Sufficient conditions of stochastic dominance for general transformations and its application in option strategy

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 (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. JEL C51 D81


Introduction
Stochastic dominance (SD) has been proved to be a powerful tool for ranking random variables and is employed in various fields such as finance, decision analysis, economics and statistics etc. (cf., Levy, 1992Levy, , 2006Chakravarty and Zoli, 2012;Jouini et al., 2013;Tsetlin et al., 2015;Post et al., 2015 andPost, 2016;Gao and Zhao, 2017). The SD rules indicate when one random variable is to be ranked higher than another by specifying a condition which the difference between their cumulative distribution functions (CDFs) must satisfy. However, economic and financial activities usually induce transformations of an initial risk, and the classical SD rules are inefficient in ranking such transformations. Transformations of random variables have been discussed in the early stochastic dominance literature, especially in the risk analysis portion. For example, Sandom (1971) has used a particular linear, risk altering, transformation in discussing the comparative statics of risk. Hadar andRussell (1971, 1974) have dealt with special cases of the transformation question, emphasizing its use in dealing with portfolios of random variables. Cheng, Magill, and Shafer (1987) have used the transformation approach to address the comparative statics of first degree stochastic dominance shifts in a random variable within a general decision model context. Meyer (1989) has proposed the first and second stochastic dominance (FSD and SSD) criteria for the increasing, continuous, and piecewise differentiable transformations on continuous random variables. Meyer goes on to analyze the transformation resulting from coinsurance, the transformation resulting from holding a stock with the corresponding call option, or even holding call and put options simultaneously. Gao and Zhao (2017) have developed FSD and SSD criteria for monotonic transformations on discrete random variables, and they apply these results in ranking transformations resulting from pension funds. These applications indicate that the transformation approach is useful in discussing comparatives statics of random variable changes and financial issues.
For the general transformations, Levy (1992) has given several sufficient conditions under which one transformation dominates another by FSD and SSD. Hereafter, some authors discuss the transformations of different random variables (cf., Trannoy, 2007, 2012;Denuit et al., 2013). To the best of our knowledge, Theorem 5 of Levy (1992) is the only result on the stochastic dominance for general transformations. However, we have found that its dominance condition for SSD is not sufficient and its dominance condition for FSD can be relaxed. Then, by restricting attention to the monotone property of the dominating transformation, we present a revised exact sufficient condition for one transformation dominating another. Next, we further extend the stochastic dominance criteria to the most general transformations.
3 Moreover, we further generalize these stochastic dominance criteria for transformations on continuous random variables to the discrete case. Finally, we employ the SD approach to analyze the transformations resulting from holding a stock with the corresponding call option.
The paper is organized as follows. Section 2 presents a counterexample to show that Levy's theorem about SSD does not hold. By discussing the monotone property of the dominating transformation, Section 3 derives the exact sufficient condition for one transformation dominating another by SSD. Section 4 deduces the stochastic dominance criteria for the most general transformations, which further perfect and improve Levy's result and extend Meyer's result to more general case. Section 5 further provides the stochastic dominance criteria for transformations on discrete random variables. Section 6 analyzes the transformations resulting from holding a stock with the corresponding call option. Section 7 concludes the paper.

Levy's sufficient conditions and its counterexample
Suppose that X is a continuous random variable with support in the finite interval [ , ] a b  . To facilitate the narrative, we will refer to transformed random variables, derived by applying transformation functions to X , as transformations on X , or shortly transformations. Obviously, the classical SD rules rely heavily on CDFs. But in most cases CDFs of transformations are difficult to compute n F x P n X x   , and the frequently-used integration by parts is invalid in this case. Thus, the classical SD rules based on the CDFs framework lose their great charm when dealing with the transformations. In order to determine the SD relations between two general transformations, Levy (1992) gives several sufficient conditions under which one transformation dominates the other by FSD and SSD, the main result is shown as follows.
Alleged Theorem 5. (Levy, 1992) Given a random variable X with the density ( ) f x and support in the Similarly, the dominance condition for SSD is given by Note: For simplicity, we assume that the range of the random variable is finite. Actually, the stochastic dominance criteria can easily be extended to the infinite range by mathematical skills (see Hanoch and Levy, 1969).

4
Although Levy's Theorem 5 only proposes the sufficient conditions for FSD and SSD relations between two transformed random variables, its really meaningful contribution is that it tries to represent the SD rules by the transformation functions and the density function of the original random variable, rather than by CDFs of the transformed random variables. To better illustrate this meaning, Figure 1 to Figure 4 show the relationship between the method of Levy's Theorem 5 and that under the framework of CDFs for the uniformly distributed random variable.
•The comparative diagrams for FSD  (2) (2) in Alleged Theorem 5. But for the increasing and concave utility n X by SSD. By carefully analyzing Theorem 5 of Levy (1992), we find that the monotone property of the dominating transformation is inevitable for stochastic dominance of transformations.
Actually, we have proved the following conclusion.

A revised sufficient condition for SSD
In this part, we will revise Theorem 5 of Levy (1992) and derive the exact sufficient condition for one transformation dominating another by SSD.
Proof. See Appendix A. 6 By restricting the monotonicity and differentiability of the dominating transformation, Theorem 1 provides the exact sufficient condition for one transformation dominating another by SSD. Compared with Theorem 5 of Levy (1992), Theorem 1 gives a revised dominance condition concerning SSD, so it can be viewed as an primary improvement of Theorem 5 in Levy (1992).
Furthermore, in the next paragraph we will prove that the FSD condition listed in Alleged Theorem 5 and the SSD condition listed in Theorem 1 can be weakened via complicated mathematics skill.

