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The fractional and mixed-fractional CEV model

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  • Axel A. Araneda

Abstract

The continuous observation of the financial markets has identified some stylized facts which challenge the conventional assumptions, promoting the born of new approaches. On the one hand, the long-range dependence has been faced replacing the traditional Gauss-Wiener process (Brownian motion), characterized by stationary independent increments, by a fractional version. On the other hand, the CEV model addresses the Leverage effect and smile-skew phenomena, efficiently. In this paper, these two insights are merging and both the fractional and mixed-fractional extensions for the CEV model, are developed. Using the fractional versions of both the Ito's calculus and the Fokker-Planck equation, the transition probability density function of the asset price is obtained as the solution of a non-stationary Feller process with time-varying coefficients, getting an analytical valuation formula for a European Call option. Besides, the Greeks are computed and compared with the standard case.

Suggested Citation

  • Axel A. Araneda, 2019. "The fractional and mixed-fractional CEV model," Papers 1903.05747, arXiv.org, revised Jun 2019.
    • Handle: RePEc:arx:papers:1903.05747
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    References listed on IDEAS

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    1. MacBeth, James D & Merville, Larry J, 1980. "Tests of the Black-Scholes and Cox Call Option Valuation Models," Journal of Finance, American Finance Association, vol. 35(2), pages 285-301, May.
    2. Sottinen Tommi & Valkeila Esko, 2003. "On arbitrage and replication in the fractional Black–Scholes pricing model," Statistics & Risk Modeling, De Gruyter, vol. 21(2/2003), pages 93-108, February.
    3. Cipian Necula, 2008. "Option Pricing in a Fractional Brownian Motion Environment," Advances in Economic and Financial Research - DOFIN Working Paper Series 2, Bucharest University of Economics, Center for Advanced Research in Finance and Banking - CARFIB.
    4. Sadique, Shibley & Silvapulle, Param, 2001. "Long-Term Memory in Stock Market Returns: International Evidence," International Journal of Finance & Economics, John Wiley & Sons, Ltd., vol. 6(1), pages 59-67, January.
    5. W. Dai & C. C. Heyde, 1996. "Itô's formula with respect to fractional Brownian motion and its application," International Journal of Stochastic Analysis, Hindawi, vol. 9, pages 1-10, January.
    6. Hsu, Y.L. & Lin, T.I. & Lee, C.F., 2008. "Constant elasticity of variance (CEV) option pricing model: Integration and detailed derivation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(1), pages 60-71.
    7. Schroder, Mark Douglas, 1989. " Computing the Constant Elasticity of Variance Option Pricing Formula," Journal of Finance, American Finance Association, vol. 44(1), pages 211-219, March.
    8. Swidler, Steve & Diltz, J. David, 1992. "Implied volatilities and Transaction Costs," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 27(3), pages 437-447, September.
    9. Tomas Björk & Henrik Hult, 2005. "A note on Wick products and the fractional Black-Scholes model," Finance and Stochastics, Springer, vol. 9(2), pages 197-209, April.
    10. Christie, Andrew A., 1982. "The stochastic behavior of common stock variances : Value, leverage and interest rate effects," Journal of Financial Economics, Elsevier, vol. 10(4), pages 407-432, December.
    11. Emanuel, David C. & MacBeth, James D., 1982. "Further Results on the Constant Elasticity of Variance Call Option Pricing Model," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 17(4), pages 533-554, November.
    12. Rubinstein, Mark, 1985. "Nonparametric Tests of Alternative Option Pricing Models Using All Reported Trades and Quotes on the 30 Most Active CBOE Option Classes from August 23, 1976 through August 31, 1978," Journal of Finance, American Finance Association, vol. 40(2), pages 455-480, June.
    13. Bender, Christian, 2003. "An Itô formula for generalized functionals of a fractional Brownian motion with arbitrary Hurst parameter," Stochastic Processes and their Applications, Elsevier, vol. 104(1), pages 81-106, March.
    14. Rafael Mendoza-Arriaga & Vadim Linetsky, 2011. "Pricing equity default swaps under the jump-to-default extended CEV model," Finance and Stochastics, Springer, vol. 15(3), pages 513-540, September.
    15. Beckers, Stan, 1980. "The Constant Elasticity of Variance Model and Its Implications for Option Pricing," Journal of Finance, American Finance Association, vol. 35(3), pages 661-673, June.
    16. Sun, Lin, 2013. "Pricing currency options in the mixed fractional Brownian motion," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(16), pages 3441-3458.
    17. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    18. Cajueiro, Daniel O & Tabak, Benjamin M, 2004. "The Hurst exponent over time: testing the assertion that emerging markets are becoming more efficient," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 336(3), pages 521-537.
    19. Lo, Andrew W, 1991. "Long-Term Memory in Stock Market Prices," Econometrica, Econometric Society, vol. 59(5), pages 1279-1313, September.
    20. Tim Bollerslev & Julia Litvinova & George Tauchen, 2006. "Leverage and Volatility Feedback Effects in High-Frequency Data," Journal of Financial Econometrics, Oxford University Press, vol. 4(3), pages 353-384.
    21. Lauterbach, Beni & Schultz, Paul, 1990. "Pricing Warrants: An Empirical Study of the Black-Scholes Model and Its Alternatives," Journal of Finance, American Finance Association, vol. 45(4), pages 1181-1209, September.
    22. C. F. Lee & Ta-Peng Wu & Ren-Raw Chen, 2004. "The Constant Elasticity of Variance Models: New Evidence from S&P 500 Index Options," Review of Pacific Basin Financial Markets and Policies (RPBFMP), World Scientific Publishing Co. Pte. Ltd., vol. 7(02), pages 173-190.
    23. Rostek, S. & Schöbel, R., 2013. "A note on the use of fractional Brownian motion for financial modeling," Economic Modelling, Elsevier, vol. 30(C), pages 30-35.
    24. Bekaert, Geert & Wu, Guojun, 2000. "Asymmetric Volatility and Risk in Equity Markets," The Review of Financial Studies, Society for Financial Studies, vol. 13(1), pages 1-42.
    25. Jackwerth, Jens Carsten & Rubinstein, Mark, 1996. "Recovering Probability Distributions from Option Prices," Journal of Finance, American Finance Association, vol. 51(5), pages 1611-1632, December.
    26. Mounir Zili, 2006. "On the mixed fractional Brownian motion," International Journal of Stochastic Analysis, Hindawi, vol. 2006, pages 1-9, August.
    27. Alan L. Tucker & David R. Peterson & Elton Scott, 1988. "Tests Of The Black-Scholes And Constant Elasticity Of Variance Currency Call Option Valuation Models," Journal of Financial Research, Southern Finance Association;Southwestern Finance Association, vol. 11(3), pages 201-214, September.
    28. Manuela Larguinho & José Carlos Dias & Carlos A. Braumann, 2013. "On the computation of option prices and Greeks under the CEV model," Quantitative Finance, Taylor & Francis Journals, vol. 13(6), pages 907-917, May.
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