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Black-Scholes Model: Formula & Examples

Instructor: David Bartosiak

Dave draws off his years of experience as a Financial Advisor and Analyst to teach others all about finance and the investing world.

The Black-Scholes model is a mathematical model for financial markets. From this larger model, the Black-Scholes formula for theoretical option value is used to determine what price put and call options should be trading at based on assumptions of volatility

A Mathematical Model for Financial Markets

Since the beginning of the stock market, investors have been trying to gain an edge. The number-crunchers out there have long sought out a mathematical model which could predict price movement. If they could somehow figure out the secret formula, they could become rich beyond their wildest dreams.

The team of Fischer Black, Myron Scholes, and Robert C Merton tried to do just that. They came up with an over-arching mathematical model for financial markets that contain derivative instruments. Lacking a creative name, this model became known as the Black-Scholes-Merton model.

From this larger model, smaller models and equations were made based on the same assumptions. After years of developing the model, Robert Merton is attributed with first mentioning the 'Black-Scholes options pricing model' in 1973. This theoretical model could help options market-makers properly price options on all types of financial instruments. Their work was so ground-breaking that twenty-four years later in 1997, Robert C. Merton and Myron Scholes won the Nobel Memorial Prize in Economic Studies for their work.

Understanding the various inputs which go into the calculation can help traders understand whether the market is overpricing, underpricing or fairly pricing an option. Here we'll discuss the various factors in detail and learn how to use the model to calculate option prices.

The Black-Scholes Formula for Options Pricing

The Black-Scholes formula is a mathematical model to calculate the price of put and call options. Since put and call options are distinctly different, there are two formulas which account for each option. Call options give the option holder the right to buy the underlying stock for an agreed upon price anytime between today and option expiration. Traders that believe the underlying stock will go up over time buy these call options in the hopes of making money.

On the flip-side, put options give the option holder the right to sell the underlying stock for an agreed upon price anytime between today and option expiration. Traders that think a stock is going to go down can buy these put options in the hopes of making money if the stock goes does.

The actual Black-Sholes formula looks complicated but is actually simple when you break it down to the basics. The main factors in the equation are:

  • T =The time to maturity: How long until the option expires, in years;
  • S =The current price: Price of the underlying stock;
  • K = The strike price : The agreed upon price of option execution;
  • r = The risk-free rate: A rate an investor could get by taking on no risk (typically the 3-Month Treasury Bill Yield);
  • σ =The price volatility: The volatility of the returns of the underlying stock, expressed as percentage.

Here is the mathematical notation of the formula:

B-Scall1

bscall2

bscall3

This formula includes the function N(x) which is the cumulative standard normal distribution function. The cumulative standard normal distribution function is defined as the probability of a random variable with normal distribution, a mean of 0 and variance of ½ falling in the range of {-x,x}. It's a complicated calculation dealing with the area under a normalized distribution curve.

Calculating Theoretical Option Value

For the sake of options calculation, you don't need to become a PhD in Math. There are hundreds of free online calculators you can use to plug in easily accessible values to calculation the Black-Scholes formula.

Let's do a sample calculation with XYZ Corp Stock and define these values.

Time to maturity (T) = 1 year

Current Price (S) = $120

Strike Price (K) = $100

Risk-Free Rate (r) = 1%

Price Volatility (?)= 50%

First you need to calculate values for d1 and d2, so you can throw these values into the cumulative standard normal distribution function.

Using these values and a calculator I found online, I was able to come up with a theoretical value of $34.20 for the call option. The price of the put option at the same strike is derived from the calculation of the call option.

Using the same input values in the above equation gives us the theoretical price of the put option at the same strike. The online calculator we used before gives us a value of $12.22 for the put option.

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