Overview
The single factor model is a model that supposes there is only on source of randomness. In the sense of Ito calculus, this means that there is only on Brownian motion, and all the rates derive from that source.Formalization
We suppose that the price of any zero coupon bond follows the given equation
{% dP(t,T) = P(t,T)[\mu(t,T)dt + \sigma(t,T) dW_t] %}
Notice that there is only source of randomness, but at any given time there will be a different equation for each bond, because
of the dependence on {% \mu %} and {% \sigma %}.
We next note the relationship between the price of zero coupon bond and the forward rates, {% F(t,s) %}
{% P(t,T) = e^{\int_t^T f(t,s)ds} %}
where {% F(t,s) %} is the forward rate from s at time t.
then
{% f(t,T) = - \frac{ \partial{} }{ \partial{T}} ln P(t,T) %}
then using Itos lemma
{% df(t,T) = \frac{ \partial{} }{ \partial{T}} (\frac{1}{2} \sigma ^2 (t,T) - \mu(t,T)) dt - \frac{ \partial{} }{ \partial{T}} \sigma(t,T) dW_t %}
this equation is of the form
{% d f(t,T) = \alpha(t,T)dt + \sigma_f(t,T)dW_t %}
restating this to integral form
{% \displaystyle f(t,T) = f(0,T) + \int_0^t \alpha(s,T)ds + \int_0^t \sigma_f(s,T) dW_s %}
It is worth noting that the short rate, {% r(t) %} is just {% f(t,t) %} or we can state:
{% \displaystyle r(t) = f(0,t) + \int_0^t \alpha(s,t) ds + \int_0^t \sigma_f(s,t) dW_s %}
Once we assume no arbitrage, the following relations can be derived:
{% \sigma_f(t,T) = - \frac{ \partial{} }{ \partial{T}} \sigma(t,T) %}
{% \displaystyle \alpha(t,T) = \sigma_f(t,T) \times \int_t^T \sigma_f(t,s) ds %}
(wilmott sect 19.5.1, hull sect 19.5.1)
The no arbitrage assumption puts us in the risk neutral world, that is the zero coupon bond price should evolve with a drift equal to the short rate.
Alternate Assumptions
The above arguments start from the assumption about zero coupon bond prices, and then we derive a formula for forward rates. Alternatively, we could first assume that forward rates follow an equation such as
{% d f(t,T) = \alpha(t,T)dt + \sigma_f(t,T)dW_t %}
The alternate derivation arrives at the same final equations.