### Discussion Paper

## Abstract

A counterexample is presented to show that the sufficient condition for one transformation dominating another by the second degree stochastic dominance, proposed by Theorem 5 of Levy (Stochastic dominance and expected utility: Survey and analysis, 1992), does not hold. Then, by restricting the monotone property of the dominating transformation, a revised exact sufficient condition for one transformation dominating another is given. Next, the stochastic dominance criteria, proposed by Meyer (Stochastic Dominance and transformations of random variables, 1989) and developed by Levy (Stochastic dominance and expected utility: Survey and analysis, 1992), are extended to the most general transformations. Moreover, such criteria are further generalized to transformations on discrete random variables. Finally, the authors employ this method to analyze the transformations resulting from holding a stock with the corresponding call option.

## Comments and Questions

The paper “Sufficient conditions of stochastic dominance for general transformations and its application in option strategy” points out the error of the second part of Theorem 5 in Levy (1992), and gives the correction to Levy’s result. As for this paper, I want to emphasize the following three points: ...[more]

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(1) It is well known that Levy (Stochastic dominance and expected utility: Survey and analysis, 1992) is a very important literature in the field of stochastic dominance. In a certain sense, this paper symbolizes the maturation of stochastic dominance theory. To the best of my knowledge, the vast majority of papers concerning on stochastic dominance have cited the paper.

(2) The counterexample in the paper is clearly illustrated the error of Theorem 5 in Levy (1992).

(3) Theorem 1, Theorem 2 and Theorem 3 could indeed be considered as the correction to Theorem 5 in Levy (1992), and the proving process of these theorems is rigorous.

In addition, I also notice that Man-Chung NG has pointed out the mistake of Theorem 4 in Levy (1992) and published his result in Management Science (A remark on third degree stochastic dominance, 2000, 46(3): 870-873).

Based on the above consideration, I recommend this paper should be published in Economics E-Journal.

Thank you for approving my job. Just as you have said, Levy (1992) is a very important literature in the field of stochastic dominance. Although it has been published for more than 20 years and has been cited about 1000 times, the mistake about Theorem 5 in Levy (1992) still ...[more]

... hasn’t been discovered. I hope that this paper will make the readers recognize the mistake.

This paper points out the mistake of Theorem 5 of Levy (Stochastic dominance and expected utility: Survey and analysis, 1992) and corrects the error by proposing Theorem1, Theorem 2 and Theorem 3. Since Levy (1992) has a very important place in the area of stochastic dominance, it is significant to ...[more]

... point out and correct the mistake of Theorem 5 of Levy (1992). In this sense, this paper should be published.

However, there is a little doubt about the paper. Notice that Levy (1992) is a survey paper that refers to earlier sources and Meyer (1989) seems to provide a correct statement. Is it possible that Levy’s Theorem 5 is just wrong for its missing out the restriction to the transformation in Meyer (1989)?

It is really worthy discussing how to distinguish the mistake of Theorem 5 of Levy (1992). For discussion purposes, the relevant contents of Levy (1992) and Meyer (1989) are listed at the end of document (it only could be shown in the attachment). In order to answer this question that ...[more]

... whether the mistake of Theorem 5 of Levy (1992) is a theoretical faultiness or it is only an error in description for missing out the restriction to the transformation in Meyer (1989), we must first ascertain that transformations mentioned in Theorem 5 of Levy (1992) are “the most general transformations” or “the increasing, continuous, and piecewise differentiable transformations”. If the answer is the former, then the discussion paper by Gao and Zhao has proved that the mistake of Theorem 5 of Levy (1992) is a theoretical faultiness.

I believe that transformations mentioned in Theorem 5 of Levy (1992) are “the most general transformations”, and the main reasons are as follows.

1. Although it seems that Theorem 5 of Levy (1992) is a summary of Meyer’s result from the sentence “Meyer's result is summarized in the following theorem”, we must notice that the sentence “Meyer (1989) and Brooks and Levy (1989) deal with the most general transformation m(X) and n(X)” indicates that transformations mentioned in Theorem 5 of Levy (1992) are “the most general transformations”.

As we have already known, Meyer (1989) indeed deals with the increasing, continuous, and piecewise differentiable transformations. The main reason for this contradiction may be that the restrictions to transformations in Meyer (1989) are displayed in neither Theorem 1 nor Theorem 2.

2. From the content point of view, transformations mentioned in Theorem 5 of Levy (1992) are “the most general transformations”.

It is easy to prove that the first part of Theorem 5 in Levy (1992) does hold for “the most general transformations”, so it is nature to think that the second part will still hold.

3. Pay attention to the fact that if transformations mentioned in Theorem 5 of Levy (1992) are increasing, continuous, and piecewise differentiable, then condition (12) is sufficient and necessary for m(X) dominating n(X) by SSD. However, Theorem 5 of Levy (1992) only shows the sufficiency. Obviously, there is a big distinguish between Levy’s Theorem 5 and Meyer’s result. This truth indicates that transformations mentioned in Theorem 5 of Levy (1992) are “the most general transformations”.

4. Let’s consider this question from the logical relationship. Levy has already pointed that Sandom (1971) and Hadar and Russell (1974) discussed the stochastic dominance relations for particular transformations, then the following part should deal with “the most general transformations”. That is, transformations mentioned in Theorem 5 of Levy (1992) should be “the most general transformations”.

After reading the four reasons carefully, I have discussed them with my colleagues. We all accept it and recognize the mistake in Theorem 5 of Levy (1992) is really a theoreticalfaultiness. Thus, we believe that this paper deservers to be published quickly.