Interest rates models with log-normally distributed rates in continuous time are known to display singular behavior. For example, Eurodollar futures prices are infinite in the Dothan and Black-Karasinski models, as shown in 1998 by Hogan and Weintraub. These singularities are usually assumed to disappear when the models are simulated in discrete time. Using a precise simulation of the BDT model, we demonstrate that this is true only for sufficiently low volatilities. Eurodollar futures prices explode for volatilities above a critical value. The explosion is due to contributions from a region in state space which corresponds to very large interest rates and is truncated off in usual simulation methods such as trees and finite difference methods. In the limit of a very small simulation time step the explosion appears for any volatility, and reproduces the Hogan-Weintraub singularity of the continuous time model.
Dan Pirjol works in the Model Risk Group at JP Morgan, covering valuation models in commodities. Previously he was with Markit and Merrill Lynch in various roles in modeling and model risk, after doing research in theoretical high energy physics. He is interested in applications of methods from mathematical physics and probability to problems in mathematical finance.
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