volatility hedge—call options expiring around 16-Aug-2014.  It’s very likely the market will have gone one way or the other by then.
  • Goal of breaking even (losses in SPY & cost of the options offset by profits) if the market drops 3% or more.

I’ll review the results from three different trades—buying calls on UVXY (2x Short term), August VIX calls (based on next to expire VIX futures or M1 futures), and VXX (1X Short term).

The Setup  (30-July-2014) UVXY M1 VIX Futures VXX SPX
The median expected multiplier vs downward SPX % moves 5X (Stdev 10.8) 3X (Stdev 10.5) 2.55X (Stdev 5.4) -1X
For a -3% move in the SPX, the expect move from the earlier analysis 15X 9X 7.65 -3%
Closing value of underlying securities on 30-July-2014 27.16 13.55 29.08 197
Target value of underlying with -3% SPX move 31.23 14.75 31.30 191
Selected option strike prices 31 15 31
Expiration dates for selected options 16-Aug 20-Aug 16-Aug
Closing value of options on 30-July-2014 1.43 0.75 0.83
Number of option purchased for $1K 7 13 16
Approximate value of positions $1000 $1000 $1000 $100K

 

The Results  (8-Aug-2014) UVXY M1 VIX Futures VXX SPX
Actual value of underlying with -3% SPX move 34.74 16 33.21 191
Actual percentage move 28% 18% 14.2% -3%
Actual percentage multiplier 9.33X 6X 4.73X 1X
Difference from predicted multiplier 1.86X 2X 1.85X
Closing value of options on 8-Aug-2014 5.05 1.4 2.57
Intraday highs of options on 8-Aug-2014 7.6 (+50%) 2.4 (+71%) 4.15 (+61%)
Value of positions at close $3535 $1820 $3084 $97K
Initial investment required for break even at close 8-Aug-2014 $1144 $2175 $1328

 

So, in spite of the underlying volatility instruments moving around 2X more than expected, the $1K spent on hedges did not achieve the goal of break even with a 3% decline in the S&P 500—although UVXY was pretty close.  During this period the VIX ramped from 13.33 to 15.77—an increase of 18.3% (the expected move was 15%).  If the correction had continued volatility would have probably increased rapidly (the intraday option prices spiked > 50% on the 8th –when the VIX climbed to 17.09), so the hedges probably would have worked well protecting the S&P 500 position against further declines.

 

One of the challenges of trading is wrestling with strategies that work until they don’t.  With short term volatility hedges you can bet on the market going up—without paying too much for insurance in case you’re wrong.

SPY-trend-break

How Does the CBOE’s VIX® Index Work?

 
Thursday, October 13th, 2016 | Vance Harwood
 

The CBOE did not create the VIX as an academic exercise, or as a service to stock market prognosticators everywhere.  They created it because they wanted to make money on volatility.  It took them two tries, but the CBOE succeeded in developing a volatility index that forms the backbone of a host of volatility products.  The CBOE offers some of these products, but other companies have built on the success of VIX to offer their own volatility based products.

To have a good understanding of how the VIX works you need to know how its value is established, what it tracks, what it predicts, and how the CBOE makes money with it.

How is VIX’s value established?

  • The VIX is a computed index, but unlike indexes such as the Dow Jones Industrial Average or the S&P 500 it’s not computed based on stock prices.  Instead it’s based on option prices.  Specifically the prices of options on the S&P 500 index (ticker SPX).
  • One component in the price of SPX options is an estimate of how volatile the S&P 500 will be between now and the option’s expiration date.  This estimate is not directly stated, but is implied in how much buyers are willing to pay.  If the market has been gyrating like mad option premiums will be high whereas in a quiet market they will be much cheaper.
  • There are various ways of extracting the volatility information from option prices.  The standard way is via the Black & Scholes model, but those equations assume that volatility will be the same for all available options—something that is definitely not the case and they also underestimate the risk of a market crash.
  • The CBOE’s approach combines the prices of many different SPX options (hundreds) to come up with an aggregate value of volatility. Their approach has some particular advantages—more on this later.
  • There are many good posts here,  here, and here on the details of the actual VIX calculation, so I won’t reinvent the wheel.
  • The VIX is an estimate of volatility for the next 30 days, but by convention volatility measures in the stock market are reported in terms of annualized volatility.  Volatility doesn’t increase linearly with time, so the annualized number is not 12 times the 30 day estimate but rather ~3.5 times the monthly number. For example, if the intermediate VIX calculation computes the expected 30 day volatility to be +-4.3%, the reported VIX will be 15%.  For more on this see Volatility and the Square Root of Time
  • There’s nothing magical about the 30 day estimate.  The CBOE uses the same methodology to compute 9 day (VXST), 93 day (VXV), and 180 day (VXMT) volatility indexes.

