A Note on the Quantile Formulation

13 Pages Posted: 10 Jun 2016

See all articles by Zuo Quan Xu

Zuo Quan Xu

Hong Kong Polytechnic University

Date Written: July 2016


Many investment models in discrete or continuous‐time settings boil down to maximizing an objective of the quantile function of the decision variable. This quantile optimization problem is known as the quantile formulation of the original investment problem. Under certain monotonicity assumptions, several schemes to solve such quantile optimization problems have been proposed in the literature. In this paper, we propose a change‐of‐variable and relaxation method to solve the quantile optimization problems without using the calculus of variations or making any monotonicity assumptions. The method is demonstrated through a portfolio choice problem under rank‐dependent utility theory (RDUT). We show that this problem is equivalent to a classical Merton's portfolio choice problem under expected utility theory with the same utility function but a different pricing kernel explicitly determined by the given pricing kernel and probability weighting function. With this result, the feasibility, well‐posedness, attainability, and uniqueness issues for the portfolio choice problem under RDUT are solved. It is also shown that solving functional optimization problems may reduce to solving probabilistic optimization problems. The method is applicable to general models with law‐invariant preference measures including portfolio choice models under cumulative prospect theory (CPT) or RDUT, Yaari's dual model, Lopes' SP/A model, and optimal stopping models under CPT or RDUT.

Keywords: portfolio choice/selection, behavioral finance, law‐invariant, quantile formulation, probability weighting/distortion function, change‐of‐variable, relaxation method, calculus of variations, CPT, RDUT, time consistency, atomic, atomless/nonatomic, functional optimization problem

Suggested Citation

Xu, Zuo Quan, A Note on the Quantile Formulation (July 2016). Mathematical Finance, Vol. 26, Issue 3, pp. 589-601, 2016, Available at SSRN: https://ssrn.com/abstract=2793598 or http://dx.doi.org/10.1111/mafi.12072

Zuo Quan Xu (Contact Author)

Hong Kong Polytechnic University ( email )

Hung Hom
Kowloon, 0
Hong Kong

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