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The problem of existence of arbitrage free and monotone CDO term structuremodels is studied. Conditions for positivity and monotonicity of thecorresponding Heath-Jarrow-Morton-Musiela equation for the $x$-forward rateswith the use of the Milian type result are formulated. Two state spaces aretaken into account - of square integrable functions and a Sobolev space. Forthe first the regularity results concerning pointwise monotonicity are proven.Arbitrage free and monotone models are characterized in terms of the volatilityof the model and characteristics of the driving L\'evy process.

Monotonicity of the collateralized debt obligations term structure model