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?

  • $\begingroup$ Can you please provide more background information? Where are these equations from? What is $V$ for each case? Is $r=r(t)$? $\endgroup$ – Gordon Feb 7 '17 at 20:56
  • 2
    $\begingroup$ @Gordon please see the updated version, and the question is in Duffy's book Finite Difference Methods in Financial Engineering Page276 $\endgroup$ – A.Oreo Feb 8 '17 at 1:57
  • $\begingroup$ See my answer to your another question. $\endgroup$ – Gordon Mar 19 '17 at 1:14

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).

| improve this answer | |
  • $\begingroup$ Conversely for a ZCB it is zero because you receive nothing (no cash flows) during the life of the ZCB. $\endgroup$ – noob2 Feb 8 '17 at 18:42
  • $\begingroup$ Sorry, you mean we receive the cash flow continuously? What I have learnt is for each time interval $[T_i,T_{i+1}]$ we receive the money at $T_{i+1}.$ Is it the discrete case? $\endgroup$ – A.Oreo Feb 9 '17 at 2:09
  • $\begingroup$ Hi @user26484. Thanks for your contribution, could please use the LaTeX formatting to make your comment clearer in the future? Thanks $\endgroup$ – Quantuple Feb 10 '17 at 9:33

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.