Consider a tenor structure (left{T_{1}, T_{2} ight}) and a bond with maturity (T_{2}) and price given at

Consider a tenor structure \(\left\{T_{1}, T_{2}\right\}\) and a bond with maturity \(T_{2}\) and price given at time \(t \in\left[0, T_{2}\right]\) by

\[P\left(t, T_{2}\right)=\exp \left(-\int_{t}^{T_{2}} f(t, s) d s\right), \quad t \in\left[0, T_{2}\right]\]

where the instantaneous yield curve \(f(t, s)\) is parametrized as

\[f(t, s)=r_{1} \mathbb{1}_{\left[0, T_{1}\right]}(s)+r_{2} \mathbb{1}_{\left[T_{1}, T_{2}\right]}(s), \quad t \leqslant s \leqslant T_{2}\]

Find a formula to estimate the values of \(r_{1}\) and \(r_{2}\) from the data of \(P\left(0, T_{2}\right)\) and \(P\left(T_{1}, T_{2}\right)\). Same question when \(f(t, s)\) is parametrized as

\[f(t, s)=r_{1} s \mathbb{1}_{\left[0, T_{1}\right]}(s)+\left(r_{1} T_{1}+\left(s-T_{1}\right) r_{2}\right) \mathbb{1}_{\left[T_{1}, T_{2}\right]}(s), \quad t \leqslant s \leqslant T_{2}\]