Intro: Stock Options


An Introduction to Stock Options

Stock Options Defined

Call Option:

  • A call option grants the owner the right to buy a share of an underlying stock for the strike price anytime before expiration.

  • A call-option seller (also known as a writer) has the obligation to sell a share of the underlying stock for the strike price, if the buyer exercises the option.

Put Option:

  • A put option grants the owner the right to sell a share of an underlying stock for the strike price anytime before expiration.

  • A put-option seller (also known as a writer) has the obligation to buy a share of the underlying stock for the strike price, if the buyer exercises the option.

American vs. European

The previous definitions were for American options. These are the type of stock options traded on most exchanges in the US.

In a European option, you can only exercise the option at expiration.

Intrinsic Value

The intrinsic value is what the option would be worth if you had to exercise it immediately. If exercising the option would result in a negative amount, intrinsic is set to $0. Denoting `I` for intrinsic value, and `S` and `X` for the stock and strike price respectively, we can write:

  • Call option: `I=max(S-X, 0)`
  • Put option: `I=max(X-S, 0)`

For example, if `S=\$53`, `X=\$50` the intrinsic value of a call is `\$3` and the intrinsic value of a put is `\$0`.

Additional Terms

  • For both calls and puts, if `I > 0` then we say the option is ‘in-the-money.’

  • If `I < 0` the option is ‘out-of-the-money.’

  • The option with a strike nearest to the present stock price is termed the ‘at-the-money.’

These terms are useful because in/at/out-of-the-money options have similarities regardless of whether they are calls or puts. For example, out-of-the-money option premiums are not sensitive to changes in the underlying stock price.

  • The market price of an option is termed the option’s premium.

  • An option’s time value is the option’s premium less its intrinsic value.

Practical Notes

  • Option contracts listed on US exchanges (such as the Chicago Board Options Exchange (CBOE)) are for 100 options.

  • Option contracts traded on US exchanges are cleared, which means you don’t have to worry about counterparty risk. This also means you’ll put up margin for each option trade.

  • Option strike prices are adjusted for stock splits, as well as for stock dividends of more than 10%.

  • Option contracts are not adjusted for cash dividends. This means call/put option premiums are lower/higher for high-dividend-paying stocks. Remember, if a stock pays a $1 dividend, its stock price is lowered by $1.

What Can We Say So Far?

An American option’s premium (both call and put) can never be less than the option’s intrinsic. Why not?

  • Arbitrage: If the intrinsic were $5, and the option premium was $3, you could buy the call option and immediately exercise it earning a riskless $2/share.
  • However, a European put’s premium could be less than the intrinsic. Also a European call option on a dividend-paying stock can have a premium less than its intrinsic.
  • In the next slide, you can graph a European call option’s intrinsic, premium, and time value for any set of inputs. Note that as you increase the stock’s dividend yield, the option premium drops below the intrinsic.

  • The risk-free rate is the rate on a zero-coupon Treasury security of the same maturity as the option

Time Value

In the previous chart, you saw that the time value is greatest for at-the-money options. Can you guess why?

  • Very in- or very out- of-the-money options are not really options. You are almost certain to not exercise, or exercise, respectively.
  • An at-the-money option roughly has a 50% chance of being exercised and a 50% chance of not being exercised. In other words, it is where there is the most option in the option.

European Put Values

We mentioned earlier that put-option premiums on European options could be less than the option’s intrinsic value. You can use the previous app to see this. Can you guess why this is so? As a hint, set both the time-to-maturity and risk-free rates very low. Then set them very high and note the difference.


  • Imagine the stock price has fallen to $0, then of course you would want to immediately exercise it. The option can never be worth more, and you would rather have your money now than later. But because it is European, you have to wait until expiration. By the simple time value of money, the value of the option is the present value of the intrinsic. As time passes and the risk-free rate increases, the option premium decreases.

Option Premium App

The following slide will allow you to calculate both call and put option premiums for your given set of inputs (via the Black-Scholes equation).

  • Use the app to see how the underlying variables affect the option premium. For example, how does an increase in interest rates effect call and put values? How about an increase in volatility?

  • Don’t worry yet about how to derive the option pricing equation – we’ll cover that in later presentations.

Credits and Collaboration

Click the following links to see the codeline-by-line contributions to this presentation, and all the collaborators who have contributed to 5-Minute Finance via GitHub.

Learn more about how to contribute here.