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

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

and then price derivatives C(t, ω) with a terminal payoff at T using risk-neutral expectation:

which results from the martingale condition

using rolled-over money-market account

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

the above question boils down to computing the dynamics of which is quite an arduous task and analytically intractable except for very few specialized ...