Issuing stock options to employees in a private closely held company can be a complicated matter, from both a legal and tax perspective. The exercise price per share should definitely not be an arbitrary number. It should be the fair market value of the shares on the day the stock options are granted to the employee. The IRS has very strict rules about setting exercise prices for employee stock options, and failure to comply can result in severe tax consequences. For non-qualified stock options, there are no immediate tax implications for the grant of options to the employee; however, when the employee does exercise the options in the future, the spread between the exercise price and the fair value of the shares at that time is reported as taxable income to the employee, which is another reason why it's important to be mindful of the exercise price when drafting these agreements. The IRS will generally require a valuation report (IRC Section 409A Report) to support the exercise price of the options. The vast majority of startups use an independent appraisal to prepare the 409A report. This also provides you some audit protection from the IRS, because they generally fall within the 409A safe harbor rules.

Generally you will start with 10M shares with a par value of 0.00001 per share which values the company at $100. This is pretty standard for a C-Corp

How long has it been since you started the company?