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For [Formula: see text] and [Formula: see text] two probability measures on the real line such that [Formula: see text] is smaller than [Formula: see text] in the convex order, this property is in general not preserved at the level of the empirical measures [Formula: see text] and [Formula: see text], where [Formula: see text] (resp., [Formula: see text]) are independent and identically distributed according to [Formula: see text] (resp., [Formula: see text]). We investigate modifications of [Formula: see text] (resp., [Formula: see text]) smaller than [Formula: see text] (resp., greater than [Formula: see text]) in the convex order and weakly converging to [Formula: see text] (resp., [Formula: see text]) as [Formula: see text]. 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 [Formula: see text] and [Formula: see text] in this set, this enables us to define a probability measure [Formula: see text] (resp., [Formula: see text]) greater than [Formula: see text] (resp., smaller than [Formula: see text]) in the convex order. We give efficient algorithms permitting to compute [Formula: see text] and [Formula: see text] (and therefore [Formula: see text] and [Formula: see text]) when [Formula: see text] and [Formula: see text] 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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We provide explicit conditions on the distribution of risk-neutral log-returns which yield sharp asymptotic estimates on the implied volatility smile. We allow for a variety of asymptotic regimes, including both small maturity (with arbitrary strike) and extreme strike (with arbitrary bounded maturity), extending previous work of Benaim and Friz [BF09]. We present applications to popular models, including Carr-Wu finite moment logstable model, Merton's jump diffusion model and Heston's model.

No abstract