Investopedia has an article here but it doesn’t seem terribly helpful to me.

Here is a document I wrote when I was trying to come to grips with it and also what my software was showing me.

### Volatility

I have been on something of a crusade to understand volatility and how it is calculated in my charting packages Optuma and Beyond Charts Plus.

First **Historical Volatility**. This is the standard deviation of the % change in price for a specified period (sometimes called the Interday return). There are a few different ways this can be calculated. In Microsoft Excel, it is calculated using either the STDEV.S or STDEV.P functions where the ‘S’ stands for Sample and the ‘P’ stands for Population. The formula for these are as follows:

STDEV.S:

where x is the sample mean AVERAGE(number1,number2,…) and n is the sample size. In statistical terms this is referred to as being non-biased.

STDEV.P:

where x is the population mean AVERAGE(number1,number2,…) and n is the sample size. In statistical terms this is referred to as biased.

So, should we use the population or the sample? It seems in my software, Optuma (Market Analyst) the Sample is used for the inbuilt Historical Volatility Indicator, HV. However, if I calculate a Standard Deviation in a script and plot it, it uses the Population Standard Deviation.

Typically, to calculate the historical volatility in something like Metastock, the following formula is used:

Historical Volatility 10 day

Std(Log(C/Ref(C,-1)),10)*Sqrt(365)*100

I don’t know if Metastock uses Population or Sample for its STD calculation.

This formula uses the natural log function and calculates the Standard Deviation of the Log of the Close Today/Close Yesterday and multiplies it by the Square Root of 365 and multiplies by 100 to get a percentage to produce an annualised percentage volatility. I think this is an error using 365 as there are not 365 trading days in a year so I prefer to use 252. It probably depends on what you want to use the calculated Historical Volatility for if this is relevant or not.

Beyond Charts Plus uses the Population Standard Deviation. You can use the following formula to plot this in BC+:

{ Historical Volatility}

Volatility:=Stdev(Log(C/Ref(C,-1)),10)*Sqrt(252)*100;

Volatility;

(Note: I have used 252 not 365. You will get a significantly larger number for HV if you use 365)

I think it is probably correct to use the Population n rather than the sample n-1 for 2 reasons. Firstly, we are using ALL the data for the period we are calculating the standard deviation. The second reason is:

The stock market does not follow a normal distribution. There are fatter tails meaning there are more stocks with large declines and large advances so the center of the curve is flatter than the normal distribution. This also means that the ‘rule’ that **about 95 percent **of observations** **are within two standard deviations (μ ± 2σ), and **about 99.7 percent** lie within three standard deviations (μ ± 3σ) might hold well if you are doing a normal distribution of, say, the height of people in a population but it doesn’t hold so well for the prices of a security.

The price distribution is not a normal distribution for security prices. The standard deviation calculation is based on a normal distribution. John Bollinger says the Standard Deviation used to calculate Bollinger bands should be based on ‘n’ not ‘n-1’ for the volatility. (That is to say it should be based on the population standard deviation, not the sample.)

So, the question then becomes (for me anyway) how is this information useful to me?

I have at times used a covered calls strategy for income on stock I own. For a couple of reasons, I sometimes don’t want to be exercised on a stock – either because I want to own the stock or because the stock is currently trading under my break-even point and I don’t want to risk being exercised. Writing a covered call is a good way to bring down your break-even price on a stock. Care needs to be taken though as if the stock keeps falling you end up in a worse and worse situation. Buying a ‘put’ option can be used as insurance to protect capital.

So, at which strike price should I write my call to minimise my chance of being exercised (or avoid needing to buy a call back before expiry)?

I have received a few suggestions.

One option is to consider the Delta of the option.

Chris Tate says:

*Delta is the most widely known option sensitivity it is defined as the amount an option price will move either up or down for a given change in the underlying share. In effect delta is measuring how sensitive to change a change in the underlying the options price is.*

Call option delta are positive and they are measured in a range from 0 to 1, the upper boundary of a call option will always be 1 since the option theoretically cannot increase at a greater rate than the underlying share. The lower boundary will also be 0 since the call cannot have a negative movement. A deep ITM call option with a delta of 1 will increase at the rate of one full tick for every full tick move in the underlying. For example, if we were trading a call on NCP that had a delta of 1 then when NCP increased by 1 cent our option would also increase by 1 cent. Likewise, should NCP fall by 1 cent then the option will fall by 1 cent.

