Black-Scholes model explained

Background:

The Black-Scholes formula was developed and introduced by Fisher Black and Myron Scholes in their 1973 Journal of Political Economy publication outlining the ability to price options and corporate liabilities. It was not until Robert Merton published his work expanding the initial formula’s application of options pricing that the use and application of the formula became an industry standard in options pricing. Merton and Scholes would later receive the Nobel Prize in Economics in 1997 for this work; Black, although integral to the foundation to the formula, had passed away two years prior and was thus ineligible to receive the prestigious award. The formula establishes a calculation to value an option based on the underlying asset and market conditions.

Before moving forward, it is recommended to refer some important Terms.

What is Black-Scholes Formula used for?
The Black-Scholes model is used to compute the theoretical price of a European option. It is important to
note that European options differ from American options in that European options can only be exercised at
the date and time of expiration. American options, however, can be exercised at any point leading up to

and including the date and time of expiration.

What are some underlying assumptions of Black-Scholes model?
Some of the underlying assumptions used in this model are summarized below:

• The underlying stock pays no dividends during the life of the option. (Note: there exists a way to modify the Black-Scholes model to account for stocks which pay dividends)
• The Market is efficient. This assumes that market behavior of the underlying stock operates with all available information about and that would affect the value of an equity and thus the current price reflects the fair valuation of the company.
• There is no commission charged to buy or sell the option.
• Interest rates remain unchanged. This assumes that the risk-free interest rate, r, is both known and constant during the life of the option.
• The returns are normally distributed. This assumes that the returns of the underlying stock follow and fit a Gaussian distribution (known as the Normal curve, or bell curve).

How do you explain the Black-Scholes Formula?

The purpose of the Black-Scholes pricing model is to calculate a theoretical value that the investor would pay for an option (either Put or Call) for a specific underlying equity. For now, lets focus on the European 

Call option as an example but is applicable to both Put and Call options (typically with inverse relationships).

In determining this theoretical price, the investor would need to know the current market stock price, S0, as well as the negotiated or given future exercise price (also called the strike price), X. It is also important to understand that the future strike price must be discounted back over the life of the option, T, at the assumed risk-free rate, r. Both the strike price and the time would be set by, and at the time of, the option contract. Finally, the investor needs to know the particular volatility, σ, of the stock as measured by the standard deviation of log returns. 

Example volatility of two stocks

Here volatility refers to the dispersion of returns above and below an asset’s mean return (see Figure right). In this figure, Stock 1 (blue) has a significantly tighter dispersion around its central value, whereas, Stock 2 (orange) has a much wider range of values around the same mean. Thus, we would conclude that Stock 2 (orange) is a more volatile stock than Stock 1 (blue). Inherently, the more volatile a stock is, the more valuable the theoretical pricing option becomes (as

discussed above).

Remember the formula for European call option:

where d1 and d2 are respectively:

Here the first term, takes the current stock price, S0, and multiplies it by the cumulative distribution function of the d1 variable. This cumulative distribution is a calculated probability from 0<=N<=1 of the normal curve. Thus, the first term of the equation is the current stock price weighted by some probability of acquiring the stock at that price.

The second term is the weighted probability of exercising the option times the discounted price you pay (discounted back over the life of the option).

Refer to d1 and d2 above, within both these terms, there are two key factors to identify. First is the ratio of the stock price to the exercise price (S0/X). Here the higher this ratio is, the larger both d1 and d2 terms are. This means that a larger resulting ratio input is used in the normal distribution function is and thus results in higher weighting probability value (output of the normal distribution function). When these higher probabilities are then multiplied in the above equation, the result is a higher chance of exercising the option.

The second key factor is the volatility, σ. From the above figure, we can note that the higher a stock’s volatility, the higher the value of the stock’s option(s) is. Now let’s examine the merit of this initial assumption. This is best explained by looking at the effects of a high volatility value. The higher the volatility is, the bigger the resulting d1 term will be because we are adding the value of volatility squared in the numerator, but only dividing by the volatility in the denominator. Thus with a high d1, we have a higher N(d1), resulting in a higher cumulative probability which is multiplied by the current stock price.Contrary to the d1 term’s behavior in response to high volatility, the d2 term will respond inversely because we are subtracting the value of volatility squared in the numerator. This results in a lower d2 term, a lower computed N(d2) and thus a lower cumulative probability multiplied by the discounted exercise price. Therefore, we are subtracting a smaller 2nd term from a larger 1st term in the above equation resulting in a high value of the European Call option. Saying it in more general terms, the higher the volatility, the higher the resulting values of the European Call option will be.