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In financial mathematics, mathematician sometimes consider financial instruments with infinite lifetime, e.g. bonds or options.

I am curious how it looks in practice. Is the case of ​​an infinite lifetime of financial instruments encountered on the real financial markets? What kind of financial instruments have this property and where we can trade them?

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The most common example of financial instruments with "infinite" lifetime are perpetual bonds (Wikipedia has details here).

Perpetual bonds do not have a contractual end date, but recent versions of these bonds are almost always callable. This means that the issuer has the right at certain points in the future to redeem the bond. A callable bond whilst having an "infinite" contractual lifetime for the issuer if the issuer so wishes has a finite expected lifetime.

Lots of recently issued perpetual bonds can be traded just like other exchange traded bonds on the usual exchanges. See the Wikipedia article for some more exotic examples of perpetual bonds which do not trade.

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