On Distribution of the Stock Market Risk with a Maximum Drawdown of a Wiener Process

Authors

  • Mohamed Abd Allah El-Hadidy Department of Mathematics, Faculty of Science, Tanta University, Tanta, Egypt https://orcid.org/0000-0002-9407-9586
  • R. Alraddadi Department of Mathematics and Statistics, College of Science in Yanbu, Taibah University, Madinah, Saudi Arabia

DOI:

https://doi.org/10.26713/cma.v16i1.2929

Keywords:

Statistical physics distributions, Multivariate distribution, Maximum drawdown distribution of a Wiener process, Basic statistical properties, Stock market risk

Abstract

In this article, we investigate the Wiener stock market risk’s maximum drawdown distribution. The danger of stochastic volatility in stock prices can be reduced by taking into account the most reliable and accurate decisions using this distribution. We extract various significant dependability aspects of this distribution, including the hazard and inverted hazard rate functions, in addition to presenting the closed-form pricing formula, which demonstrated the precise maximum drawdown distribution of the price path from the perspective of mathematical analysis. Additionally, a Wiener stock market risk’s estimated value is calculated. This predicted value can be used to forecast future risk. Furthermore, we present a multivariate distribution of a maximum drawdown for an m-dimensional Wiener process and its key reliability aspects when this risk depends on the n separate and distinct primary stock market risks.

Downloads

Download data is not yet available.

References

Z. Bergman, Pricing path contingent claims, Research in Finance 5 (1985), 229 – 241.

C. E. Bonferroni, Elementi di Statistica Generale, Seeber, Firenze (1930).

M. El-Hadidy, Discrete distribution for the stochastic range of a Wiener process and its properties, Fluctuation and Noise Letters 18(04) (2019), 1950024, DOI: 10.1142/s021947751950024x.

M. El-Hadidy, On the random walk microorganisms cells distribution on a planar surface and its properties, Journal of Computational and Theoretical Transport 49(4) (2020), 145 – 161, DOI: 10.1080/23324309.2020.1785892.

M. El-Hadidy, Study of water pollution through a Lévy flight jump diffusion model with stochastic jumps of pollutants, International Journal of Modern Physics B 33(19) (2019), 1950210, DOI: 10.1142/s0217979219502102.

M. El-Hadidy and A. Alfreedi, Internal truncated distributions: Applications to Wiener process range distribution when deleting a minimum stochastic volatility interval from its domain, Journal of Taibah University for Science 13(1) (2019), 201 – 215, DOI: 10.1080/16583655.2018.1555020.

M. El-Hadidy and A. Alzulaibani, On multivariate distribution of n-dimensional Brownian diffusion particle in the fluid, Journal of Computational and Theoretical Transport 52(4) (2023), 314 – 322, DOI: 10.1080/23324309.2023.2254951.

M. El-Hadidy and R. Alraddadi, On bivariate distributions with N deleted areas: Mathematical definition, Communications in Mathematics and Application 15(3) (2024), 1181 – 1190, DOI: 10.26713/cma.v15i3.2834.

M. El-Hadidy and R. Alraddadi, Studying the influence of antimicrobial resistance on the probability distribution of densities for synchronization growing of different kinds of bacteria, Journal of Computational and Theoretical Transport 53(1) (2024), 51 – 68, DOI: 10.1080/23324309.2024.2319237.

W. Feller, The asymptotic distribution of the range of sums of independent random variables, The Annals of Mathematical Statistics 22(3) (1951), 427 – 432, DOI: 10.1214/aoms/1177729589.

M. Goldman, H. B. Sosin and M. A. Gatto, Path dependent options; “Buy at the low, Sell at the High”, The Journal of Finance 34(5) (1979), 1111 – 1127, DOI: 10.2307/2327238.

A. G. Z. Kemna and A. C. F. Vorst, A pricing method for options based on average asset values, Journal of Banking and Finance 14(1) (1990), 113 – 129, DOI: 10.1016/0378-4266(90)90039-5.

Z. Liu, M. D. Moghaddam and R. A. Serota, Distributions of historic market data – stock returns, The European Physical Journal B 92 (2019), article number 60, DOI: 10.1140/epjb/e2019-90218-8.

M. O. Lorenz, Methods of measuring the concentration of wealth, Publications of the American Statistical Association 9(70) (1905), 209 – 219, DOI: 10.2307/2276207.

M. Magdon-Ismail, A. F. Atiya, A. Pratap and Y. Abu-Mostafa, On the maximum drawdown of a Brownian motion, Journal of Applied Probability 41 (2004), 147 – 161, DOI: 10.1017/s0021900200014108.

M. Milev and A. Tagliani, Entropy convergence of finite moment approximations in Hamburger and Stieltjes problems, Statistics & Probability Letters 120 (2017), 114 – 117, DOI: 10.1016/j.spl.2016.09.017.

M. Milev, P. N. Inverardi and A. Tagliani, Moment information and entropy valuation for probability densities, Applied Mathematics and Computations 218(9) (2012), 5782 – 5795, DOI: 10.1016/j.amc.2011.11.093.

Y. Nagahara, Cross-sectional-skew-dependent distribution models for industry returns in the Japanese stock market, Financial Engineering and the Japanese Markets 2 (1995), 139 – 154, DOI: 10.1007/bf02425170.

D. T. Nguyen, S. P. Nguyen, U. H. Pham and T. D. Nguyen, A calibration-based method in computing Bayesian posterior distributions with applications in stock market, in: Predictive Econometrics and Big Data (TES 2018), V. Kreinovich, S. Sriboonchitta and N. Chakpitak (editors), Studies in Computational Intelligence, Vol. 753, Springer, Cham., DOI: 10.1007/978-3-319-70942-0_10.

L. Nowakowska, Dynamic discrete model for electricity price forecasting, in: 2015 5th International Youth Conference on Energy (IYCE, Pisa, Italy, 2015), pp. 1-6, (2015), DOI: 10.1109/iyce.2015.7180798.

A. El-M. A. Teamah and M. A. A. El-Hadidy, On bounded range distribution of a Wiener process, Communications in Statistics – Theory and Methods 51(4) (2022), 919 – 942, DOI: 10.1080/03610926.2016.1267766.

B. H. Wang and P. M. Hui, The distribution and scaling of fluctuations for Hang Seng index in Hong Kong stock market, The European Physical Journal B – Condensed Matter and Complex Systems 20(4) (2001), 573 – 579, DOI: 10.1007/pl00022987.

C. S. Withers and S. Nadarajah, The distribution and quantiles of the range of a Wiener process, Applied Mathematics and Computations 232 (2014), 766 – 770, DOI: 10.1016/j.amc.2014.01.147.

Downloads

Published

01-07-2025
CITATION

How to Cite

El-Hadidy, M. A. A., & Alraddadi, R. (2025). On Distribution of the Stock Market Risk with a Maximum Drawdown of a Wiener Process. Communications in Mathematics and Applications, 16(1), 301–314. https://doi.org/10.26713/cma.v16i1.2929

Issue

Vol. 16 No. 1 (2025)

Section

Research Article

License

Creative Commons Licence 4.0 by  

Authors who publish with this journal agree to the following terms:

  • Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a CCAL that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
  • Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
  • Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work.