Overview
Definition
The value of the short term bond that is re-invested at every period is given by.
{% \displaystyle B(n \Delta t) = (1+r_1)(1+r_2)...(1+r_n) %}
Notice that this formula gives the value of the bond at the end of the {% n^{th} %}
period. Typically, the short term bond is chosen as a numeraire when the asset being
priced has well defined values at the end of one of the periods.
Pricing the Fixed Income Curve
The short term bond model can be used to obtain a no-arbitrage yield curve consistent with the dynamics of the short term bond. In particular, to price the value of a zero coupon bond maturing in n-periods from a short term bond that has a length of one period.
{% Discount = \mathbb{E}_{\mathbb{Q}}[1/(1+r_1)(1+r_2)...(1+r_n)] %}
That is, if one can construct a short term bond model that has a period that divides the period of the maturity of
the zero coupon bond that one wishes to price, one can derive the arbitrage free discount rate of that zero coupon bond,
consistent with the short term bond model.