The previous lessons have introduced the basics of historical and implied volatility. We will now build upon this knowledge and apply volatility to the to the normal distribution theory so that volatility can be further analyzed. Specifically, standard deviations (σ) will be introduced as a way to quantify volatility in terms of expected fluctuations of the stock price.
Standard Deviation (σ)
The statistical measurement of the dispersion of data from the mean.
The random-walk theory, upon which the normal distribution theory is based, suggests that stock prices are random. The theory further proposes that future prices typically adhere to what is known as lognormal distribution- that prices are likely to occur around the mean (average) and less likely to occur further away from the mean.
In the previous lesson, we noted that Wal-Mart had a 13% historical volatility. According to this, at $50 per share, a 13% volatility level meant the price was expected to fluctuate between $44.50 and $56.50 per share over the next year based on the past volatility of the stock.
Using this information we can
apply it to the normal distribution
theory. Using $50 as the mean, the distribution theory states that at
13% volatility there is a 68% chance that Wal-Mart will be trading
between $44.50 and $56.50 per share in one year.
This price swing of $6.50 above and below the current stock price, which represents the 13% volatility, is called one standard deviation from the current mean.
Also according to the theory, two standard
deviations from the mean would represent a 95% chance that the
stock
price in one year will be between $37 and $63 per share in one year.
EXAMPLE: The current stock price (mean) is $50 per share. 30-day
historical volatility is annualized at 13%.
Based on the data in the previous example, we can draw the
conclusion that:
- There is a 68% chance that in one year price will fall within 1
standard deviation of either side of the mean ($44.50 to $56.50 per
share).
- Also, there is a 95% chance that in one year price will fall within 2
standard deviations of either side of the mean ($37 to $63 per share).
Investors can graph the expected distribution of a stock using historical volatility to analyze past price expectations. This data can then be compared to the expected future volatility to assess the investment risk.
Understanding normal distribution and standard deviation is the first step to properly understanding risk and probability.
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