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Interest Rate Swaps and Their Derivatives: A Practitioner's Guide by AMIR SADR

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CHAPTER 8
Short-Rate Models
Having milked Black′s formula dry for flow products, we need models to price more complicated interest-rate options, prominent among them Bermudan (flexible exercise time) and Asian (path-dependent) options. For non-rate-based assets like equities, closed-form formulae (some exact, and some approximations) exist for some of these options. However, since interest-rate options depend on multiple underlyings (zero-coupon bonds), these formulae are hard to adapt to rate options.
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:
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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:
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which is loosely called ″ensuring no-arbitrage.″ At all future times t > 0, and specifically at option expiry te, discount factors can be recovered as
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from which we can compute the payoffs of the contingent claim, C(te , ω), as it is a function of the discount ...

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