Explosion in the quasi-Gaussian HJM model

Abstract

We study the explosion of the solutions of the SDE in the quasi-Gaussian HJM model with a CEV-type volatility. The quasi-Gaussian HJM models are a popular approach for modeling the dynamics of the yield curve. This is due to their low dimensional Markovian representation which simplifies their numerical implementation and simulation. We show rigorously that the short rate in these models explodes in finite time with positive probability, under certain assumptions for the model parameters, and that the explosion occurs in finite time with probability one under some stronger assumptions. We discuss the implications of these results for the pricing of the zero coupon bonds and Eurodollar futures under this model.

Small-noise limit of the quasi-Gaussian log-normal HJM model

“…The solutions of the process (5) may explode with non-zero probability. This will be discussed in [14]. When the volatility σ = 0, there is no explosion.…”

The investor problem based on the HJM model

“…Note that in affine models (for the definition see the next section) taking ε → 0 is an instantenous operation and does not need separate justification. For more information about such models we refer to Pirjol [29].…”