How to Determine the Right Volatility Inputs in Black-Scholes for Section 409A Appraisals

A small change in the volatility assumption can affect the outcome of a Black-Scholes-based 409A valuation. For private companies, determining the right volatility input can be challenging because their shares have no public trading history.

Instead, valuation professionals typically estimate volatility using the historical data from comparable public companies. The challenge is deciding which companies to use, how much historical data to consider, and which volatility statistic best represents the company being valued.

This article explains how to determine an appropriate Black-Scholes volatility input for a Section 409A valuation.

Key Takeaways

• In a 409A valuation, volatility is the key Black-Scholes input that relies on subjective choices and uses comparable public company data.
• The lookback period should match the expected time to liquidity. In parallel, the analyst should consider the trade-off between a stable volatility assumption and current market relevance.
• The peer set should reflect the company’s industry, sector, stage, geographical diversification, and business model.
• Volatility is calculated for each peer using the standard deviation of logarithmic returns over the selected period and is annualized by multiplying by the square root of the number of trading days in a year.
• Based on the volatilities of the comparable company set, the analyst should select a representative figure that limits the effect of outliers while capturing important trends across the peer group.

How Black-Scholes Used in a 409A Valuation?

Black-Scholes can be used as part of an option pricing model (OPM) to allocate a private company’s equity value among different classes of securities.

The model uses several assumptions to estimate how value may be distributed under different future outcomes. Depending on the valuation methodology and the company’s capital structure, relevant inputs can include equity value, exercise or threshold prices, expected time to liquidity, risk-free rate and volatility.

Volatility is particularly important because it represents the uncertainty associated with potential future outcomes. Since a private company does not have an observable market price for its shares, its volatility must generally be estimated using relevant market data.

Why does volatility matter in a 409A valuation?

Volatility is one of the most judgment-sensitive inputs in a Black -Scholes based 409A analysis. A higher volatility assumption can affect the value allocated to option-like interests under the model.

For a private company, the analyst looks to comparable public companies to develop a supportable volatility assumption. This requires judgement in areas such as:

• Selecting an appropriate lookback period
• Identifying relevant comparable companies
• Calculating historical volatility consistently
• Choosing a representative statistic from the comparable company data

Because these choices can influence the valuation outcome, the assumptions should be documented and supported by relevant data.

How to accurately calculate volatility input?

Typically, a valuation analyst will set the appropriate volatility in a B-S model-based 409A valuation by taking the following steps.

Step	What the analyst determines	Why it matters
1. Lookback period	The historical period used to measure volatility	Balances historical stability with current market relevance
2. Peer set	Comparable public companies	Provides a relevant market benchmark
3. Historical volatility	Volatility for each comparable company	Creates a consistent data set for analysis
4. Representative input	Mean, median, or another appropriate statistic	Produces the volatility assumption used in the model

1. Select the lookback period

The first step is to select a lookback period that matches the expected time to liquidity, represented by t in the B-S model.

A long lookback period captures more market cycles. This can result in a more stable volatility estimate. However, there is also a higher chance that the period will include market conditions that do not align with the company’s current situation.

A shorter lookback period is more responsive to current market conditions. At the same time, it can overfit recent market noise. This makes it less suitable during periods of extreme market disruption. Hence, the judgment of an experienced valuation analyst becomes necessary in 409A valuations.

2. Choose an appropriate peer set

Once the lookback period is established, the next step is to identify comparable public companies. The selected companies should match the company being valued as closely as possible in terms of industry, sector, stage, geographical diversification, and business model.

The peer companies should also have been public for the entire lookback period. This helps avoid unnecessary adjustments when calculating the historical volatility. A peer set that does not reflect the company being valued can produce a volatility estimate that does not properly represent its risk.

3. Calculate volatility for each company

After selecting the peer set, the analyst calculates the volatility for each comparable company over the selected lookback period.

The formula used is: Volatility for each company= σ(ln Price/Price on the previous day )×√252

Here, the volatility is calculated based on the change in the company’s price over the relevant trading periods. The standard deviation of these returns is then annualized by multiplying by the square root of 252. The root of 252 is used because there are 252 trading days in a year. Applying this factor allows the analyst to convert the observed volatility over the trading period into an annualized volatility figure that can be used as an input in the B-S model.

4. Select the representative volatility figure

The analyst may select the mean, the median, or another appropriate figure based on the characteristics of the data to represent the company being valued.

This decision is important because comparable companies can show different levels of volatility. A few companies with unusually high or low volatility can distort the average. The median can help minimize the impact of such outliers. At the same time, the selected representative figure should still capture the important trends present across the peer group.

The objective is therefore not simply to select the highest or lowest number. The analyst must choose a figure that reasonably reflects the volatility supported by the comparable company data.

Common Mistakes When Selecting Volatility

Several issues can weaken a volatility analysis.

Using an unrelated peer group

Selecting public companies simply because they operate in the same broad industry may not provide a sufficiently relevant benchmark. Business model, stage, geographic exposure, and other characteristics should also be considered.

Choosing a lookback period without considering the valuation horizon

A lookback period should not be selected solely because it produces a convenient volatility figure. The historical period should be considered in relation to the relevant expected term and current market conditions.

Allowing extreme observations to dominate the analysis

A single company with unusually high or low volatility can affect the mean. Analysts should examine outliers and determine whether the selected statistical measure appropriately represents the overall peer group.

Treating volatility as a standalone assumption

Volatility is only one component of a broader valuation analysis. The assumption should be considered alongside the other inputs and the company’s overall facts and circumstances.

FAQs

Selecting a volatility input for a Black-Scholes-based 409A valuation can raise several practical questions. The answers below explain how volatility is estimated, how analysts select comparable companies and lookback periods, and how they determine which statistical measure best represents the company being valued.

What volatility input is used in a Black-Scholes 409A valuation?

The volatility assumption is generally estimated using historical volatility data from an appropriately selected group of comparable public companies. The analyst considers the lookback period, peer group, calculation methodology, and representative statistic when determining the input.

Why is volatility important in a 409A valuation?

Volatility shows how uncertain future results might be and can change the value given to option-like interests using a Black-Scholes option pricing model.

How is historical volatility calculated?

Historical volatility is commonly calculated using the standard deviation of logarithmic returns over a selected historical period and then annualized using an appropriate number of trading periods.

How are comparable companies selected for volatility analysis?

Comparable companies are generally evaluated based on factors such as industry, business model, company stage, geographic exposure, and other characteristics relevant to the company being valued.

Is the median or mean better for volatility?

Neither measure is automatically better. The appropriate statistic depends on the characteristics of the comparable-company data and the facts and circumstances of the valuation.

Determine the Right Volatility Input for your 409A Valuation

Selecting a Black-Solve volatility input requires more than choosing a number from a market-data source. The analysis should consider the lookback period, comparable company selection, historical return data, and the appropriate representative statistic.

A well-supported volatility analysis can help make the valuation methodology more consistent, transparent and defensible.

Eqvista provides 409A valuation services to help private companies determine FMV using appropriate valuation methodologies and supporting analysis.

Contact Eqvista to learn more about 409A valuations and how Eqvista can support your next valuation.