Approximate Bermudan option pricing based on the r\'eduite or cubature: soundness and characterisation of perpetual prices as fixed points

In this paper, it is shown that Bermudan option pricing based on either the r\'eduite (in a one-dimensional setting: piecewise harmonic interpolation) or cubature -- is sensible from an economic vantage point: Any sequence of thus-computed prices for Bermudan options with increasing sets of exercise times is increasing. Furthermore, under certain regularity assumptions on the payoff function and provided the exercise times are equidistant of exercise mesh size $h$, it has a supremum which coincides with the least fixed point of the approximate pricing algorithm -- this algorithm being perceived as a map that assigns to any real-valued function $f$ (on the basket of underlyings) the approximate value of the European option of maturity $h$ and payoff function $f$.