Shares can be traded in the bank by talking to the bank manager, Mr. Smith. In Airline Tycoon 1 all airlines start with 10,000 issued shares at the beginning of the game, while they own 80% of their own shares. New shares can be issued, if the market needs new shares, to gain money. Shares can also be bought from the other companies.
From time to time, rival players will issue new shares and these can be bought from them through the bank. If you own at least 50% of the shares of another company, and they in turn own less than 30% of your shares, you can choose to either take over the company or to liquidate it.
If you choose to take it over, you'll inherit all planes and personnel from the other company, as well as the money they had in their account or possible loans. You will also get their gates, check-in desks, outstanding contracts and shares they own in other companies.
The other option is to liquidate the airline, in which case the bank will sell all assets of the airline and fire all staff. The player will then receive a percentage of the profits that is equal to the amount of shares they own in the liquidated airline. The gates will be up for auction in the Airport Manager's office.
If you need money, you can choose to sell shares that you may own from other companies. You can also choose to sell your own shares, although you shouldn't sell too many shares, as you could lose your company's majority.
New shares can be issued from time to time. The player can choose how many new shares they want to issue and can also decide how expensive they should be. The higher the price, the more money you'll make. A lower price makes the stock price drop, which will generate less money. The other share owners have to be compensated for the dropping stock price.
When issuing new shares, the player should make sure they don't lose their company's majority—this is also what barman Rick says when he tells you there is demand for your stocks. Oftentimes, in free play, Honey Airlines will issue many new shares very early on in the game, leaving him with about 35% of his own company. This allows the player to take over the rival, usually within 25-30 days after the game starts.