A Note on the Quantile Formulation

Abstract

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.

Quantile Optimization Under Derivative Constraint

“…To the best of our knowledge, we have never seen anyone else has done this. This also allows us to give a similar ODE interpretation for the optimal quantiles obtained in [25] and [27].…”

“…We remark that the above argument is invertible, so solving the insurance contract design problem (2.7) reduces to solving the quantile optimization problem (2.10) subject to the constraint (2.11). Xia and Zhou [25] and the author [27] respectively studied the same type of quantile optimization problems as follows.…”

“…In the following sections, we will further develop the author's [27] relaxation method and link this problem to an ordinary differential equation that has been well studied in literature. The optimal quantile will be expressed via the solution of the ODE.…”

“…They demonstrated their approach via solving a quantile optimization problem within the RDUT framework. Shortly after, the author [27] provided another simple way, that is, making change-ofvariable and relaxation method, to solve the same type of problems as in [25]. In this type of problems, the constraints on the decision variables (namely, quantiles) are almost minimum: additional to the monotonicity constraint (which is a must for quantile optimization problems as mentioned earlier), one only requires the so-called budget constraint, which is, mathematically speaking, a one-dimensional integral constraint that can be removed by employing a Lagrange multiplier.…”

Optimal Redeeming Strategy of Stock Loans Under Drift Uncertainty

“…the underlying process. In the context of the classical Black-Scholes framework, in which the underlying asset process follows the geometric Brownian motion with known drift, the theory of financial optimal stopping problems has been well established (see, e.g., [5], [11], [6], [9], [16], [25], [27], [42], [47], [52], [51], and [55]).…”