Interest Rate Models - From SDE to PDE

After our review of different equity models in the physics's friday blog post series we now turn to the family of interest rate models. Today's model I want to show the derivation of a PDE (partial differential equation) for a short rate model, where the SDE (stochastic differential equation) is given by

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

Choosing

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. 

Happy Easter