Decoding Black-Scholes: Valuing Employee Stock Options

Decoding Black-Scholes: Valuing Employee Stock Options

Employee stock options (ESOs) are a common form of compensation provided by employers to their employees. ESOs are contracts that allow employees to purchase a specific number of shares of their company’s stock at a predetermined price within a specified time frame. ESOs can become very valuable for employees if the company’s stock price increases over time. However, valuing ESOs can be a challenging task due to their complexity. In this article, we will decode Black-Scholes and explain how it can be used to value ESOs.

Understanding ESOs

Before we jump into the details of how to value ESOs, it’s important to understand how they work. ESOs generally have a vesting period, which means they can’t be exercised immediately after being granted. Once the vesting period is over, the employee can exercise the option, purchasing stock at the pre-determined price. If the company’s stock price rises above the option price, the employee can sell the purchased stock for a profit. If the stock price falls below the option price, the employee may choose not to exercise the option.

The Black-Scholes Model

The Black-Scholes model is a well-known mathematical model used to calculate the theoretical value of European call and put options. It was introduced by Fischer Black and Myron Scholes in 1973 and later extended by Robert Merton. The model is based on the following assumptions:

  • The stock price follows a lognormal distribution
  • The option can only be exercised at expiration
  • The risk-free rate and volatility of the stock price are known and constant
  • There are no transaction costs or taxes
  • The underlying asset pays no dividends

While these assumptions may not hold true in reality, the model provides a good starting point for valuing ESOs.

The Inputs

The Black-Scholes model requires several inputs to calculate the theoretical value of the option. These inputs include:

  • Stock price: The current market price of the company’s stock
  • Strike price: The pre-determined price at which the employee can purchase the stock
  • Time to expiration: The time remaining until the option can no longer be exercised
  • Volatility: The degree to which the stock price fluctuates
  • Risk-free rate: The rate of return on a risk-free investment over the option’s life

Calculating the Value

Using the above inputs, the Black-Scholes model calculates the theoretical value of the option. The value of the option can be calculated using the following formula:

C = S*N(d1) - X*e^(-r*t)*N(d2)

– C is the theoretical value of the call option
– S is the stock price
– X is the strike price
– r is the risk-free rate
– t is the time to expiration
– N refers to the cumulative normal distribution
– d1 and d2 are calculated as follows:
d1 = (ln(S/X) + (r + 0.5 * σ^2) * t) / (σ * sqrt(t))
d2 = d1 - σ * sqrt(t)

Additional Considerations

While the Black-Scholes model provides a good starting point for valuing ESOs, there are several other factors that can impact the value of the option. For example, the vesting period, the stock’s dividend yield, and the cost of exercising the option can all impact its value. Additionally, market conditions, such as changes in interest rates or volatility, can also impact the option’s value.


ESOs are a valuable form of compensation that can be difficult to value. The Black-Scholes model provides a useful framework for valuing ESOs, but it is important to consider additional factors that may impact the value of the option. By understanding the inputs and calculations involved in the valuation process, employers and employees alike can make informed decisions about the value of ESOs.