Stochastic dominance criteria for general transformations
Theorem 5 in Levy(1992) and Theorem 1 of this paper give the dominance condition under which one transformation dominates another by FSD and SSD. In the following, we will prove that these conditions can be relaxed to a more general case. That is, the restrictions to the dominating transformation in Theorem 1 can be further relaxed.
Proof. See Appendix B.
Theorem 2 provides two dominance conditions under which one transformation dominates another by FSD or SSD for the most general transformations. Compared with Theorem 5 of Levy (1992), in Theorem 2(1) points are permitted to violate the dominance condition (1) only if they constitute a set of measure zero. So, Theorem 2(1) reduces the dominance condition for FSD in Theorem 5 of Levy (1992). Compared with Theorem 1, Theorem 2(2) only requires the dominating transformation to be increasing, and the property of differentiability is not necessary.
Moreover, only the increasing property is considered in Theorem 2, and we can derive a similar conclusion if the dominating transformation is decreasing. Obviously, Theorem 2 and Theorem 3 extend Meyer's result to a more general case. Either the dominating or the dominated transformation in Meyer (1989) is assumed to be increasing, continuous, and piecewise differentiable. However, in Theorem 2 and Theorem 3, only the dominating transformation is assumed to be monotonous, and there are no any other restrictions to both the dominating and the dominated transformation.
Apparently, the differentiability is redundant. Furthermore, Theorem 3 considers the decreasing transformation that is absence in Meyer's result.
Remark 2. The concept of increasing risk and increasing n th degree risk, introduced by Rothschild and Stiglitz (1970) and Ekern (1980), play an important role in risk analysis. It requires that all the random variables to be compared have the same mathematical expectations. Given this supposition, we can easily induce the following conclusion from Theorem 2 and Theorem 3.

Corollary. Given a random variable
(2) Supposing that ( ) m x is decreasing in [ , ] a b , ( ) m X has more increasing risk than ( ) n X if From this corollary, we can easily deduce that there exist a kind of risk transformations which lead to the SSD relation which is completely opposite to the conclusion of Theorem 5 in Levy (1992).
Example 2. Assume that the random variable X satisfies standard normal distribution. Define Obviously, ( ) m x and ( ) n x satisfy condition (2). According to Theorem 5 of Levy (1992), it should be concluded that ( ) m X dominates ( ) n X by SSD. But, the truth is on the opposite side. Actually, by Theorem 3, it is easy to prove the fact that ( ) n X dominates ( ) m X by SSD since

Stochastic dominance criteria for general transformations on discrete random variables
By discussing Levy's dominance conditions for one transformation dominating another by FSD or SSD, we obtain several stochastic dominance criteria for transformations, which perfect and improve Levy and Meyer's results. It must be pointed out that all the conclusions, whether Levy and Meyer's results or the stochastic dominance criteria developed in the paper, are concentrating on transformations of continuous random variables. Actually, there exist similar stochastic dominance criteria for transformations on discrete random 8 variables. Gao and Zhao (2017) have discussed the stochastic dominance relationship between two transformations on discrete random variables, and presents several sufficient conditions for ranking transformations on discrete random variables by FSD or SSD. Such conclusions can be summarized in the following theorem.
Theorem 4. Let X be a discrete random variable whose prospects are characterized by The proofs of the first three items in Theorem 4 are in Gao and Zhao (2017), and the proofs of the last two items follow from them and are omitted. Theorem 4 presents several dominance conditions for ranking transformations on discrete random variables by FSD or SSD, and it overcomes the drawbacks of Meyer and Levy's results that cannot deal with transformations on discrete random variables.

Applications in the option strategy
It is well-known that put and call option contracts can modify the value of common stock. These contracts provides the buyer of the option with the right to either buy (call) or sell (put) shares of common stock at a fixed price referred to as the striking price. On the other hand, the seller of such an option contract incurs the obligation to either sell or buy the common stock at the agreed upon striking price if the contract purchaser 9 decides to exercise the option. To model one such option transaction using the transformation notation, let X represent the random value of 100 shares of a given common stock and assume that its support is the interval Of course, experience in choosing option strategies with varying sizes for the striking price indicates that it is unlikely for the option price charged to be smaller with lower striking price. Furthermore, it is typical for the reduction in the option price to be a fraction of the increase in the striking price. Thus it is further assumed Theorem 2 we deduce that ( ) m X dominates ( ) n X by SSD. That is, if the mean value of ( ) m X is at least large as the mean value of ( ) n X , then ( ) m X is a better choice for all risk-averse investors.
10 While this example deals with the selling of a call option, the purchase of a put option contract can also be modeled using a similar transformation. One can also model the simultaneous purchase or sale of put or call contracts with different striking prices, although the transformations involved become cumbersome.

Conclusion
We first present a counterexample to show that Levy's result with respect to SSD does not hold. Then, we give the revised exact dominance condition for one transformation dominating another by SSD. Next, we propose several stochastic dominance criteria for the most general transformations, which can be viewed as a further improvement of Theorem 5 in Levy (1992). Moreover, we further generalize these stochastic dominance criteria for transformations on continuous random variables to the discrete case. Finally, we employ the SD approach to analyze the transformations resulting from holding a stock with the corresponding call option.
Whether on theory or in applications, much can still be done concerning transformations and stochastic dominance. It would be useful to extend such stochastic dominance criteria to transformations on more than one random variables and to consider higher-degree SD rules for transformations. In addition, we will further apply these results to the analysis of transformations resulting from economic and financial issues. derive the conclusion that ( ( ( )) ( ( ( )) 0 E u m X E u n X   . □