## The Greek Letters

Since the price of an option depends on several variables, it will change with changes in the value of any of those variables. Option investors and traders as well as those who use options to hedge their portfolios need to understand and measure the impact of changes in the different variables on the price of options. At the simplest level, the holder of an option wants to know how much the value of his option will change if the stock price changes by one dollar or how fast the value of his option will deteriorate with the passage of time. At more complex levels, these sensitivities are used to measure and control the risks of large portfolios with respect to the different variables. For example, financial institutions who hold large portfolios that include enormous positions in derivatives need to estimate how the values of their portfolios would change if volatility went up or down sharply and decide how much of that risk they should hedge and how they should do so.

In mathematical terms, these sensitivities are measured by partial derivatives of the option price with respect to the different variables that determine option prices, and they are represented by certain Greek letters. This is why they are generally referred to as the Greek letters or simply "the Greeks." If you do not understand what partial derivatives are, what you need to know and must remember is that all these measures quantify the sensitivities only for very small changes in the underlying variables, while the other variables remain unchanged. I will explain this a little more in the next section on deltas.

A good way to develop some appreciation for the magnitudes and the behaviors of the Greeks is to look at charts that show how the price of an option changes with changes in the values of the different variables. We will be able to create such charts using some of the models we develop.

In the following sections, we will define the various Greeks and the equations to calculate their values for European options on dividend-paying stocks, which can be derived from the BSM equations. Note that the Greeks apply to portfolios of derivatives as well, and are most often used to measure the risks of portfolios and to hedge those risks.

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