Short Rate Model

Let us define the cash in the bank at time t, as B(t), paying interest at the rate r(t). After a short time later, defined as ∆(t).

Rearranging equation (1)

As ∆t becomes infinitesimally small, in the limit we can write equation (2) as

Integrating from t to T and applying change of base

which results in

Applying the limits and taking the inverse of the log

On the right hand side of equation (6), if r(t) is deterministic and a constant, the integrand reduces to r[T-t].

However, if r(t) is stochastic (something we will delve into later), then equation (6) becomes stochastic.

This then expands into single factor short rate models (Merton et al) or multi factor short rate models (Hull-White et al).

Again we will expand into these in later sections.