Overview
The money market account is an account that is invested in the overnight rate on money balances. In a mathematical framework, the rate obtained in the money market is an instantaneous rate that is coninuously compounded.Definition
The rate obtained by the money market account is denoted {% r(t) %}. Then the value of $1 invested at time {% t=0 %} is given by
{% \displaystyle B(t) = exp(\int_0^t r(u)du) %}
If the instantaneous rate is assumed to be a constant, the value of the account is computed to be
{% B(t) = exp(rt) %}
Discounting
When used in risk neutral pricing, the discount appears as
{% \displaystyle exp(\int_0^t -r(u)du) X(t) %}
or in the case of constant rate
{% exp(-rt) X(t) %}