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

Aurélien Alfonsi 1, 2 Jacopo Corbetta 1 Benjamin Jourdain 1, 2
2 MATHRISK - Mathematical Risk Handling
UPEM - Université Paris-Est Marne-la-Vallée, ENPC - École des Ponts ParisTech, Inria de Paris
Abstract : 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 i=1 δX i and νJ = 1 J J j=1 δY j , 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 and 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.
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Dernière modification le : mercredi 26 février 2020 - 19:06:17
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Aurélien Alfonsi, Jacopo Corbetta, Benjamin Jourdain. Sampling of one-dimensional probability measures in the convex order and computation of robust option price bounds. International Journal of Theoretical and Applied Finance, World Scientific Publishing, 2019, 22 (3), ⟨10.1142/S021902491950002X⟩. ⟨hal-01963507⟩

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