A multivariate model for the Wiener process range, with statistical properties under stochastic volatility

Rawiyah Muneer Alraddadi; Mohamed Abd Allah El-Hadidy; Rawiyah Muneer Alraddadi; Mohamed Abd Allah El-Hadidy

doi:10.3934/math.2025980

AIMS Mathematics

2025, Volume 10, Issue 9: 22023-22052. doi: 10.3934/math.2025980

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Research article

A multivariate model for the Wiener process range, with statistical properties under stochastic volatility

Rawiyah Muneer Alraddadi ^{1
,},
Mohamed Abd Allah El-Hadidy ^{2
,
,}

1.
Department of Mathematics and Statistics, College of Science in Yanbu, Taibah University, Madinah 46423, Saudi Arabia; rmraddadi@taibahu.edu.sa
2.
Mathematics Department, Faculty of Science, Tanta University, Tanta 31527, Egypt; melhadidi@science.tanta.edu.eg

Received: 25 July 2025 Revised: 05 September 2025 Accepted: 15 September 2025 Published: 22 September 2025
MSC : 60J65, 60J70

In this paper, we presented a mathematical model to analyze financial instruments that are sensitive to the difference between the highest and lowest prices of n independent stocks in a random volatility environment. This model relies on the multivariate distribution of the ranges of n independent Wiener processes, describing the difference between the highest and lowest stock prices for a known time period. In addition to deriving the statistical characteristics of this distribution and its truncated version, including reliability properties, moments, the stress–strength parameter, and order statistics; we considered Bonferroni and Lorenz curves and the Gini index of the proposed model, as well as assessed its robustness in turbulent market environments. The proposed distribution enhances the modeling of range-based financial products to enable the construction of more efficient risk management and hedging strategies. Simulations with real financial data also confirmed its effectiveness in modeling range-based products and reducing volatility in markets.
- multivariate distribution,
- Wiener process range,
- truncated distribution,
- stock price,
- stress-strength parameter
Citation: Rawiyah Muneer Alraddadi, Mohamed Abd Allah El-Hadidy. A multivariate model for the Wiener process range, with statistical properties under stochastic volatility[J]. AIMS Mathematics, 2025, 10(9): 22023-22052. doi: 10.3934/math.2025980

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Abstract

In this paper, we presented a mathematical model to analyze financial instruments that are sensitive to the difference between the highest and lowest prices of n independent stocks in a random volatility environment. This model relies on the multivariate distribution of the ranges of n independent Wiener processes, describing the difference between the highest and lowest stock prices for a known time period. In addition to deriving the statistical characteristics of this distribution and its truncated version, including reliability properties, moments, the stress–strength parameter, and order statistics; we considered Bonferroni and Lorenz curves and the Gini index of the proposed model, as well as assessed its robustness in turbulent market environments. The proposed distribution enhances the modeling of range-based financial products to enable the construction of more efficient risk management and hedging strategies. Simulations with real financial data also confirmed its effectiveness in modeling range-based products and reducing volatility in markets.

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