 

What does VIX track?

  • The moves of the VIX track prices on the SPX options market, not the general stock market—this is a key point.  The SPX options market is big, with a notional value greater than $100 billion, and is dominated by institutional investors. A single SPX put or call option has the leverage of around $200K in stock value—too big for most retail investors.
  • In general option premiums move inversely to the market.  In a rising market, stock prices tend to be less volatile and option premiums low—hence a lower VIX.  Declining markets are volatile (the old saying is that the market takes the stairs up and the elevator down) and option premiums increase.  Much of this increase occurs when worried investors pay a large premium on puts to protect their positions.
  • While S&P 500 option premiums generally move opposite to the S&P 500 itself they sometimes go their own way.  For example, if the market has been on a long bull run without a pullback institutional investors will become increasingly concerned that a correction is overdue and start bidding up the price of puts—leading to a rising VIX in spite of a rising S&P.   Historically 20% of the time the VIX moves in the same direction as the S&P 500—so please don’t claim the VIX is “broken” when you see the two markets move in tandem.
  • The daily percentage moves of the VIX tend to be around 4 times the percentage moves of the S&P 500, but unlike the stock market, the VIX stays within a fairly limited range.  The all-time intraday high is 89.53 (24-Oct-2008) and the all-time intraday low is 9.39 (15-Dec-2006) with the current methodology.  Within this 10 to 1 range option premiums run from incredibly expensive to dirt cheap.  It’s unlikely that the VIX will go much below 9 because option market makers won’t receive enough premium to make it worth their while.  At the high-end things go could go higher (if the VIX had been available in the October 1987 crash it would have peaked around 120), but at some point investors refuse to pay the premium and switch to alternatives (e.g., just selling their positions if they can).  The chart below shows the historical distribution of VIX values since 2002.VIX-histo Data Source: CBOE

 

  • Another way to look at the moves of the VIX is to recognize that it’s almost always a few percentage points higher than the recent historical volatility of the S&P 500.  It’s a good assumption that the future volatility of the market will be the same as recent volatility—but obviously, this relationship doesn’t always hold.   Option market makers demand a premium to justify the risk they assume in buying / selling options in the face of this uncertainty—and this premium shows up as a VIX value greater than historical volatility.

How does VIX trade?

  • So far, no one has figured out a way to directly buy or sell the VIX index.  The CBOE offers VIX options, but they follow the CBOE’s VIX Futures of the same expiration date, not the VIX index itself.  VIX futures usually trade at a significant premium to the VIX.  The only time they reliably come close to the VIX is at expiration, but even then they can settle up to +-5% different from the VIX level at the time (see this post for upcoming expirations)
  • There are around 25 volatility Exchange Traded Products (ETPs) that allow you to go long, short, or shades in-between on volatility (see here for the complete list), but none of them do a good job of matching the VIX over any span of time.   For more on ways to trade volatility see  How to Go Long on the VIX, and How to Go Short on the VIX.

What does the VIX predict?

  • In my opinion, nothing.  I think it does a good job of reflecting the current emotional state of the overall market (e.g., fearful, optimistic), but I don’t think the SPX options market is any better at forecasting the future than any other market or index.  We don’t take the value of the Dow Jones Industrial Average as a predictor of the future, so why should the value of the VIX be any different?

How does the CBOE make money on the VIX?

