0.1 Structure
Due to the challenges of modeling bond prices, denoted \(p(t,T)\), we instead of to modeling the forward rate \(f(t,T)\) and short rate \(r(t)\)
Modeling short rate \(r(t)\) is simpler since it does not contain \(T\). Typically, the dynamics of \(\{r_{t}\}\) is given by \[ dr_{t}=\mu(t,r_{t})dt+\sigma(t, r_{t})dW_{t} \] from which we can recover the price of the bond through \[ p(t,T)=\mathbb{E}_{\mathbb{Q}}\left( \exp\left( -\int _{t}^Tr_{s} \, ds \right) \middle| \mathcal{F}_t \right) \] which is the discounted expected payoff where \(\mathbb{Q}\) is a martingale measure.