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A new closed-form solution as an extension of the Black-Scholes formula allowing smile curve plotting

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  • Yacin Jerbi

Abstract

In this paper, as a generalization of the Black-Scholes (BS) model, we elaborate a new closed-form solution for a uni-dimensional European option pricing model called the J -model. This closed-form solution is based on a new stochastic process, called the J -process, which is an extension of the Wiener process satisfying the martingale property. The J -process is based on a new statistical law called the J -law, which is an extension of the normal law. The J -law relies on four parameters in its general form. It has interesting asymmetry and tail properties, allowing it to fit the reality of financial markets with good accuracy, which is not the case for the normal law. Despite the use of one state variable, we find results similar to those of Heston dealing with the bi-dimensional stochastic volatility problem for pricing European calls. Inverting the BS formula, we plot the smile curve related to this closed-form solution. The J -model can also serve to determine the implied volatility by inverting the J -formula and can be used to price other kinds of options such as American options.

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  • Yacin Jerbi, 2015. "A new closed-form solution as an extension of the Black-Scholes formula allowing smile curve plotting," Quantitative Finance, Taylor & Francis Journals, vol. 15(12), pages 2041-2052, December.
    • Handle: RePEc:taf:quantf:v:15:y:2015:i:12:p:2041-2052
    DOI: 10.1080/14697688.2012.762458
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    Cited by:

    1. Yacin Jerbi, 2016. "Early exercise premium method for pricing American options under the J-model," Financial Innovation, Springer;Southwestern University of Finance and Economics, vol. 2(1), pages 1-26, December.
    2. Haiyang Fang & Dali Jiang & Tinghong Yang & Ling Fang & Jian Yang & Wu Li & Jing Zhao, 2018. "Network evolution model for supply chain with manufactures as the core," PLOS ONE, Public Library of Science, vol. 13(1), pages 1-28, January.

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