  • The “O” in CBOE stands for options.  In the early 90’s the CBOE wanted to sell options on volatility, but there was a problem—options need to be based on an underlying tradable security to function, and there wasn’t one.   To address that gap the CBOE created an index that could form the basis of a volatility futures market.  Once that market was functioning then options could be introduced.
  • Version one  of the VIX index (now named VXO) was introduced in 1992, but futures based on it were never available.   For a futures market to function, the market makers need to be able to cost effectively hedge their positions.  Hedging the 1992 version of the VIX required frequent rebalancing of SPX options that was too expensive to implement.
  • Undeterred the CBOE introduced version 2 in 2003.  The new methodology allowed market makers to hedge their positions with a static portfolio of SPX options that could be held until the VIX futures expired.  VIX Futures started trading in 2004 and in 2006 options on VIX futures were rolled out.
  • VIX futures and options have been very successful with recent daily volumes in the hundreds of thousands. The CBOE is generating hundreds of millions of dollars in annual revenues from these products—primarily from highly profitable transaction fees.

The VIX frustrates a lot of investors.  It’s complicated, you can’t directly trade it, and it’s not useful for predicting future moves of the market.  In spite of that, the investment community has adopted it, both as a useful second opinion on the markets, and as the backbone  for a growing suite of volatility based products.

But what impresses me is the vision and persistence of the people at the CBOE in advancing the highly theoretical concept of stock market volatility from an academic exercise to an effective commercial product.  It was a multi-decade project and they were successful.

For more information:

Volatility and the Square Root of Time

 
Friday, April 17th, 2015 | Vance Harwood
 

It’s not obvious (at least to me) that volatility theoretically scales with the square root of time (sqrt[t]).  For example if the market’s daily volatility is 0.5%, then the volatility for two days should be the square root of 2 times the daily volatility (0.5% * 1.414 = 0.707%), or for a 5 day stretch 0.5% * sqrt(5) = 1.118%.

This relationship holds for ATM option prices too.  With the Black and Scholes model if an option due to expire in 30 days has a price of $1, then the 60 day option with the same strike price and implied volatility should be priced at sqrt (60/30) = $1 * 1.4142 = $1.4142  (assuming zero interest rates and no dividends).

Underlying the sqrt[t] relationship of time and volatility is the assumption that stock market returns follow a Gaussian distribution (lognormal to be precise).  This assumption is flawed (Taleb, Derman, and Mandelbrot lecture us on this), but general practice is to assume that the sqrt[t] relationship is close enough.

I decided to test this relationship using actual S&P 500 data.  Using an Excel based Monte-Carlo simulation1 I modeled 700 independent stock markets, each starting with their index at 100 and trading continuously for 252 days (the typical number of USA trading days in a year).  For each day and for each market I randomly picked an S&P 500 return for a day somewhere between Jan 2, 1950 and May 30, 2014 and multiplied that return plus one times the previous day’s market result.  I then made a small correction by subtracting the average daily return for the entire 1950 to 2014 period (0.0286%) to compensate for the upward climb of the market over that time span.  Plotting 100 of those markets on a chart looks like this:

100Mrkt


Notice the outliers above 160 and below 60.

Volatility is usually defined as being one standard deviation of the data set, which translates into a plus/minus percentage range that includes 68% of the cases.  I used two handy Excel functions: large(array,count) and small(array,count) to return the boundary result between the upper 16% and the rest of the results and the lowest 16% for the full 700 markets being simulated. The 16% comes from splitting the remaining 32% outside the boundaries into a symmetrical upper and lower half. Those results are plotted as the black lines below.

100Mrkt+1sig


The next chart compares those two lines to the theoretical result which takes the annualized standard deviation of the S&P 500 daily returns from 1950 to 2014 and divides it by the square root of time.

Standard Deviation (N) = Annualized Standard Deviation/ sqrt (252/N)

Where N is the Nth day of the simulation.

100Mrkt-vstheory-labels


Impressively close.

Since the simulated boundaries vary some from run to run I collected 32 runs and determined the mean

100Mrkt-vstheory-mean


Very, very close.

So, in spite of the S&P 500’s distribution of results not being particularly normally distributed (see chart below), the general assumption that volatility scales with the square root of time is very appropriate.

SP-Dist


Notes:

  1. Returns are expressed as the natural log of the current day’s close divided by the previous day’s close.  The specific daily return used is selected by randomly choosing a number between 0 and 16204 (Trunc(Rand()*16205)) and then using that number to index into the table of SPX returns.  The 16205 constant is the number of trading days from 3-Jan-1950 to 30-May-2014.  As mentioned in the post, the overall daily mean for that period (0.0286%) is subtracted from the result to compensate for the general upward bias of the market over that period.