On some semi-parametric estimates for European option prices

We show that an estimate by de la Pena, Ibragimov, and Jordan for ${\mathbb{E}}(X-c)<^>+$ , with c a constant and X a random variable of which the mean, the variance, and $\mathbb{P}(X \leqslant c)$ are known, implies an estimate by Scarf on the infimum of ${\mathbb{E}}(X \wedge c)$ over the set of positive random variables X with fixed mean and variance. This also shows, as a consequence, that the former estimate implies an estimate by Lo on European option prices.