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"1. When a company buys stock on open market, I assume they pay
cash. What asset account do they credit?"

Since they are paying cash (99% of the time)for the stock, then cash is the credited account and a contra-equity account called Treasury stock is debited. A contra-equity account is tread similarly to a contra-asset account like accumulated depreciation.

"2. What happens to actual shares purchased? Do they simply keep
the actual certificate? Or is there a procedure used to
eliminate the stock? Does this have any bearing on the
number of shares authorized to issue, but not issued?"

Normally the stock is either retired (done away with)or held as treasury stock for a while. A while can be years to decades, ultimately the treasury stock is usually retired or issued back out. The total amount of shares authorized normally only changes when the company goes back to the state of incorporation (most often Delaware) and changes their charter to issue more shares. On some balance sheet (the more detailed balance sheets) you will sometimes see the following
authorized x, issued y, and outstanding z. The difference between Y and Z is treasury shares.

"3. If, at a later date, share options are exercised at a price
much lower than market price, but at the same price company
paid when they purchased shares on open market, can
these shares be involved in that transaction? If so, how
would this be done on companies books?"

If the shares are sold (maybe the same rules apply for issuing for stock options) at a price over the price paid for the treasury stock the entry will be :

Treasury Stock
PIC from treasury stock

If the shares were issued at prices below the what was paid for the treasury stock then the entries are:

Paid in capital from treasury stock (or see note 1)
Treasury stock

Note 1: If there is no paid in capital from treasury stock then retained earnings are debited (reduced).

If you go to a lot of tech stocks Cisco, Nortel, etc., you will very little retained earnings for this reason.

4. If company must purchase shares to transfer to option
taker, do the just debit cash and credit expense account
for the difference in value?"

The company doesn't have to purchase the shares they issue due to options, although some do. Most certainly in the past haven't expensed anything, but more will in the future. The accounts involved were cash and various owners equity accounts. The total effect in both actuallity and the accounting, in the past, was a transfering of ownership from the stockholders to the option recepients.

My financial accounting text book is 100 miles away at present and this is the best I can do from memory. If you have any other questions reply and I will look it up.

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