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?
Finite Difference Methods in Financial Engineering
Page276 $\endgroup$ – A.Oreo Feb 8 '17 at 1:57