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When a warrant is exercised, a new share of stock is created, whereas when an option is exercised, the owner of the option receives an existing share that is delivered by a counterparty (except in the case of employee stock options, where new shares are created and issued by the company upon exercise).
Traditional warrants are issued in conjunction with a bond (known as a warrant-linked bond), and represent the right to acquire shares in the entity issuing the bond.
Warrants are issued in this way as a 'sweetener' to make the bond issue more attractive, and to reduce the interest rate that must be offered in order to sell the bond issue.