# Stock Option

## Definition

Stock option is a derivative security that gives its holder the right to buy or sell a share of stock on or before a predefined date at a predetermined price.

## What It Means

Stock options enable investors to change the profit pattern of the underlying stock. This can help the investor to hedge existing investment positions, profit from stock price movements with relatively little investment, and devise a number of strategies to capitalize on the market value changes in the options contracts.

### Stock option terminology

Stock options come in two basic flavors:

- Call option that grants the holder the right to buy stock at a known price on or before a certain date.
- Put option that entitles its holder to sell a share of stock at an agreed upon price on or before a certain date.

The price the underlying stock can be bought or sold at is known as the option exercise or strike price. The term of the option runs until the expiration date. The option price is the amount the holder pays for the privilege of being able to exercise the option.

A stock option is in the money if positive cash flow can be obtained by exercising the option. The option is at the money if the current stock price equals the option strike price. If the option exercise today would result in a loss, the option is known to be out of the money.

### Option exercise time: American and European options

Timing of option exercise is important and leads to two different types of options:

- American option can be exercised on or before its expiration date.
- European option can only be exercised on its expiration date.

### Parties to a stock option contract

The purchaser of a stock option is known as the holder. The seller of the option is called the option writer. Note that, unlike the holder, the option writer has the obligation to sell or buy a share of stock should the holder choose to exercise the option. To bear this risk the writer receives an up-front sum equal to the option price from the holder.

### Stock option payoffs – profiting from options

Comparing profit patterns of the underlying stock to an option shows clearly why you may wish to consider investing in stock options.

First, suppose you buy a share of stock in XYZ corporation currently worth $50. If in three months the stock price rises to $80, you can sell your stock and realize a profit of $30. On the other hand, if the stock price drops to $30, you will have a loss of $20.

Now consider purchasing a call option on the same stock for $2 with a strike price of $50. If the stock price is $80 three months out, you can exercise the option and pocket $28, the difference between the current stock price and the sum of the option price and its strike price. If, however, the stock price goes down to $30, you pass on the option and are out only of your original $2 investment.

This ability to significantly change the profit pattern is what makes the stock options an attractive investment strategy.

### Stock option pricing models

Choosing how much to pay for a stock option is a key task for the investor. There are several pricing models widely used in option valuation. Here are the major ones:

- Binomial model
- Black-Scholes option pricing model
- Merton model, an extension of Black-Scholes to dividend paying stocks and currency options
- Monte Carlo model

The binomial option pricing model is a widely used numerical technique to calculate the option value based on the so-called state prices. The Black-Scholes model enables you to determine the values of most common or “plain vanilla” stock options by applying a set of closed form formulas.

The Merton option pricing model is actually an extension of the standard Black-Scholes technique to options on dividend paying stocks. In addition, the Merton model can also be easily adapted to pricing currency options.

For more advanced types of options, such as the Asian or Barrier options, there is no closed form pricing solution. To value such options, the investors resort to the statistical methods such as the Monte Carlo simulation. Since the prices of Asian or Barrier options are path dependent, you will need to run a number of Monte Carlo simulations and then average the results to arrive at the option price.

### Option Greeks: option price sensitivity tools

It is very useful to study how the option price changes with respect to the option parameters. By convention, Greek alphabet letters are used to denote the most common types of option price sensitivity relationships. Here is the list:

- Delta indicates the change in the option price relative to the underlying stock price.
- Gamma denotes how quickly the change in the option price takes place with respect to the underlying.
- Vega shows how sensitive the option price is to the standard deviation of the underlying stock’s return.
- Theta is the change in the option value as the time to option expiration draws near.
- Rho measures how sensitive the option price is to changes in the interest rate.

Option Greeks can be easily calculated for options that can be priced using the Black-Scholes or Merton models.