Sampling of one-dimensional probability measures in the convex order and computation of robust option price bounds

For μ and ν two probability measures on the real line such that μ is smaller than ν in the convex order, this property is in general not preserved at the level of the empirical measures μI = 1/I∑ _i=1 ^I δ Xi and νJ = 1/J∑ _j=1 ^J δ Yj, where (Xi)1≤i≤I (resp., (Yj)1≤j≤J) are independent and identically distributed according to μ (resp., ν). We investigate modifications of μI (resp., νJ) smaller than νJ (resp., greater than μI) in the convex order and weakly converging to μ (resp., ν) as I,J →∞. According to Kertz & Rösler(1992), the set of probability measures on the real line with a finite first order moment is a complete lattice for the increasing and the decreasing convex orders. For μ and ν in this set, this enables us to define a probability measure μ ∨ ν (resp., μ ∧ ν) greater than μ (resp., smaller than ν) in the convex order. We give efficient algorithms permitting to compute μ ∨ ν and μ ∧ ν (and therefore μI ∨ νJ and μI ∧ νJ) when μ and ν have finite supports. Last, we illustrate by numerical experiments the resulting sampling methods that preserve the convex order and their application to approximate martingale optimal transport problems and in particular to calculate robust option price bounds.