Definitions
C is the current call option value. This is the cost to purchase one European-type call option of a certain stock.





































S is the current stock price. This is the price that the stock underlying the call option is currently trading at.





































N(d1) is a fraction (whose value is between 0 and 1) determined by the price of the stock; the exercise price; the risk-free interest rate (the annualized continuously compounded rate on a safe asset with the same maturity as the option); the time to maturity of the call option, and the volatility of the underlying stock price. See an explanation of d1.

N(d) is the probability that a random draw from a standard normal distribution will be less than d.





































L is the exercise price, the price at which you have the right to buy the stock when the call option expires.





































e-rT is a term that adjusts the exercise price, L, by taking into account the time value of money. Here, "e" is 2.718, the base for the natural logarithm, used for continuous compounding; "r" is the risk-free interest rate; and "T" is the time until expiration of the call option.





































N(d1) sigma sqrt(t) is a fraction (whose value is between 0 and 1) determined by the price of the stock; the exercise price; the risk-free interest rate (the annualized continuously compounded rate on a safe asset with the same maturity as the call option), the time to maturity of the option, and the volatility of the underlying stock price. It differs from N(d1) in that one subtracts sigma sqrt(t) (the volatility of the stock times the square root of the time till the call option expires) from d1 before the N function is performed. See an explanation of d1.

N(d) is the probability that a random draw from a standard normal distribution will be less than d.





































What are stock options?
A stock option is a contract that gives you the right—but not the obligation—to buy or sell a stock at a pre-specified price (the exercise price) for a pre-specified time, that is, until the option "expires." If the option gives you the right to buy shares of a stock, it is a call option. If the option gives you the right to sell shares of a stock, it is a put option. Exactly how much you should pay for these contracts is determined using the Black-Scholes Formula.

Options are usually sold in sets of 100 (which would allow you to buy or sell 100 shares of the underlying stock at a certain price for the duration of the option).

There are two kinds of stock options: American-type and European-type. American-type stock options allow you to buy or sell the shares of the underlying stock at the exercise price ("exercise" the option) any time until the option expires. European-type stock options allow you to exercise the option only at its expiration date. The formula we provide here applies to a European-type call option. You can buy and sell options just like stocks; their value is determined by the likelihood that they will be "exercised" for a payoff ("in the money"). You can calculate the exact value of the call option using the Black-Scholes Formula (if you know what you're doing, of course).

A Successful European-Type Call Option
Stock ABC is currently trading at $20/share. You pay a premium of $300 and purchase one European-type stock call option—a contract that enables you to buy 100 shares of ABC at $25/share three months from now. Let's say that three months from now, the stock is trading at $30/share. At that moment, you may choose to exercise your call: Buy 100 shares at $25/share. You can then immediately sell those shares at $30/share. Your payoff in this exchange is $500. Subtract $300, the purchase price of the call option, and you made a net profit of $200 - not bad for an investment of only $300. In this scenario, you exercised your option "in the money": You were able to buy your stocks at a price lower than their current value. You also may sell the call to someone else before it expires. Let's say that one month after purchasing the call option, the stock is trading at $29. The value of the call option is now much greater than when you bought it, and you could sell it to someone.

An Unsuccessful European-Type Call Option
Stock ABC is currently trading at $20/share. You pay a premium of $300 and purchase one European call option—a contract that enables you to buy 100 shares of ABC at $25/share three months from now. In three months, however, the stock is only trading at $23/share. In this case, buying the stock at $25/share would not be to your advantage. You can only hope that during the three months, you were able to sell the call option to another investor. As the expiration date approaches, however, the value of the call option decreases. If you don't sell the call option, and it expires in three months "out of the money," you would simply lose the $300 you had spent on the premium.