Conversely a deep OTM call with a delta of for example 0.1 will only move 0.1 of a cent for every 1 cent move made by NCP. All call options will have a delta that falls somewhere between the range of 0 and 1. As a rule of thumb, very deep ITM calls will have a delta very close to 1, ATM calls will have a delta of approximately 0.5 and OTM calls will have delta’s that move toward 0 the further OTM they become.

As a rule-of-thumb, if you multiply the delta by 100 that will give you a percentage chance of the option reaching the strike price by expiry. On the basis of that, I look at a delta of 0.3 or less.

The Delta is usually found in an Options ticket or information screen like this:

Notice also the implied volatility, in this case 15.63%. The volatility in the discussion above was Historical Volatility based on the standard deviation but the Implied Volatility is not calculated. It is a made-up number representing market sentiment. Implied volatility has no direction. Generally, when prices are falling, volatility will be high and when prices are rising, volatility will be low but this is not a rule. You can also look at the VIX – a volatility index for a feel of the sentiment on S&P500 index options. Implied Volatility is a psychological quantity, not a physical quantity. The implied volatility is directly related to an option premium via the Black-Scholes formula.

So, if we have an implied volatility of 15.63%, how can I use that to determine the probable or expected price movement to help me select a strike price where I am unlikely to be exercised?

The date of writing is the 1^{st} of November. Options for November 2016 will expire on 18^{th} November – 14 days away. If I divide 15.63% by SQRT(252) and multiply by SQRT(14) this will give me the expected % change in the underlying price at option expiry. This evaluates to 3.69% So AAPL closed yesterday at $113.54 so at Option expiry on 18/11/2016 AAPL * could *be expected to close at $113.54 +/- 3.69% ($4.19) $109.35-$117.73. So, I

*write a call at $118 and be fairly sure I won’t be exercised.*

**could**Notice here that the implied volatility is different for the different strike price. (16.38% vs 15.63%)

The option premium for this is $0.32 and the delta is 0.15 so only a 15% chance of losing my shares. (UPDATE: Closed at $110.06 on expiry)

Other ways to set a strike price could be to use the ATR and write a strike above that. I am unsure how I would convert a daily ATR to work out an expected move in 14 days. Perhaps the same calculation could be applied to Historical Volatility as shown above for Implied Volatility. HV is currently 13.68% which is not dissimilar to the IV calculation shown. The other option would be to just write a call a couple of strikes above the current share price or simply stand aside. The option premium for 115 strike is currently $1.

Whichever option/strike is selected you would need to continue to monitor the underlying so you can close the position prior to expiry if necessary.

Regarding Bollinger Bands, I was wanting to plot Bollinger Envelopes which are reportedly more useful for instruments like the FX market which doesn’t really have any fixed session times (trades 24/7 5.5 days a week). In the process of doing this I found that the Middle Band of the Bollinger Band is a Simple Moving average of 20 periods (which of course I already knew) But I then came to grief calculating the upper and lower bands until I realised that the Standard Deviation I needed to use was the standard deviation of the high for the upper band and the standard deviation of the low for the lower band, – not the standard deviation I calculated when I was looking at the Volatility (which was calculated from the interday return based on the close.)

To cut a long story short, I discovered that Optuma and BC+ both use the sample standard deviation (n-1) for their inbuilt Bollinger Bands function, but if you use the Standard Deviation in a formula they both use the Population (n). Incidentally, it should be noted that BC+ and Optuma are wrong to use n-1 – John Bollinger uses n on his website www.bbands.com I suspect it doesn’t make any practical difference unless you use a short period Bollinger Band.

You may find it instructive sitting down with an excel spreadsheet and entering prices and calculating this by hand. I learnt a lot by doing this.

Originally posted 2017-05-04 13:09:42.