Research Article Open Access

Drawdown and Drawup of Bi-Directional Grid Constrained Stochastic Processes

Aldo Taranto1 and Shahjahan Khan1
  • 1 University of Southern Queensland, Australia
Journal of Mathematics and Statistics
Volume 16 No. 1, 2020, 182-197

DOI: https://doi.org/10.3844/jmssp.2020.182.197

Submitted On: 14 May 2020 Published On: 14 September 2020

How to Cite: Taranto, A. & Khan, S. (2020). Drawdown and Drawup of Bi-Directional Grid Constrained Stochastic Processes. Journal of Mathematics and Statistics, 16(1), 182-197. https://doi.org/10.3844/jmssp.2020.182.197

Abstract

The Grid Trading Problem (GTP) of mathematical finance, used in portfolio loss minimization, generalized dynamic hedging and algorithmic trading, is researched by examining the impact of the drawdown and drawup of discrete random walks and of Itô diffusions on the Bi-Directional Grid Constrained (BGC) stochastic process for profit Pt and equity Et over time. A comprehensive Discrete Difference Equation (DDE) and a continuous Stochastic Differential Equation (SDE) are derived and proved for the GTP. This allows fund managers and traders the ability to better stress test the impact of volatility to reduce risk and generate positive returns. These theorems are then simulated to complement the theoretical models with charts. Not only does this research extend a rich mathematical problem that can be further researched in its own right, but it also extends the applications into the above areas of finance.

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Keywords

  • Grid Trading Problem (GTP)
  • Bi-Directional Grid Constrained (BGC)
  • Random Walks
  • Itô Diffusions
  • Probability of Ruin
  • Maximal Drawdown
  • Maximal Drawup
  • Discrete Difference Equation (DDE)
  • Stochastic Differential Equation (SDE)