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What's the difference between a derivative and a contingent claim? What is an example of a derivative which isn't a contingent claim?

Since options or swaps are examples of derivatives that are contingent claims.

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Quoting from wikipedia

"In finance, a contingent claim is a derivative whose future payoff depends on the value of another “underlying” asset, or more generally, that is dependent on the realization of some uncertain future event. These are so named, since there is only a payoff under certain contingencies. Any derivative instrument that is not a contingent claim is called a forward commitment. The prototypical contingent claim is an option, the right to buy or sell the underlying asset at a specified exercise price by a certain expiration date; whereas (vanilla) swaps, forwards, and futures are forward commitments, since these grant no such optionality. Contingent claims are applied under financial economics in developing models and theory, and in corporate finance as a valuation framework."

See also references quoted in the original article. This also seems to be the definition used by the CFA Institute (just google "contingent claim vs forward comittment" along with CFA keyword).

So according to these definitions:

  • options, swaps, forwards/future contracts are all derivatives
  • options are contingent claims
  • (vanilla) swaps, forwards/future contracts are forward comittments
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    $\begingroup$ +1. I would only add that "contingent" is a word used by lawyers very similar to the word "conditional" used in math. For ex. a conditional statement is a statement that is only executed if the preceding "if clause" is true. A claim is a statement that someone owes you money which you can take to court to collect the money. A contingent/conditional claim is something where you may be owed money in some future scenario but we don't know yet, it depends on some condtion. $\endgroup$ – Alex C Mar 20 '18 at 14:10
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Usually a contingent claim is regarded as another name of derivatives. But it may be more generic. The payoffs of contingent claims may depend on other assets or some events, such as credit events, corporate actions, etc.

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