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A Theory of Non_Gaussian Option Pricing

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  • Lisa Borland

Abstract

Option pricing formulas are derived from a non-Gaussian model of stock returns. Fluctuations are assumed to evolve according to a nonlinear Fokker-Planck equation which maximizes the Tsallis nonextensive entropy of index $q$. A generalized form of the Black-Scholes differential equation is found, and we derive a martingale measure which leads to closed form solutions for European call options. The standard Black-Scholes pricing equations are recovered as a special case ($q = 1$). The distribution of stock returns is well-modelled with $q$ circa 1.5. Using that value of $q$ in the option pricing model we reproduce the volatility smile. The partial derivatives (or Greeks) of the model are also calculated. Empirical results are demonstrated for options on Japanese Yen futures. Using just one value of $\sigma$ across strikes we closely reproduce market prices, for expiration times ranging from weeks to several months.

Suggested Citation

  • Lisa Borland, 2002. "A Theory of Non_Gaussian Option Pricing," Papers cond-mat/0205078, arXiv.org, revised Dec 2002.
    • Handle: RePEc:arx:papers:cond-mat/0205078
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    1. Hirsa, Ali & Neftci, Salih N., 2013. "An Introduction to the Mathematics of Financial Derivatives," Elsevier Monographs, Elsevier, edition 3, number 9780123846822.
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    Cited by:

    1. Arismendi, Juan & Genaro, Alan De, 2016. "A Monte Carlo multi-asset option pricing approximation for general stochastic processes," Chaos, Solitons & Fractals, Elsevier, vol. 88(C), pages 75-99.
    2. Nikola Gradojevic, 2021. "Brexit and foreign exchange market expectations: Could it have been predicted?," Annals of Operations Research, Springer, vol. 297(1), pages 167-189, February.
    3. Dufera, Tamirat Temesgen, 2024. "Fractional Brownian motion in option pricing and dynamic delta hedging: Experimental simulations," The North American Journal of Economics and Finance, Elsevier, vol. 69(PB).
    4. Zhao, Pan & Pan, Jian & Yue, Qin & Zhang, Jinbo, 2021. "Pricing of financial derivatives based on the Tsallis statistical theory," Chaos, Solitons & Fractals, Elsevier, vol. 142(C).
    5. Zhang, Qingye & Gao, Yan, 2016. "Optimal consumption—portfolio problem with CVaR constraints," Chaos, Solitons & Fractals, Elsevier, vol. 91(C), pages 516-521.
    6. Tsallis, Constantino & Borges, Ernesto P., 2021. "Comment on “Pricing of financial derivatives based on the Tsallis statistical theory” by Zhao, Pan, Yue and Zhang," Chaos, Solitons & Fractals, Elsevier, vol. 148(C).
    7. Borland, Lisa, 2016. "Exploring the dynamics of financial markets: from stock prices to strategy returns," Chaos, Solitons & Fractals, Elsevier, vol. 88(C), pages 59-74.
    8. Marco Airoldi & Vito Antonelli & Bruno Bassetti & Andrea Martinelli & Marco Picariello, 2004. "Long Range Interaction Generating Fat-Tails in Finance," GE, Growth, Math methods 0404006, University Library of Munich, Germany, revised 27 Apr 2004.
    9. Zhao, Pan & Xiao, Qingxian, 2016. "Portfolio selection problem with liquidity constraints under non-extensive statistical mechanics," Chaos, Solitons & Fractals, Elsevier, vol. 82(C), pages 5-10.
    10. De Domenico, Federica & Livan, Giacomo & Montagna, Guido & Nicrosini, Oreste, 2023. "Modeling and simulation of financial returns under non-Gaussian distributions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 622(C).
    11. Viktor Stojkoski & Trifce Sandev & Lasko Basnarkov & Ljupco Kocarev & Ralf Metzler, 2020. "Generalised geometric Brownian motion: Theory and applications to option pricing," Papers 2011.00312, arXiv.org.
    12. Devi, Sandhya, 2021. "Asymmetric Tsallis distributions for modeling financial market dynamics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 578(C).
    13. Gangadhar Nayak & Amit Kumar Singh & Dilip Senapati, 2021. "Computational Modeling of Non-Gaussian Option Price Using Non-extensive Tsallis’ Entropy Framework," Computational Economics, Springer;Society for Computational Economics, vol. 57(4), pages 1353-1371, April.
    14. Federica De Domenico & Giacomo Livan & Guido Montagna & Oreste Nicrosini, 2023. "Modeling and Simulation of Financial Returns under Non-Gaussian Distributions," Papers 2302.02769, arXiv.org.
    15. Sandhya Devi & Sherman Page, 2022. "Tsallis Relative entropy from asymmetric distributions as a risk measure for financial portfolios," Papers 2205.13625, arXiv.org.
    16. Arias-Calluari, Karina & Najafi, Morteza. N. & Harré, Michael S. & Tang, Yaoyue & Alonso-Marroquin, Fernando, 2022. "Testing stationarity of the detrended price return in stock markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 587(C).

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