We are interested in the change dV of the value of an interest rate instrument V(r , t) in an infinitesimally short time interval dt. We set up a self-replicating portfolio containing two interest rate instruments with different maturities T1 and T2 and corresponding values V1 and V2 and apply Ito's Lemma to obtain
we can get rid of the stochastic terms in the equation above. To avoid arbitrage we use the risk free rate
and with some rearranging
This equality only holds when both sides of the equation only depend on r and t and not on product specific quantities, thus we get the following PDE for our short rate models
In our next blog post we will take a look on the different terms of this equation.