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Scaling and long-range dependence in option pricing V: Multiscaling hedging and implied volatility smiles under the fractional Black–Scholes model with transaction costs

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  • Wang, Xiao-Tian

Abstract

This paper deals with the problem of discrete time option pricing using the fractional Black–Scholes model with transaction costs. Through the ‘anchoring and adjustment’ argument in a discrete time setting, a European call option pricing formula is obtained. The minimal price of an option under transaction costs is obtained. In addition, the relation between scaling and implied volatility smiles is discussed.

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  • Wang, Xiao-Tian, 2011. "Scaling and long-range dependence in option pricing V: Multiscaling hedging and implied volatility smiles under the fractional Black–Scholes model with transaction costs," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(9), pages 1623-1634.
    • Handle: RePEc:eee:phsmap:v:390:y:2011:i:9:p:1623-1634
    DOI: 10.1016/j.physa.2010.12.021
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    1. 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.
    2. Robert J. Elliott & John Van Der Hoek, 2003. "A General Fractional White Noise Theory And Applications To Finance," Mathematical Finance, Wiley Blackwell, vol. 13(2), pages 301-330, April.
    3. Erzgräber, Hartmut & Strozzi, Fernanda & Zaldívar, José-Manuel & Touchette, Hugo & Gutiérrez, Eugénio & Arrowsmith, David K., 2008. "Time series analysis and long range correlations of Nordic spot electricity market data," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(26), pages 6567-6574.
    4. Leland, Hayne E, 1985. "Option Pricing and Replication with Transactions Costs," Journal of Finance, American Finance Association, vol. 40(5), pages 1283-1301, December.
    5. Ozdemir, Zeynel Abidin, 2009. "Linkages between international stock markets: A multivariate long-memory approach," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(12), pages 2461-2468.
    6. Thomas Lux & Michele Marchesi, 1999. "Scaling and criticality in a stochastic multi-agent model of a financial market," Nature, Nature, vol. 397(6719), pages 498-500, February.
    7. Shleifer, Andrei, 2000. "Inefficient Markets: An Introduction to Behavioral Finance," OUP Catalogue, Oxford University Press, number 9780198292272, Decembrie.
    8. Nicolas P. B. Bollen & Robert E. Whaley, 2004. "Does Net Buying Pressure Affect the Shape of Implied Volatility Functions?," Journal of Finance, American Finance Association, vol. 59(2), pages 711-753, April.
    9. L. C. G. Rogers, 1997. "Arbitrage with Fractional Brownian Motion," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 95-105, January.
    10. Robert F. Engle & Jeffrey R. Russell, 1998. "Autoregressive Conditional Duration: A New Model for Irregularly Spaced Transaction Data," Econometrica, Econometric Society, vol. 66(5), pages 1127-1162, September.
    11. Paolo Guasoni, 2006. "No Arbitrage Under Transaction Costs, With Fractional Brownian Motion And Beyond," Mathematical Finance, Wiley Blackwell, vol. 16(3), pages 569-582, July.
    12. 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.
    13. Daniel Kahneman & Amos Tversky, 2013. "Prospect Theory: An Analysis of Decision Under Risk," World Scientific Book Chapters, in: Leonard C MacLean & William T Ziemba (ed.), HANDBOOK OF THE FUNDAMENTALS OF FINANCIAL DECISION MAKING Part I, chapter 6, pages 99-127, World Scientific Publishing Co. Pte. Ltd..
    14. De Long, J Bradford, et al, 1990. "Positive Feedback Investment Strategies and Destabilizing Rational Speculation," Journal of Finance, American Finance Association, vol. 45(2), pages 379-395, June.
    15. Cajueiro, Daniel O. & Tabak, Benjamin M., 2008. "Testing for time-varying long-range dependence in real state equity returns," Chaos, Solitons & Fractals, Elsevier, vol. 38(1), pages 293-307.
    16. Cajueiro, Daniel O. & Tabak, Benjamin M., 2009. "Testing for long-range dependence in the Brazilian term structure of interest rates," Chaos, Solitons & Fractals, Elsevier, vol. 40(4), pages 1559-1573.
    17. Salopek, D. M., 1998. "Tolerance to arbitrage," Stochastic Processes and their Applications, Elsevier, vol. 76(2), pages 217-230, August.
    18. Tabak, Benjamin M. & Cajueiro, Daniel O., 2005. "The long-range dependence behavior of the term structure of interest rates in Japan," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 350(2), pages 418-426.
    19. Ayache, Antoine & Lévy Véhel, Jacques, 2004. "On the identification of the pointwise Hölder exponent of the generalized multifractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 111(1), pages 119-156, May.
    20. Jean-Philippe Bouchaud & Didier Sornette, 1994. "The Black-Scholes option pricing problem in mathematical finance: generalization and extensions for a large class of stochastic processes," Science & Finance (CFM) working paper archive 500040, Science & Finance, Capital Fund Management.
    21. Ming-Shiun Pan, 2010. "Autocorrelation, return horizons, and momentum in stock returns," Journal of Economics and Finance, Springer;Academy of Economics and Finance, vol. 34(3), pages 284-300, July.
    22. Muniandy, S.V. & Lim, S.C. & Murugan, R., 2001. "Inhomogeneous scaling behaviors in Malaysian foreign currency exchange rates," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 301(1), pages 407-428.
    23. 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.
    24. Alvarez-Ramirez, Jose & Alvarez, Jesus & Rodriguez, Eduardo & Fernandez-Anaya, Guillermo, 2008. "Time-varying Hurst exponent for US stock markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(24), pages 6159-6169.
    25. Robert F. Engle, 2000. "The Econometrics of Ultra-High Frequency Data," Econometrica, Econometric Society, vol. 68(1), pages 1-22, January.
    26. Mariani, M.C. & Florescu, I. & Beccar Varela, M.P. & Ncheuguim, E., 2009. "Long correlations and Levy models applied to the study of memory effects in high frequency (tick) data," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(8), pages 1659-1664.
    27. De Bondt, Werner F M & Thaler, Richard H, 1987. "Further Evidence on Investor Overreaction and Stock Market Seasonalit y," Journal of Finance, American Finance Association, vol. 42(3), pages 557-581, July.
    28. 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.
    29. Cajueiro, Daniel O. & Tabak, Benjamin M., 2004. "Evidence of long range dependence in Asian equity markets: the role of liquidity and market restrictions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 342(3), pages 656-664.
    30. Potters, Marc & Bouchaud, Jean-Philippe & Sestovic, Dragan, 2001. "Hedged Monte-Carlo: low variance derivative pricing with objective probabilities," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 289(3), pages 517-525.
    31. Kumar, Sunil & Deo, Nivedita, 2009. "Multifractal properties of the Indian financial market," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(8), pages 1593-1602.
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