Explosive Behavior in the Black–Derman–Toy Model

Abstract

We consider the simulation of the Black–Derman–Toy (BDT) model with log-normally distributed rates in the spot measure, in discrete time and with a continuous state variable. We note an explosive behavior in the Eurodollar futures convexity adjustment at a critical value of the volatility, which depends on maturity, rate tenor, and simulation time step size. In the limit of a very small time step, this singularity appears for any volatility, and reproduces the Hogan–Weintraub singularity, which is generic for short-rate interest rate models with lognormally distributed rates. The singular behavior arises from a region in the state space which is usually truncated off in finite difference and tree methods, or is very poorly sampled in Monte Carlo methods, and thus is not observed under usual simulation methods.