A Modified Crank-Nicolson Method for Valuing Option Embedded Bonds Using the Hull-White Interest Rate Model

We demonstrate a modified Crank-Nicolson finite-difference diffusion algorithm for valuing option-embedded bonds using the Hull-White model of the short rate process. In particular, this method allows the practitioner to maintain model stability with relatively large values of bond tenor even when inputting both relatively large values of the annualised short-rate standard deviation and vanishingly small values of the mean reversion constant. Indeed, typical finite-difference diffusion methods such as the fully explicit 3–1 and CrankNicolson 3–3 algorithms are demonstrated here to be unstable for values of short-rate standard deviation, σ, and bond tenor, T, such that σ 2T 3 > 3.6 and σ 2T 3 > 17.1 respectively, given sensible bond market tolerance requirements for the model’s accuracy. In what follows, we show how the practitioner must modify these algorithms so that instability only occurs above an arbitrary user-determined value of the product σ 2T 3 . As a concrete example, using a short-rate standard deviation of two points, σ = 0.02, the typical Crank-Nicolson algorithm does not accurately value bonds of tenor greater than 35 years if the number of modelling time steps is less than N ≈ 200. We then demonstrate the general modifications one can make to the Crank-Nicolson algorithm in order to model bonds of any short-rate deviation and any tenor. Our principal result is that for practitioners valuing long-dated option-embedded bonds in the Hull-White paradigm, modifying the diffusion algorithm, either fully-explicit or CrankNicolson, to achieve the desired accuracy is unavoidable