Pricing Fractal Derivatives under Sub-Mixed Fractional Brownian Motion with Jumps
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918076492021760 |
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| author | Karimi, Nader |
| author_facet | Karimi, Nader |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_24111 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Pricing Fractal Derivatives under Sub-Mixed Fractional Brownian Motion with Jumps Karimi, Nader Pricing of Securities 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. |
| title | Pricing Fractal Derivatives under Sub-Mixed Fractional Brownian Motion with Jumps |
| topic | Pricing of Securities |
| url | https://arxiv.org/abs/2506.24111 |