A General Method for Analysis and Valuation of Drawdown Risk under Markov Models

Drawdown risk is a major concern in financial markets. We develop a novel algorithm to solve the first passage problem of the drawdown process of general one-dimensional time-homogeneous Markov processes. We compute its Laplace transform based on continuous time Markov chain (CTMC) approximation and numerically invert the Laplace transform to obtain the first passage probabilities and the distribution of the maximum drawdown. We prove convergence of our algorithm for general Markovian models and provide sharp estimate of the convergence rate for a general class of jump-diffusion models. We apply the algorithm to calculate the Calmar ratio for investment analysis, price maximum drawdown derivatives and hedge the risk of selling such derivatives with a highly volatile asset as the underlying. Various numerical experiments document the computational efficiency of our method. We also develop extensions to solve the drawdown problem in models with time dependence or stochastic volatility or regime switching and for portfolio drawdown analysis.

[1]  Peter A. Forsyth,et al.  Dynamic Hedging Under Jump Diffusion with Transaction Costs , 2009, Oper. Res..

[2]  L. Ballotta,et al.  Multivariate asset models using Lévy processes and applications , 2012 .

[3]  Dean S. Clark,et al.  Short proof of a discrete gronwall inequality , 1987, Discret. Appl. Math..

[4]  J. Vecer.  Preventing Portfolio Losses by Hedging Maximum Drawdown , 2007 .

[5]  J. Vecer.  Maximum Drawdown and Directional Trading , 2006 .

[6]  Paul Glasserman,et al.  Monte Carlo Methods in Financial Engineering , 2003 .

[7]  Aleksandar Mijatovi'c,et al.  Markov chain approximations for transition densities of Lévy processes , 2012, 1211.0476.

[8]  Olympia Hadjiliadis,et al.  Drawdowns and the Speed of Market Crash , 2012 .

[9]  Libor Pospisil,et al.  Formulas for stopped diffusion processes with stopping times based on drawdowns and drawups , 2009 .

[10]  B. Eriksson,et al.  American Option Valuation under Continuous-Time Markov Chains , 2015, Advances in Applied Probability.

[11]  Duy Nguyen,et al.  A General Framework for Discretely Sampled Realized Variance Derivatives in Stochastic Volatility Models with Jumps , 2017, Eur. J. Oper. Res..

[12]  Jose Luis Menaldi,et al.  Second Order Elliptic Integro-Differential Problems , 2002 .

[13]  A General Approach for Parisian Stopping Times under Markov Processes , 2019, 2107.06605.

[14]  P. Carr,et al.  The Variance Gamma Process and Option Pricing , 1998 .

[15]  Marc Yor,et al.  On Probability Characteristics of "Downfalls" in a Standard Brownian Motion , 2000 .

[16]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[17]  J. Vecer,et al.  Portfolio sensitivity to changes in the maximum and the maximum drawdown , 2010 .

[18]  Aleksandar Mijatović,et al.  On the drawdown of completely asymmetric Lévy processes , 2011, 1103.1460.

[19]  Vadim Linetsky,et al.  Pricing and Hedging Path-Dependent Options Under the CEV Process , 2001, Manag. Sci..

[20]  J. Lehoczky.  FORMULAS FOR STOPPED DIFFUSION PROCESSES WITH STOPPING TIMES BASED ON THE MAXIMUM , 1977 .

[21]  S. Ethier,et al.  Markov Processes: Characterization and Convergence , 2005 .

[22]  Aleksandar Mijatović,et al.  CONTINUOUSLY MONITORED BARRIER OPTIONS UNDER MARKOV PROCESSES , 2009, 0908.4028.

[23]  Bin Li,et al.  On Magnitude, Asymptotics and Duration of Drawdowns for Levy Models , 2015, 1506.08408.

[24]  Lingfei Li,et al.  Analysis of Markov Chain Approximation for Diffusion Models with Non-Smooth Coefficients , 2019, SSRN Electronic Journal.

[25]  Jakša Cvitanić,et al.  On portfolio optimization under "drawdown" constraints , 1994 .

[26]  Lingfei Li,et al.  Pricing American Drawdown Options under Markov Models , 2020, Eur. J. Oper. Res..

[27]  V. Linetsky.  Spectral Methods in Derivatives Pricing , 2007 .

[28]  Steven Kou,et al.  Computable Error Bounds of Laplace Inversion for Pricing Asian Options , 2018, INFORMS J. Comput..

