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

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CHAPTER 10
Full Term-Structure Interest-Rate Models
Short-rate models were the first systematic attempt to break away from Black′s formula for pricing interest-rate derivatives. The basic setup was to posit some dynamic for the short rate, r(t, ω) as
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and then price derivatives C(t, ω) with a terminal payoff at T using risk-neutral expectation:
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which results from the martingale condition
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using rolled-over money-market account
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as numeraire.
While serving as a consistent arbitrage-free, risk-neutral framework, short-rate models were lacking in providing a clear picture as to the dynamics of the implied discount factors, forward rates, and swap rates. For example, considering the instantaneous forward rate f (t, T, ω), that is, the forward rate at time t for a forward deposit over [T, T + dT], what can be said about its dynamics? Recalling that
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the above question boils down to computing the dynamics of which is quite an arduous task and analytically intractable except for very few specialized ...

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