Assume the short rate rt (see Chapter 4) is modeled by an Itô process,
drt = μ(r, t)dt + σ>(r, t)dWt,
where we again use the notation r(T) = rt, Wt 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(rt, 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 instruments1 with different maturities T1 and T2 and corresponding values V1 and V2 (Hull, 2002). By applying the Itô Lemma for an infinitesimal change dπt = dV1 − ΔdV2, we obtain
Choosing, the stochastic terms in the equation above can be eliminated. To avoid arbitrage, we have to use the risk-free rate,
This equality only holds ...