One-factor short-rate models

Abstract

The theory of interest-rate modeling was originally based on the assumption of specific one-dimensional dynamics for the instantaneous spot rate process r. Modeling directly such dynamics is very convenient since all fundamental quantities (rates and bonds) are readily defined, by no-arbitrage arguments, as the expectation of a functional of the process r. Indeed, the existence of a risk-neutral measure implies that the arbitrage-free price at time t of a contingent claim with payoff H_T at time T is given by

$$ H_t = E_t \{ D(t,T)H_T \} = E_t \left\{ {e^{ - \int_t^T {r(s)ds} } H_T } \right\}, $$

(3.1)

with E_t denoting the time t-conditional expectation under that measure. In particular, the zero-coupon-bond price at time t for the maturity T is characterized by a unit amount of currency available at time T, so that H_T = 1 and we obtain

$$ P(t,T) = E_t \left\{ {e^{ - \int_t^T {r(s)ds} } } \right\}. $$

(3.2)