Estimating Historical Volatility from Log Returns for Black-Scholes

Summary

The document gives a basic historical method for estimating volatility from a time series of observed prices. Calculate consecutive log returns as the natural logarithm of each price divided by the previous price, then take the standard deviation of those returns. A short price sequence illustrates the calculation and reports a sample standard deviation.

The question concerns valuing warrants in a private company, where public comparable firms may be scarce or poorly matched. The response provides a way to calculate volatility when relevant historical price data exist, but it does not explain how to choose a suitable proxy when such data are unavailable. It also cautions that this simple estimate does not model skewness or kurtosis, while noting that the Black-Scholes framework itself does not capture those features. The example does not discuss annualizing volatility or selecting an observation interval, both of which matter when applying the estimate in practice.

Key ideas

• Compute each log return as the natural logarithm of the current price divided by the preceding price.
• Estimate historical volatility by taking the standard deviation of the log-return series.
• The method requires a usable history of observed prices, which may be difficult to obtain for a private company.
• The simple estimate does not account for skewness or kurtosis in returns.
• The document does not specify an observation frequency or annualization procedure.

Tags

# How to determine volatility for private company for Black-Scholes


# How to determine volatility for private company for Black-Scholes












I am trying to determine the volatility to use Black-Scholes to value some warrants for a private company. Very few comps are public or they are large diversified businesses. Any thoughts on how to do this or ranges to start from? Thanks Connie Dubois

## Answer by andrew.paul.acosta (score -4)

https://quant.stackexchange.com/a/23192

There is an easy method to calculate volatility if you have a historic time series of price data. First, obtain the standard deviation of the log returns.

Imagine you have these observed prices,

> {30.00, 31.70, 27.38, 27.50, 23.96, 23.30, 30.63, 24.04}

Calculate the log return,

> ln(31.70/30.00), ln(27.38/31.70), . . . ln(24.04/30.63).

Calculate the standard deviation of the series,

> sd = 0.07325

Keep in mind that this is a simple method, and it does not allow for skewness or kurtosis in the frequency distribution of returns, but neither does the Black-Scholes option pricing model.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.