Forward rate models with linear volatilities

The existence of solutions to the Heath?CJarrow?CMorton equation of the bond market with linear volatility and general Lévy random factor is studied. Conditions for the existence and non-existence of solutions in the class of bounded fields are presented. For the existence of solutions, the Lévy process should necessarily be without a Gaussian part and without negative jumps. If this is the case, then necessary and sufficient conditions for the existence are formulated either in terms of the behavior of the Lévy measure of the noise near the origin or the behavior of the Laplace exponent of the noise at infinity.