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The following is an excerpt from Introduction to the Mathematics of Finance by Roman:

As a more concrete example, suppose that IBM is selling for $\$100$ per share at this moment. A $3$ month call option on IBM with strike price $\$102$ is a contract between the buyer and the seller of the option that says that the buyer may (but is not required to) purchase $\color{blue}{\bf 100}$ shares of IBM from the seller for $\$102$ per share at any time during the next $3$ months.

In this example is the number $\color{blue}{100}$ in blue arbitrarily picked by the author? Is it relevant to the concept of "call option"?

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Call options are usually standardized product: in the contract you can (but are not obliged to) buy a certain amount, which is specified. The most common quantity is 100 shares (see for example the description by J. Hull, Options, Futures, and Other Derivatives).

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Yes it's arbitrary, when you agree on the condition you define the how many shares and the Strike price if the contract is not standard

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