PORTFOLIO SELECTION PROBLEMS VIA THE BIVARIATE CHARACTERIZATION OF STOCHASTIC DOMINANCE RELATIONS¹

Abstract

Stochastic dominance (SD) is a very useful tool in various areas of economics and finance. the purpose of this paper is to provide the results of SD relations developed in other areas such as applied probability which, we believe, are useful for many portfolio selection problems. In particular, the bivariate characterization of SD relations given by Shanthikumar and Yao (1991) is a powerful tool for the demand and the shift effect problems in optimal portfolios. the method enables one to extend many results that hold for the case where the underlying lying assets are statistically independent to the dependent case directly. Copyright 1996 Blackwell Publishers.

A Multivariate Extension of Equilibrium Pricing Transforms: The Multivariate Esscher and Wang Transforms for Pricing Financial and Insurance Risks

“…This ordering is stronger than the first order stochastic dominance. See Kijima and Ohnishi (1996) for details. On the other hand, from (1.2), we have…”

An algebraic theory of portfolio allocation

“…There exists a literature on using groups to "build" objects such as functions (Brown, 1989;Ronan, 1989). In the case of a bivariate distribution, Kijima and Ohnishi (1996) and Lapan and Hennessy (2001) have used the most elementary group, reflection through a line, to appended a functional asymmetry to a distribution function such that order could be induced on the optimal portfolio allocation. Perhaps, after some thought, this constructive approach may be extended to the -variate context?…”

“…The classical CAPM model, due to Sharpe (1964) among others, imposes the assumption of multivariate normality so that all uni-dimensional marginals are symmetric up to location and scale parameters and all stochastic interactions are also linear in form. Samuelson (1967aSamuelson ( , 1967b, Brumelle (1974), Hadar, Russell and Seo (1977), McEntire (1984), Landsberger and Meilijson (1990), Kijima and Ohnishi (1996), Kijima (1997), and Lapan and Hennessy (2001) have all identified symmetries of various forms and strengths that are necessary, sufficient, or both when seeking to assert something about the optimal allocation vector for a risk averse expected utility maximizing investor.…”

Pricing of CDOs based on the multivariate Wang transform

“…The CDO quotes are available on the five benchmark tranches trading on the Dow Jones iTraxx index, consisting 15 This is one of advantages in our model. In the standard copula approach, the existence of implied correlations is not guaranteed in general.…”

“…Appendix A shows that the base lambda λ D always exists and is unique for any market price of CDO's. 15 We note that, for standard copula models such as the one-factor Gaussian copula model, default correlation among the underlying names is the only unobservable element and, as a result, the correlation plays a role of risk aversion of investors. The industry convention for this purpose is to assume that ρ i = ρ for all i, and the parameter ρ implied by the market prices, called the implied correlation, is used as in much the same way as implied volatilities for the Black-Scholes model.…”