# Existence and Uniqueness for the Multivariate Discrete Terminal Wealth Relative

## TL;DR

This paper proves the existence and uniqueness of an optimal multivariate fractional trading strategy in money management, extending previous univariate results and ensuring numerical computability via steepest ascent methods.

## Contribution

It generalizes the univariate fractional trading results to the multivariate case, establishing existence, uniqueness, and computational methods for optimal strategies.

## Key findings

- Proves existence and uniqueness of multivariate optimal f
- Extends univariate fractional trading results
- Ensures numerical solutions via steepest ascent

## Abstract

In this paper the multivariate fractional trading ansatz of money management from Ralph Vince (Portfolio Management Formulas: Mathematical Trading Methods for the Futures, Options, and Stock Markets, John Wiley & Sons, Inc., 1990) is discussed. In particular, we prove existence and uniqueness of an optimal f of the respective optimization problem under reasonable assumptions on the trade return matrix. This result generalizes a similar result for the univariate fractional trading ansatz. Furthermore, our result guarantees that the multivariate optimal f solutions can always be found numerically by steepest ascent methods.