0.1 Structure & Bond Price
Te Vasicek Model picks [[ornstein-uhlenbeck-process|OU]] dynamics to model the short rate under risk-free measure \(\mathbb{Q}\) as \[ dr_{t}=\alpha(m-r_{t})dt+\sigma dW_{t}^\mathbb{Q} \] where \(W_{t}^\mathbb{Q}\) is \(\mathbb{Q}\)-BM and \(m\) is the risk-adjusted mean reversion level \[ m=m_{\infty}- \frac{\lambda \sigma}{a}, \] where \(m_{\infty}\) is the long-run mean interest level and \(\lambda\) is the assumed constant market price of interest rate risk.
The bond price under this model is given by \[ \begin{align} p(t,T) & = \mathbb{E}^\mathbb{Q}\left[ e^{-\int _{t}^Tr_{s} \, ds }\middle| r_{t}=x \right] \\ & = \exp\left\{ - \frac{1}{\alpha}(1-e^{\alpha(t-T)})r_{t}+\left( m- \frac{\sigma^2}{2\alpha^2} \right)\left[ \frac{1}{\alpha}(1-e^{\alpha(t-T)}) -(T-t)\right] - \frac{\sigma^2}{4\alpha}\left[ \frac{1}{\alpha}(1-e^{\alpha(t-T)}) \right]^2\right\}. \end{align} \] For the derivation see pg25 of Haoshengs notes: [[pstat223b-revision-notes-haosheng-pdf]].
0.2 Results
To be updated.
0.3 Comments