[29]  Steven Kou,et al.  A Jump Diffusion Model for Option Pricing , 2001, Manag. Sci..

[30]  Michael C. Fu,et al.  Option Pricing for a Jump-Diffusion Model with General Discrete Jump-Size Distributions , 2017, Manag. Sci..

[31]  Zhenyu Cui,et al.  A General Valuation Framework for SABR and Stochastic Local Volatility Models , 2018, SIAM J. Financial Math..

[32]  Jan Oblój,et al.  Portfolio optimisation under non-linear drawdown constraints in a semimartingale financial model , 2011, Finance Stochastics.

[33]  Ward Whitt,et al.  The Fourier-series method for inverting transforms of probability distributions , 1992, Queueing Syst. Theory Appl..

[34]  R. Leal,et al.  Maximum Drawdown , 2005 .

[35]  Tim Leung,et al.  Stochastic Modeling and Fair Valuation of Drawdown Insurance , 2013, 1310.3860.

[36]  Bin Li,et al.  A unified approach for drawdown (drawup) of time-homogeneous Markov processes , 2017, Journal of Applied Probability.

[37]  M. Yor,et al.  The Fine Structure of Asset Retums : An Empirical Investigation ' , 2006 .

[38]  S. Asmussen,et al.  Russian and American put options under exponential phase-type Lévy models , 2004 .

[39]  Larry A Shepp,et al.  The Russian Option: Reduced Regret , 1993 .

[40]  Olympia Hadjiliadis,et al.  MAXIMUM DRAWDOWN INSURANCE , 2011 .

[41]  K. Miller.  On the Inverse of the Sum of Matrices , 1981 .

[42]  Zhenyu Cui,et al.  Magnitude and Speed of Consecutive Market Crashes in a Diffusion Model , 2016 .

[43]  Zhenyu Cui,et al.  Omega Diffusion Risk Model with Surplus-Dependent Tax and Capital Injections , 2016 .

[44]  Gianluca Fusai,et al.  Estimation of Multivariate Asset Models with Jumps , 2018, Journal of Financial and Quantitative Analysis.

[45]  Zhenyu Cui,et al.  A General Framework for Time-Changed Markov Processes and Applications , 2018, Eur. J. Oper. Res..

[46]  Sanford J. Grossman,et al.  OPTIMAL INVESTMENT STRATEGIES FOR CONTROLLING DRAWDOWNS , 1993 .

[47]  J. Vecer,et al.  PDE Methods for the Maximum Drawdown , 2008 .

[48]  Hongzhong Zhang,et al.  Occupation Times, Drawdowns, and Drawups for One-Dimensional Regular Diffusions , 2013, Advances in Applied Probability.

[49]  Gang George Yin,et al.  Weak Convergence Methods for Approximation of the Evaluation of Path-Dependent Functionals , 2013, SIAM J. Control. Optim..

[50]  Olympia Hadjiliadis,et al.  Drawdowns and Rallies in a Finite Time-horizon , 2010 .

[51]  Steven Kou,et al.  A General Framework for Pricing Asian Options Under Markov Processes , 2015, Oper. Res..

[52]  S. Uryasev,et al.  Drawdown Measure in Portfolio Optimization , 2003 .

[53]  Lingfei Li,et al.  Error analysis of finite difference and Markov chain approximations for option pricing , 2017 .

[54]  Martin Eling,et al.  Sufficient Conditions for Expected Utility to Imply Drawdown-Based Performance Rankings , 2010 .

[55]  Lingfei Li,et al.  Analysis of Markov Chain Approximation for Option Pricing and Hedging: Grid Design and Convergence Behavior , 2017, Oper. Res..

[56]  R. Serfozo.  Basics of Applied Stochastic Processes , 2012 .

[57]  Florin Avram,et al.  Exit problems for spectrally negative Levy processes and applications to (Canadized) Russian options , 2004 .

[58]  Olympia Hadjiliadis,et al.  Drawdowns preceding rallies in the Brownian motion model , 2006 .

[59]  Chi-Hoon Lee,et al.  Single-Transform Formulas for Pricing Asian Options in a General Approximation Framework under Markov Processes , 2015, Eur. J. Oper. Res..

[60]  Amir F. Atiya,et al.  The maximum drawdown of the Brownian motion , 2003, 2003 IEEE International Conference on Computational Intelligence for Financial Engineering, 2003. Proceedings..