# 5
Upwinding Techniques for Short Rate Models

## 5.1 DERIVATION OF A PDE FOR SHORT RATE MODELS

Assume the short rate *r*_{t} (see Chapter 4) is modeled by an Itô process,

*dr*_{t} = μ(*r, t*)*dt + σ*>(*r, t*)*dW*_{t},

where we again use the notation *r*(*T*) *= r*_{t}, W_{t} is a Brownian motion, and μ and *σ*> are functions to be defined later.

We are interested in the change *dV* of the value of an interest rate instrument *V*(*r*_{t}, *T*) in an infinitesimally short time interval *dt.* Again, we utilize Itô’s Lemma and try to use the same kind of analysis that has been successful in deriving the Black-Scholes PDE (PS-PDE), where we got rid of the stochastic terms in the BS-PDE by Δ-hedging.

We set up a self-replicating portfolio *π* containing two interest rate instruments^{1} with different maturities *T*_{1} and *T*_{2} and corresponding values *V*_{1} and *V*_{2} (Hull, 2002). By applying the Itô Lemma for an infinitesimal change *dπ*_{t} *= dV*_{1} − Δ*dV*_{2}*,* we obtain

Choosing, the stochastic terms in the equation above can be eliminated. To avoid arbitrage, we have to use the risk-free rate,

Rearranging equations (5.3)-(5.5) yields

This equality only holds ...