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There are two types of warrants: "traditional" warrants and so-called naked warrants.
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. In other words, the writer of a traditional warrant is also the issuer of the underlying instrument. 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.
Naked warrants are issued without an accompanying bond, and like traditional warrants, are traded on the stock exchange. They are typically issued by banks and securities houses. The writer of a naked warrant need not be the issuer of the underlying instrument. A naked warrant is essentially an option with a very long time to expiry. Therefore an employee stock option is also equivalent to a warrant.
Also, when a government agency issues checks which they are unable to pay (due to lack of money) but are redeemable at some point in the future, usually with interest, these are also called warrants. A few years ago, when the State of California had a budget crisis due to a disagreement between the governor and the legislature, the state treasurer was forced to issue warrants paying 18% interest in lieu of being able to pay the state's bills with real money. The state had not had to rely on this practice since before the Depression of the 1930s Many places were accepting them at face value because of the interest provision. It was interesting that during this period the Controller of Los Angeles County was buying state warrants to invest the county's surplus funds because the county, on the other hand, actually had money and the interest rate was better than any bank would pay.
In the military, payroll checks for soldiers are also referred to as warrants.