Option Pricing with Periodic Volatility : A Modified Black-Scholes Model Using Jacobi Elliptic Functions
DOI:
https://doi.org/10.64021/Keywords:
Black-Scholes, Volatility, Jacobi Elliptic Functions, Crank-Nicolson, Option-PricingAbstract
In this work, a modified Black-Scholes model for European option pricing is presented, in which the volatility function exhibits a periodic pattern determined by the Jacobi elliptic sine function. This method maintains smoothness and mathematical tractability while capturing organized, oscillatory activity frequently seen in financial markets.
We develop the relevant parabolic partial differential equation and confirm regularity and uniform ellipticity criteria to confirm that it is well-posed. The complete model is solved nu- merically using a Crank-Nicolson finite difference method after a formal solution is derived using the Fourier transform under some assumptions. The findings of the simulation show how volatility patterns and option prices are impacted by changes in the elliptic modulus. Specifi- cally, for short-term and at-the-money options, the model produces rippling price surfaces and observable deviations from traditional Black-Scholes pricing.
This methodology, which remains rooted in the traditional option pricing model while offering insight into periodic risk dynamics, provides a useful and understandable alternative for stochastic volatility models.
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Copyright (c) 2026 Chinwe N. Obi, Bright O. Osu , Promise Azor (Author)
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