Why there is some inhomogeneous term in the PDE of fixed income

Question

We consider one factor driving model of fixed income product say short-term interest $r(t)=\lim\limits_{T\rightarrow t} R(t,T),$ $R(t,T)$ is yield i.e $$B(t,T)e^{(T-t)R(t,T)} = 1$$ Then we see several PDE of contingent claim

Zero-coupon bond $B(t,T)$ $$\dfrac{\partial B}{\partial t} + LB -r(t)B = 0$$ here $L$ is the differential operator in Feynman-Kac equation.

Swap of fixed rate $r^*$ $$\dfrac{\partial V}{\partial t} + LV -r(t)V + (r - r^*) = 0$$

Caplet at rate $r*$ $$\dfrac{\partial V}{\partial t} + LV -r(t)V + \min(r,r^*) = 0$$

Floorlet at rate $r*$ $$\dfrac{\partial V}{\partial t} + LV -r(t)V + \max(r,r^*) = 0$$

Here $r = r(t)$ and $V(t,T,r(t))$ is the value of contingent claim which is the function of $t$ and $r$ e.g, for zero-coupon bond $V=B.$

I couldn't understand when the dynamic of $r(t)$ is given, why there are some inhomogeneous terms in the Black-Scholes equation? Can some one explain one of later three?

Answer 1

Check out https://en.wikipedia.org/wiki/Feynman%E2%80%93Kac_formula, it is the $f(X_r,r)$ in the formula. For the swap you receive $(r-r^*) dt$ (assume notional of 1) which translates into your $f$ (inhomogeneous term).