Can we change the price of previously issued stock options?

Technically you can reprice any options that are still outstanding. Though many private companies early on are not calculating stock comp expense on options, particularly if it is possible that you will go public in the near future, you need to be aware of the implications.

For awards that are repriced, in addition to the stock comp expense related to the original award, the new award (for accounting purposes it is considered a new award as of the reprice date) should carry an "incremental" expense. This expense is the Fair Value of the awards the day of the reprice (with the new price) - the Fair Value of the Award the day before the reprice. This results in a per-share incremental expense. For any shares vested on the awards as of the reprice date, companies generally take a one-time lump sum expense in the period of the reprice. The incremental expense on the shares unvested as of the reprice date can be amortized over the remaining vesting period for those shares. Some companies will reduce the number of shares in the new awards to make the exchange accounting neutral.

The reprice itself does not create any tax implications to the participant if the options are Non-Qualified Options. For companies repricing Incentives Stock Options (ISOs), the new options can still be ISOs. However, the canceled options that were scheduled to vest in the year when cancellation occurs still count against the 100K rule (i.e., that one participant may not have more than $100K worth of ISOs vesting in a single year, where value = exercise price X shares), so this has to be taken into account when calculating whether the new shares will cause a participant to cross the 100K rule. If so, then any shares over the 100K will need to be issued as NQs.

In public companies, where there is the issue of shareholder scrutiny, often the repriced awards offered will have some loss of benefit, such as change in vesting or share amount.

With regards to 409A, a single reprice does not usually pose a problem. If a company reprices multiple times, however, it can create the perception of a variable exercise price that could be treated as a discount from the original grant and lead the company afoul of 409A.