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Modeling Maximum drawdown Records with Piecewise Deterministic Markov Processe in Capital Markets

Rolando Rubilar-Torrealba, Lisandro Fermin, Soledad Torres

TL;DR

This work develops a PDMP-based framework to model the sequence of maximum drawdown records in capital markets, connecting record theory with regime-switching dynamics. It derives analytical expressions for the mean and variance of the drawdown-record process, and provides simulation and maximum-likelihood estimation procedures to infer PDMP parameters from financial data. The methods are illustrated with a two-regime analysis of the S&P 500, demonstrating regime-dependent jump behavior and the practical viability of PDMP-based risk modeling. The approach offers a principled way to quantify drawdown risk and to simulate and estimate the stochastic structure underlying drawdown records, with potential extensions to more general waiting-time distributions and jump specifications.

Abstract

We propose to model the records of the maximum Drawdown in capital markets by means a Piecewise Deterministic Markov Process (PDMP). We derive statistical results such as the mean and variance that describes the sequence of maximum Drawdown records. In addition, we developed a simulation study and techniques for estimating the parameters governing the stochastic process, using a practical example in the capital market to illustrate the procedure.

Modeling Maximum drawdown Records with Piecewise Deterministic Markov Processe in Capital Markets

TL;DR

This work develops a PDMP-based framework to model the sequence of maximum drawdown records in capital markets, connecting record theory with regime-switching dynamics. It derives analytical expressions for the mean and variance of the drawdown-record process, and provides simulation and maximum-likelihood estimation procedures to infer PDMP parameters from financial data. The methods are illustrated with a two-regime analysis of the S&P 500, demonstrating regime-dependent jump behavior and the practical viability of PDMP-based risk modeling. The approach offers a principled way to quantify drawdown risk and to simulate and estimate the stochastic structure underlying drawdown records, with potential extensions to more general waiting-time distributions and jump specifications.

Abstract

We propose to model the records of the maximum Drawdown in capital markets by means a Piecewise Deterministic Markov Process (PDMP). We derive statistical results such as the mean and variance that describes the sequence of maximum Drawdown records. In addition, we developed a simulation study and techniques for estimating the parameters governing the stochastic process, using a practical example in the capital market to illustrate the procedure.

Paper Structure

This paper contains 17 sections, 3 theorems, 60 equations, 8 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Model and preliminaries
  3. Piecewise Deterministic Markov Process
  4. Markov Process
  5. Definition of record process
  6. Statistical properties of record process
  7. Mean
  8. Case of one-state
  9. Case of two-state
  10. Variance estimation
  11. Case of one-state
  12. Case of two-state
  13. Simulation and estimation in the record process
  14. Simulation of record process
  15. Estimation of the parameters of PDMP record process
  16. ...and 2 more sections

Key Result

Lemma 1

The records process $(R_t)_{t \in \mathbb{R}_+}$ defined in eq.1.2 with initial value $R_0\in[0,1[$ satisfy where the process $(R^0_t)_{t \in \mathbb{R}_+}$ given by is the records process obtained when the initial condition is zero, $R_0^0=0$.

Figures (8)

  • Figure 1: Sample path of the PDMP process $(R_t, \nu_t)$.
  • Figure 2: Evolution of the expected value of $R_t$ process.
  • Figure 3: Evolution of variance of $R_t$ process.
  • Figure 4: Sample path of $R_t$ process for a sample of 10 simulate paths and a large sample of 10,000 simulate paths.
  • Figure 5: Estimated variance of $R_t$ process, $10,000$ simulations
  • ...and 3 more figures

Theorems & Definitions (5)

  • Remark 1
  • Lemma 1
  • Proposition 1
  • Proposition 2
  • Remark 2