Calculating the volatility of the underlying stock is more complicated. Volatility is the standard deviation is the price of underlying stock. The…

Beta is a measure of systematic, or non-diversifiable, risk. It quantifies the volatility of a specific stock's returns in relation to the returns of the overall market portfolio. A beta of 1.0 indicates the stock's price moves in line with the market. A beta greater than 1.0 indicates it is more volatile than the market, and less than 1.0 indicates it is less volatile. The Black-Scholes model requires total volatility (standard deviation), which includes both systematic risk (measured by beta) and unsystematic, company-specific risk. The question correctly distinguishes between these two measures of risk.

B. That the stock’s price changes are relatively constant describes low volatility, not beta, which measures volatility relative to the market.

C. Implied volatility is the volatility derived from an option's current market price; it is an output of the model, not what beta measures.

D. The number of outstanding shares is a factor in a company's market capitalization, not a measure of its price risk relative to the market.

1. Pratt, S. P., & Grabowski, R. J. (2014). Cost of Capital: Applications and Examples (5th ed.). Wiley.

Chapter 10, "Beta: The Capital Asset Pricing Model's Measure of Systematic Risk," p. 169. The text states, "Beta is a measure of the systematic risk of a security or a portfolio in comparison to the market as a whole." This directly supports that beta measures the relative movement of a stock's price to the market.

2. Damodaran, A. (2012). Investment Valuation: Tools and Techniques for Determining the Value of Any Asset (3rd ed.). Wiley. (NYU Stern School of Business courseware basis).

Chapter 4, "The Basics of Risk," p. 81. "The beta of an investment is a standardized measure of this risk [market risk]... By standardizing the risk, we can compare the market risk in an investment to the risk in an average investment."

Chapter 29, "The Options to Delay and Abandon: The Real Options Approach," p. 778. In discussing inputs for option pricing, it clarifies, "The variance in the value of the underlying asset is a key input in option pricing models... Note that this variance is of the total value of the asset and not just the market-related portion of the value." This distinguishes the total volatility used in Black-Scholes from the market-relative risk measured by beta.

3. National Association of Certified Valuators and Analysts (NACVA). (2021). Business Valuations: Fundamentals, Techniques and Theory (FT&T).

Chapter 10, "Cost of Capital/Rates of Return," Section on Capital Asset Pricing Model (CAPM). The official NACVA training material defines beta as a measure of the volatility, or systematic risk, of a security or a portfolio in comparison to the market as a whole, reinforcing the definition in option A.