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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2511.10371 |
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| _version_ | 1866915615634096128 |
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| author | Zhang, Shuaiqi Chen, Zhen-Qing |
| author_facet | Zhang, Shuaiqi Chen, Zhen-Qing |
| contents | We propose a novel Black-Scholes model under which the stock price processes are modeled by stochastic differential equations driven by sub-diffusions. The new framework can capture the less financial activity phenomenon during the bear markets while having the classical Black- Scholes model as its special case. The sub-diffusive spot market is arbitrage-free but is in general incomplete. We investigate the pricing for European-style contingent claims under this new model. For this, we study the Girsanov transform for sub-diffusions and use it to find risk-neutral probability measures for the new Black-Scholes model. Finally, we derive the explicit formula for the price of European call options and show that it can be determined by a partial differential equation (PDE) involving a fractional derivative in time, which we coin a time-fractional Black-Scholes PDE. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_10371 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sub-diffusive Black-Scholes model and Girsanov transform for sub-diffusions Zhang, Shuaiqi Chen, Zhen-Qing Probability We propose a novel Black-Scholes model under which the stock price processes are modeled by stochastic differential equations driven by sub-diffusions. The new framework can capture the less financial activity phenomenon during the bear markets while having the classical Black- Scholes model as its special case. The sub-diffusive spot market is arbitrage-free but is in general incomplete. We investigate the pricing for European-style contingent claims under this new model. For this, we study the Girsanov transform for sub-diffusions and use it to find risk-neutral probability measures for the new Black-Scholes model. Finally, we derive the explicit formula for the price of European call options and show that it can be determined by a partial differential equation (PDE) involving a fractional derivative in time, which we coin a time-fractional Black-Scholes PDE. |
| title | Sub-diffusive Black-Scholes model and Girsanov transform for sub-diffusions |
| topic | Probability |
| url | https://arxiv.org/abs/2511.10371 |