### Valuation of Stock Options-Black Scholes Model

November 11, 2010

*(*Published by *QuickRead NACVA*, January 22, 2014)

**The Black Scholes Model **

The true value of a stock option is often greater than its intrinsic value. This article takes a theoretical approach to valuation that focuses on the time value of money with the Black-Scholes Option Pricing Model.

An employee stock option is a contract between the employer and the individual employee providing the right to purchase company stock at a designated price for a designated length of time. For tax purposes, employee stock options are either classified as incentive stock options or nonqualified stock options. The fixed purchase price granted by the option is called the exercise price or the striking (“strike”) price. Often, a waiting period is assigned before the option can be exercised. This time period is known as the vesting period. An employee stock option has an expiration date, after which the employee loses the right to exercise the option. This date is referred to as the expiration date. The difference between the strike price and the market price stock is commonly referred to as the intrinsic value. Although simple to calculate, the intrinsic value is rarely considered the true value of the option because it ignores the time value of the stock option. In fact, the true value of the option is often greater than the intrinsic value. This is because the longer the term of the option, the greater the time value of the option. This is due to the fact that a longer time period increases the likelihood that the underlying shares will rise above the strike price of the option. Therefore, to determine the true value of stock options, it is necessary to use a theoretical approach to valuation that considers the time value of money. We employ the Black-Scholes Option Pricing Model.

The use of the Black-Scholes model is also supported within the accounting and financial reporting sectors. According to the Financial Accounting Standards Board Statement No.123 (revised 2004), a public entity is required “to measure the cost of employee services received in exchange for an award of equity instruments based on the grant-date fair value of the award. The grant-date fair value of employee share options and similar instruments will be estimated using option-pricing models adjusted for the unique characteristics of those instruments.” The statement further clarifies that “for stock options, fair value is determined using an option-pricing model that takes into account the stock price at the grant date, the exercise price, the expected life of the option, the volatility of the underlying stock and the expected dividends on it, and the risk-free rate over the expected life of the option.” These are the factors that are the key components used in the Black-Scholes option pricing model.

**BLACK-SCHOLES OPTION PRICING MODEL**

**Options Theory**

The economic theory on which we rely is Options Pricing Theory. The paradigm options pricing model is the Black-Scholes Options Pricing Model (“Black-Scholes” or “BSOPM”), developed by University of Chicago Professors Fisher Black and Myron Scholes, the latter of whom received the Nobel Prize in Economics for developing the model (Black had already died).

A call option is a contract enabling one to buy a specific number of shares of a company at a specific price and time. For example, one might buy an option to purchase 100 shares of IBM at $100 per share on a specific date. A European option is such that one can buy only on that date, while an American option allows one to buy anytime up to and including that date. The original Black-Scholes model works on the assumption of a European option. A put option is the opposite of a call. It enables one to sell the stock at a specific price and time. Let us examine a call option.

We do not know what the price of the stock will be. Black-Scholes assumes a normal probability distribution (the bell-shaped curve) of prices on the expiration date of the option. The bell shaped curve is symmetric and peaks in the center, which is the statistical mean, median, and mode, these being three different types of averages, which are not identical for asymmetric distributions.[1]

All normal distributions are measured by two and only two parameters: the mean and the standard deviation. The mean is the average, and the standard deviation is a statistical measure of the width of the curve.

The stock volatility is the main determinant of the value of the option. The more volatile the stock, the shorter and fatter is the normal curve and the greater is the probability of making a lot of money on the investment.

*Conclusion*

The Black-Scholes Option Pricing Model is one of the most widely accepted methods to value stock options. Given the similarities between publicly traded options and the employee stock options, the court frequently considers the results of the BSOPM as an appropriate proxy of value.

[1] Technically it is the natural logarithm of prices that is normally distributed, but for a more intuitive explanation, we speak in terms of prices rather than log prices.