Three ways to solve for bond prices in the Vasiček model (Q705419)

One of the classical stochastic models for interest rates is the Vasiček model. It allows the short-term interest rate to follow a random walk. Though not too realistic (the interest rate can go negative) it has become very popular for its simplicity and tractability, i.e., it produces closed-form solutions for the prices of bonds and many other interest rate derivatives. This article presents three reviewed ways of obtaining an explicit formula for the bond price in the Vasiček model. One takes advantage of the distributional properties of the short-rate process. The second revisits the original derivation obtaining the bond price PDE (partial differential equation) using a martingale-oriented methodology. Then the PDE is solved by integrating ordinary differential equations. The third alternative relies on the Heath-Jarrow-Morton pricing framework describing the bond price dynamics under the forward measure.