In options trading, arithmetic mean and standard deviation are used to estimate the future price of an underlying asset, such as a stock or an index. The arithmetic mean is based on the historical or implied volatility of the asset, the current price, and the time to expiration. The standard deviation is used to calculate the probability of the asset price being within a certain range by a certain date. For example, if the arithmetic mean of an asset is $100 and the standard deviation is $10, then there is a 68% chance that the asset price will be between $90 and $110 by the expiration date1. This information can help options traders to choose the right strike price and expiration date for their options contracts.
Basic Theory
Arithmetic Mean
The arithmetic mean is a measure of central tendency that represents the average value of a set of numbers. In the context of options, it can be used to calculate the average return or price over a specific period.
Formula: Arithmetic Mean = (Sum of values) / (Number of values)
Standard Deviation
Standard deviation measures the dispersion of values from their average. It is crucial in options trading to assess the volatility of returns.
Formula: Standard Deviation = sqrt((Σ(Xi - X̄)2) / N)
Where 
is each value, 
is the mean, and 
is the number of values.
Procedures
- Gather Data: Collect data on option prices or returns for a specific period.
- Calculate Arithmetic Mean: Use Excel’s AVERAGE function to find the mean of the data set.
- Calculate Standard Deviation: Utilize Excel’s STDEV.P function to calculate the standard deviation.
Scenario: Analyzing Monthly Returns of an Option
Suppose you have collected monthly returns of a call option over a year:
| Month | Returns (%) |
|---|---|
| Jan | 2.5 |
| Feb | 1.8 |
| Mar | -0.5 |
| Apr | 3.2 |
| May | 0.7 |
| Jun | -1.4 |
| Jul | 2.9 |
| Aug | 1.1 |
| Sep | -0.8 |
| Oct | 2.3 |
| Nov | -1.9 |
| Dec | 1.5 |
Excel Formulas
- Arithmetic Mean:
=AVERAGE(B2:B13) - Standard Deviation:
=STDEV.P(B2:B13)
Result:
Arithmetic Mean: 1.025%
Standard Deviation: 1.672%
Interpretation
The arithmetic mean provides the average monthly return of 1.025%, indicating the general direction of the option’s performance. The standard deviation of 1.672% gives insight into the volatility, helping assess the risk associated with the returns.
Other Approaches
- Excel Data Analysis Tool: Utilize the Data Analysis Tool in Excel to generate descriptive statistics, including mean and standard deviation.
- Historical Volatility: Instead of monthly returns, calculate historical volatility based on daily price changes for a more granular analysis.
