An Empirical Study of the Convergence Properties of the Non-Recombining HJM Forward Rate Tree in Pricing Interest Rate Derivatives

Convergence rate and accuracy of the non-recombining HJM forward rate tree are tested by constructing a tree for the forward rate process equivalent to the Hull-White short rate process. Put option values on a ten-year discount bond from the forward rate tree are compared to the accurate values obtained from a recombining short rate lattice. European option values from the HJM tree converge to the true value in twelve steps for all option maturities up to twenty-five years. American option values are within a cent or two of the accurate values for one and five-year options, but do not converge to the accurate value in twenty-four steps, the maximum attempted, for higher maturities. At-the-money options are underpriced by one percent for ten-year maturity and by more than three percent for twenty-five year maturity. Out-of-the-money options are underpriced by up to nine percent. Results are independent of the shape of the initial term structure. Using an HJM tree with equal stepsizes leads to more accurate values for European options, but a tree with linearly increasing stepsizes performs better in the case of American options. It is found optimal to have the same number of forward rates maturing per year beyond option maturity as the number of steps per year through option maturity