Paper: arXiv 2506.00552

Authors: Pingping Zeng, Gongqiu Zhang, Weinan Zhang

Abstract

Drawdown risk, an important metric in financial risk management, poses significant computational challenges due to its highly path-dependent nature. This paper proposes a unified framework for computing five important drawdown quantities introduced in Landriault et al. (2015) and Zhang (2015) under general Markov models. We first establish linear systems and develop efficient algorithms for such problems under continuous-time Markov chains (CTMCs), and then establish their theoretical convergence to target quantities under general Markov models. Notably, the proposed algorithms for most quantities achieve the same complexity order as those for path-independent problems: cubic in the number of CTMC states for general Markov models and linear when applied to diffusion models. Rigorous convergence analysis is conducted under weak regularity conditions, and extensive numerical experiments validate the accuracy and efficiency of the proposed algorithms.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 4.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical framework establishing computational algorithms and convergence proofs for complex path-dependent drawdown and occupation time metrics using Continuous-Time Markov Chains (CTMCs). While mathematical complexity and methodological novelty are high, empirical rigor is limited primarily to numerical efficiency experiments rather than empirical market data validation.

Research Flowchart

  flowchart TD
    A["<b>Research Goal</b><br>Compute 5 path-dependent drawdown quantities under general Markov models"] --> B["<b>Framework & Modeling</b><br>Formulate linear systems under Continuous-Time Markov Chains (CTMCs)"]
    
    C["<b>Inputs</b><br>Model parameters for General Markov, Diffusion, and Multi-Asset models"] --> B
    
    B --> D["<b>Numerical Algorithms</b><br>Solve linear systems with high efficiency<br><i>(O(N³) for CTMCs, O(N) for Diffusions)</i>"]
    
    D --> E["<b>Convergence Analysis</b><br>Prove theoretical convergence under weak regularity conditions"]
    
    E --> F["<b>Outcomes & Validation</b><br>Fast, accurate computations matching path-independent complexity"]

Cited References (59)

  1. Pricing and hedging autocallable products by M arkov chain approximation (2024)
  2. The F ourier-series method for inverting transforms of probability distributions (1992)
  3. The pricing of options and corporate liabilities (1973)
  4. The variance gamma process and option pricing (1998)
  5. A jump-diffusion model for option pricing (2002)
  6. The constant elasticity of variance option pricing model (1996)
  7. Pricing and hedging path-dependent options under the CEV process (2001)
  8. A general valuation framework for rough stochastic local volatility models and applications (2025)
and 51 more
  1. A general framework for time-changed M arkov processes and applications (2019)
  2. A general valuation framework for SABR and stochastic local volatility models (2018)
  3. A general framework for discretely sampled realized variance derivatives in stochastic volatility models with jumps (2017)
  4. A unified framework for regime-switching models (2019)
  5. Analysis of M arkov chain approximation for option pricing and hedging: Grid design and convergence behavior (2019)
  6. Error analysis of finite difference and M arkov chain approximations for option pricing (2018)
  7. Simulation of multidimensional diffusions with sticky boundaries via M arkov chain approximation (2023)
  8. Markov chain approximation of one-dimensional sticky diffusions (2021)
  9. Single-transform formulas for pricing A sian options in a general approximation framework under M arkov processes (2018)
  10. A general framework for pricing A sian options under M arkov processes (2015)
  11. American option valuation under continuous-time M arkov chains (2015)
  12. Portfolio optimisation under non-linear drawdown constraints in a semimartingale financial model (2013)
  13. Optimal investment strategies for controlling drawdowns (1993)
  14. On portfolio optimization under ``drawdown" constraints (1995)
  15. Drawdown measure in portfolio optimization (2005)
  16. Exit problems for spectrally negative L 'e vy processes and applications to ( C anadized) R ussian options (2004)
  17. The R ussian option: reduced regret (1993)
  18. Russian and A merican put options under exponential phase-type L 'e vy models (2004)
  19. Magnitude and speed of consecutive market crashes in a diffusion model (2018)
  20. Omega diffusion risk model with surplus-dependent tax and capital injections (2016)
  21. On magnitude, asymptotics and duration of drawdowns for L 'e vy models (2017)
  22. Portfolio sensitivity to changes in the maximum and the maximum drawdown (2010)
  23. PDE methods for the maximum drawdown (2008)
  24. Preventing portfolio losses by hedging maximum drawdown (2007)
  25. Option pricing: Maximum drawdown and directional trading (2006)
  26. Stochastic modeling and fair valuation of drawdown insurance (2013)
  27. A unified approach for drawdown (drawup) of time-homogeneous M arkov processes (2017)
  28. Drawdowns and rallies in a finite time-horizon: drawdowns and rallies (2010)
  29. Drawdowns and the speed of market crash (2012)
  30. Sufficient conditions for expected utility to imply drawdown-based performance rankings (2011)
  31. Maximum drawdown (2004)
  32. Formulas for stopped diffusion processes with stopping times based on drawdowns and drawups (2009)
  33. Drawdowns preceding rallies in the B rownian motion model (2006)
  34. On the drawdown of completely asymmetric L 'e vy processes (2012)
  35. On the maximum drawdown of a B rownian motion (2004)
  36. On probability characteristics of ``downfalls" in a standard B rownian motion (2000)
  37. Formulas for stopped diffusion processes with stopping times based on the maximum (1977)
  38. A stopped B rownian motion formula (1975)
  39. Inverting Modified Matrices (1950)
  40. Weak convergence methods for approximation of the evaluation of path-dependent functionals (2013)
  41. Limit Theorems for Stochastic Processes (2013)
  42. Continuously monitored barrier options under M arkov processes (2013)
  43. On the frequency of drawdowns for B rownian motion processes (2015)
  44. Occupation times, drawdowns, and drawups for one-dimensional regular diffusions (2015)
  45. A general method for analysis and valuation of drawdown risk (2023)
  46. Speed and duration of drawdown under general M arkov models (2024)
  47. Pricing A merican drawdown options under M arkov models (2021)
  48. A general approach for lookback option pricing under M arkov models (2023)
  49. Analysis of M arkov Chain Approximation for Regime-Switching Jump Diffusions with Nonsmooth Coefficients: Deformed Contour Integration Approach (2023)
  50. Analysis of M arkov chain approximation for diffusion models with nonsmooth coefficients (2022)
  51. A general approach for P arisian stopping times under M arkov processes (2023)

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