For more complicated interest-rate options, we have to go back to our original framework: risk-neutral valuation, where all prices (underlyings and contingent claims) must satisfy:

This compact formula is surprisingly all we need for contingent claim valuation, and is the basis of all interest-rate option models. When applied to interest-rate underlyings, it imposes the following constraint on today′s (*t* = 0) discount factors:
which is loosely called ″ensuring no-arbitrage.″ At all future times *t >* 0, and specifically at option expiry *t*_{e}, discount factors can be recovered as
from which we can compute the payoffs of the contingent claim, *C*(*t*_{e} *, ω*), as it is a function of the discount ...

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