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:
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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.
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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.
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Impressively close.
Since the simulated boundaries vary some from run to run I collected 32 runs and determined the mean
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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.
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Notes:
- 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.