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One of the most widely used formulas for pricing non-dividend paying European stock options is the Black-Scholes Option Pricing Formula.  The formula was published in a famous paper written in 1973 by Fischer Black and Myron Scholes titled ”The Pricing of Options and Corporate Liabilities”.

Following is a breakdown of the formula for both Puts and Calls. Pseudo-code will be used to show the formula for calculating the value of an option using the Black-Scholes formula::

Puts

Put Option Price = A * e^-rT * N(-q₂) - P * N(-q₁)

Symbol Expansion:

q₁ = q_1a / q_1b
q_1a = ln(P / A) + (r + sigma² / 2) * Y
q_1b = sigma * square_root(Y)

q₂ = q_2a / q_2b
q_2a = ln(P / A) + (r - sigma² / 2) * Y
q_2b = sigma * square_root(Y)

Where:

A = strike price of the Option
P = price of the Stock
r = the current risk-free interest rate (e.g. - long-term bonds)
sigma = Annual volatility of the stock price
Y = the number of years before the Option expires
N(x) = The cumulative normal distribution function
e = exponential function
 

Calls:

Call Option Price = P * N(q₁) - A * e^-rY * N(q₂)

The symbol expansions for the Call option pricing formula are the same as it is for Puts (see above).

Analysis:

As you can see the annual volatility of the stock price is of tremendous importance in calculating the value of a Put or Call option. This is be because the holder of the option wants the option price to reflect the probability that the underlying asset, the stock, will reach the option strike price. In more volatile times, option sellers will charge more because the probability is greater that the stock might reach the option strike price than in quiet times.

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