A Note on Stochastic Dominance and the Omega Ratio

We first show that second-order stochastic dominance (SSD) and/or second-order risk-seeking stochastic dominance (SRSD) alone for any two prospects is not sufficient to imply the Omega ratio of one asset is always greater than that of the other one. We then extend the theory of risk measures by proving that the preference of second-order stochastic dominance implies the preference of the corresponding Omega ratios only when the return threshold is less than the mean of the higher-return asset. On the other hand, the preference of second-order risk-seeking stochastic dominance implies the preference of the corresponding Omega ratios only when the return threshold is bigger than the mean of the smaller-return asset. Nonetheless, the preference of first-order stochastic dominance does imply the preference of the corresponding Omega ratios for any return threshold