Discount-Bond Derivatives on a Recombining Binomial Tree

Interest-rate derivative models governed by parabolic partial differential equations (PDEs) are studied with discrete-time recombining binomial trees. For the Buehler-Kaesler discount-bond model, the expiration value of the bond is a limit point of tree sites. Representative calculations give a close approximation to the continuum results. Next, situations are considered in which spatial inhomogeneity of the drift velocity can cause binomial jump probabilities to become negative. When the continuous-time boundary conditions are applied near the tree points at which this occurs, good agreement is obtained with Hull and White's explicit-finite-difference treatment of the Cox- Ingersoll-Ross model. Finally, to mimic the effect of a drift-velocity divergence which prevents interest rates from becoming negative, Neumann boundary conditions are applied in the Vasicek model. Discrete-time computations are performed for a mean-reverting situation and for a case with constant negative short-rate drift; the ensuing bond values have nonnegative interest rates and forward rates. The results are compared with the Vasicek solution and with the leading term in a spectral expansion.