Hogan–Weintraub singularity and explosive behaviour in the Black–Derman–Toy model

Abstract

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