Document Type : Research Article
Authors
1 Department of Operations Management, University of South Africa, Pretoria, South Africa.
2 Department of Decision Sciences, University of South Africa, Pretoria, South Africa.
10.22054/jmmf.2026.90131.1246
Abstract
In practice, uncertainty and randomness are two common, basic forms of indeterminacy. These two forms of indeterminacy are logically modelled by two distinct mathematical systems. Randomness is modelled by probability theory and uncertainty is modelled by uncertainty theory. However, in some cases, uncertainty and randomness concurrently surface in a multifaceted mathematical system. This study develops an American call option pricing model with random jumps for Itô-Liu financial markets. Uncertain stochastic differential equations (USDEs) incorporating Poisson jumps are employed to model the underlying asset dynamics. These jump-diffiusion USDEs are driven by a Brownian motion, a canonical Liu process, and a Poisson jump process. To validate the proposed American call option pricing model, a numerical example is given. The findings of the study show that the developed model is capable of pricing American call options in Itô-Liu financial markets. This study adds a voice to the discourse of option pricing in Itô-Liu financial markets.
Keywords
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[56] H. Zhao, J. Zhang, J. Lu, and J. Hu, Approximate controllability and optimal control in fractional differential equations with multiple delay controls, fractional Brownian motion with Hurst parameter in 0 < H < 12 , and Poisson jumps, Communications in Nonlinear Science and Numerical Simulation, (2024) 128:107636.
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[59] S. J. Tabatabaei, Stochastic portfolio optimization by diversity-weighted portfolio approach, Journal of Mathematics and Modeling in Finance (JMMF), (2024) 4, No. 2, 65–82.
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[52] X. C. Yu, A stock model with jumps for uncertain markets, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, (2012) 20(03):421-432.
[53] X. Yu, A stock model for uncertain European finance markets, Journal of Mathematics and Informatics, (2013) 1:13-24.
[54] J. Yu, Catastrophe options with double compound Poisson processes, Economic Modelling, (2015) 50:291-297.
[55] S. Yu, and Y. Ning, An interest-rate model with jumps for uncertain financial markets, Physica A: Statistical Mechanics and its Applications, (2019) 527:121424.
[56] H. Zhao, J. Zhang, J. Lu, and J. Hu, Approximate controllability and optimal control in fractional differential equations with multiple delay controls, fractional Brownian motion with Hurst parameter in 0 < H < 12 , and Poisson jumps, Communications in Nonlinear Science and Numerical Simulation, (2024) 128:107636.
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[58] L. Zhou, Z. He, J. Liu, and X. Yin, Option Pricing and Parameter Estimation for Uncertain Mean-Reverting Currency Model, Computational Economics, (2025) 66(6):4781-4811.
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