Existence of solutions to the Heath-Jarrow-Morton equation of the bond market with linear volatility and general Lévy random factor is studied. Conditions for 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 the 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.

The paper studies the Heath-Jarrow-Morton-Musiela equation of the bond market. The equation is analyzed in weighted spaces of functions defined on [0, +∞). Sufficient conditions for local and global existence are obtained. For equation with the linear diffusion term the conditions for global existence are close to the necessary ones.

The problem of quantile hedging for basket derivatives in the Black-Scholes model with correlation is considered. Explicit formulas for the probability maximizing function and the cost reduction function are derived. Applicability of the results for the widely traded derivatives as digital, quantos, outperformance and spread options is shown.