Volatility

There are two kinds of volatility. Historical volatility is a statistical measurement of past price movements. Implied volatility measures whether option premiums are relatively expensive or inexpensive. Implied volatility is calculated based on the currently traded option premiums.

Ideally, what traders would like to know is what the future volatility is going to be. If we knew what the future volatility would be, we could make a fortune quite easily. Because we don't know what future volatility is going to be, we try to guess what it will be.

The beginning point for this guess is historical volatility. What has the volatility been for this stock or other security, over a given period of time. Generally, when evaluating volatility, we look at several different periods. We may look at what the volatility has been for the past week, for the past month, for the past three months, for the past six months, and so forth. The longer time period will yield more of an average volatility. However, don't expect large changes in volatility over time. Stocks or other securities which are volatile on a daily or weekly basis usually remain that way over time. When evaluating the purchase of an option, it is the historical volatility of the underlying security we are looking at. For instance, when deciding whether or not to purchase an option on XYZ, we would look at the historical volatility of XYZ. Basically, what historical volatility boils down to is, what is the probability that this particular underlying security will move a particular distance measured in price on a given day, week, month, etc.

However, there is a different interpretation of volatility not associated with the underlying security. This is implied volatility. There are many different models for pricing options. But most will yield a price relatively close to each other.

However, what if we use our historical volatility in the formula and come up with a price far away from where the option is trading? What if we do this using several different option pricing formulas and we still come up with a price which is not close to where the option is trading? Why would we come up with a price which can't be accounted for? We're all using the same inputs. We all use the price of the underlying security, the time until expiration, the strike price, dividends to be paid by the stock, the current risk free interest rate, and volatility. All of these inputs are known. Or are they?

The one input which is not known and for which we have to take a guess at is volatility. What has happened is that the marketplace is assuming a different volatility other than historical volatility. The way to solve for this implied volatility is to use our option pricing model in reverse. We know the price of the option and all the other variables except the volatility the marketplace is using. Therefore, instead of using the equation to solve for the option's price, we use the model to solve for the option's volatility. We insert the price into the model, leave out the volatility (which we are looking for), and keep the other variables the same. It is then that we will find out what volatility will yield the current market price.

Most traders refer to implied volatility as premium. To be precise, the word premium refers to the option price relative to the underlying security. Nevertheless, traders will say things like, "Premium levels are high." or "Premium levels are low." What the trader is really referring to is the implied volatility. The implied volatility is high or the implied volatility is low.

The first thing that one thinks about when trying to evaluate historical volatility is that the standard deviation should be used. And if a person is looking for a simple way to measure volatility, the simple standard deviation will work well enough. However, use of the standard deviation assumes that there is a normal distribution. If stock prices were normally distributed, the implication would be that there could be negative prices. This we know is impossible. The furthest a stock's price can fall is to zero. However, the stock price can rise infinitely. Therefore, we take the standard deviation of the logarithmic price changes measured at regular intervals of time.

Formula For Calculating A Stock's Volatility Using a Lognormal Distribution

Xi = ln[Pi divided by ((1+r)/52))Pi-1]

Where:

Xi = each price change

Pi = price of the underlying security at the end of the i-th period

r = risk free interest rate (We are using weekly data in this formula. For daily data use 253)

Step 2 is to calculate the standard deviations lognormal for the data series. This would be the price changes for the period under consideration. For example, using weekly data, we would calculate the above calculation for each week for at least 14 weeks.

Step 3 is to sum the answer for each calculation in step 2 and divide by 14. This gives us the mean.

In Step 4 we subtract each calculation from the mean.

In Step 5 we square each number from step 4 and add them all together.

Step 6 The annualized historic volatility is the answer from step 4 X the square root of (365/7)

This entire calculation is easily set up in a spreadsheet like Excel. Additionally, good technical analysis software will usually perform this calculation. In OptionVue 4, which is a software product dealing specifically with options and futures, the historical and implied volatility are both displayed for each strike.

An easier way to calculate volatility is simply to take the standard deviation of the lognormal return. In Excel the formula is easily calculated.

If column A has the closing prices, column B should contain the equation, =ln(B3/B2)

Column C should then use at least 10 periods and it would look like this: =stdev(B3:B13)

The resulting answer will be the standard deviation of the lognormal return.

In both columns B and C you can drag the formula all the way down the column

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