# A mathematical approach to fractional trading : using the terminal wealth relative with discrete and continuous distributions

• Eine mathematische Herangehensweise an das Fractional Trading unter Verwendung des Terminal Wealth Relative mit diskreten und stetigen Verteilungen

Hermes, Andreas; Maier-Paape, Stanislaus (Thesis advisor); Zhu, Qiji Jim (Thesis advisor)

Aachen (2016, 2017)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2016

Abstract

The "optimal f" trading model based on discrete historical trading returns was introduced by Ralph Vince [Vin90, Vin08, Vin09] as an optimization approach for the "fixed fractional" money management strategy. Here a trader wants to invest a fixed percentage of his current capital for future investments. Since the percentage of the current capital used per trade is fixed, the absolute height of the capital to invest for each new trade depends on the outcomes of past investments. To determine an optimal fraction for this strategy, Vince maximizes the Terminal Wealth Relative (TWR) defined on a discrete set of historical trading returns. The TWR for a given fraction f represents the gain or loss obtained after the occurrence of the given historical trade returns when risking a biggest loss of a percentage of f of the current capital per trade.In this thesis the univariate and multivariate TerminalWealth Relative are analyzed and existence and uniqueness proofs for the optimal solutions of the optimization problems for discrete and continuous settings are given.The thesis is split into four parts. The first part (Chapter 2) recapitulates the univariate (i.e. based on one underlying trading system) discrete model and cites an existence and uniqueness proof of an optimal solution by Maier-Paape [MP13]. The TWR is extended to a generalized version based on continuous distribution functions for the (historical) trades similar to the approach of Zhu [Zhu07]. The existence and uniqueness of an optimal solution for the optimization problem for this extended TWR is proved. The connection between the discrete and the continuous models is established, justifying the approximation of the optimal solution of the continuous model by using the discrete model. Chapter 3 transfers the results from the previous chapter to a risk-averse modification of the univariate continuous model using a constraint for the Deepest Drawdown. Again the connection between the discrete and the continuous approach is demonstrated using stochastic calculus. Chapters 4 and 5 cover the multivariate models, i.e. the TWR is defined based on several underlying trading systems. In Chapter 4 the existence and uniqueness of an optimal solution of the optimization problem for the discrete multivariate TWR is proved, whereas Chapter 5 extends the TWR to a generalized version based on continuous distribution functions. This model is introduced using multidimensional Stieltjes integration (cf. [Pro12] and [Owe05]). An existence and uniqueness result is proved for this multivariate continuous model as well.