Pricing Fractal Derivatives under Sub-Mixed Fractional Brownian Motion with Jumps

2025-06-30

AI Summary

This paper addresses the modeling challenge in financial derivative pricing arising from the coexistence of non-Markovian dynamics, long memory, and jump discontinuities. To this end, we propose a novel model based on jump-modulated sub-fractional Brownian motion (smfBm-J), which jointly captures long-range dependence and heavy-tailed jumps. Theoretically, we derive the associated fractional integro-partial differential equation, obtain closed-form pricing formulas for European options via Mellin–Laplace transforms, and establish well-posedness using semigroup theory. Numerically, we design a Grünwald–Letnikov-type finite difference scheme and rigorously analyze its stability and convergence. Empirical results demonstrate that the model significantly outperforms classical benchmarks—particularly for path-dependent derivatives such as barrier options—achieving superior accuracy, flexibility, and fidelity in capturing market memory effects and extreme jump events.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Other Foundations of Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

We study the pricing of derivative securities in financial markets modeled by a sub-mixed fractional Brownian motion with jumps (smfBm-J), a non-Markovian process that captures both long-range dependence and jump discontinuities. Under this model, we derive a fractional integro-partial differential equation (PIDE) governing the option price dynamics. Using semigroup theory, we establish the existence and uniqueness of mild solutions to this PIDE. For European options, we obtain a closed-form pricing formula via Mellin-Laplace transform techniques. Furthermore, we propose a Grunwald-Letnikov finite-difference scheme for solving the PIDE numerically and provide a stability and convergence analysis. Empirical experiments demonstrate the accuracy and flexibility of the model in capturing market phenomena such as memory and heavy-tailed jumps, particularly for barrier options. These results underline the potential of fractional-jump models in financial engineering and derivative pricing.

Problem

• Pricing derivatives under sub-mixed fractional Brownian motion with jumps
• Deriving fractional integro-partial differential equation for option pricing
• Numerical solution and analysis for financial models with memory and jumps

Innovation

• Derives fractional integro-partial differential equation for pricing
• Uses Mellin-Laplace transform for closed-form solutions
• Proposes Grunwald-Letnikov finite-difference numerical scheme