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Option pricing using a skew random walk pricing tree

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  • Yuan Hu
  • W. Brent Lindquist
  • Svetlozar T. Rachev
  • Frank J. Fabozzi

Abstract

Motivated by the Corns-Satchell, continuous time, option pricing model, we develop a binary tree pricing model with underlying asset price dynamics following It\^o-Mckean skew Brownian motion. While the Corns-Satchell market model is incomplete, our discrete time market model is defined in the natural world; extended to the risk neutral world under the no-arbitrage condition where derivatives are priced under uniquely determined risk-neutral probabilities; and is complete. The skewness introduced in the natural world is preserved in the risk neutral world. Furthermore, we show that the model preserves skewness under the continuous-time limit. We provide numerical applications of our model to the valuation of European put and call options on exchange-traded funds tracking the S&P Global 1200 index.

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  • Yuan Hu & W. Brent Lindquist & Svetlozar T. Rachev & Frank J. Fabozzi, 2023. "Option pricing using a skew random walk pricing tree," Papers 2303.17014, arXiv.org.
    • Handle: RePEc:arx:papers:2303.17014
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    1. Dilip B. Madan & Peter P. Carr & Eric C. Chang, 1998. "The Variance Gamma Process and Option Pricing," Review of Finance, European Finance Association, vol. 2(1), pages 79-105.
    2. Valeri Zakamouline, 2009. "The Best Hedging Strategy In The Presence Of Transaction Costs," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 12(06), pages 833-860.
    3. Leippold, Markus & Wu, Liuren, 2002. "Asset Pricing under the Quadratic Class," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 37(2), pages 271-295, June.
    4. Pan, Jun, 2002. "The jump-risk premia implicit in options: evidence from an integrated time-series study," Journal of Financial Economics, Elsevier, vol. 63(1), pages 3-50, January.
    5. Dilip B. Madan & Frank Milne, 1991. "Option Pricing With V. G. Martingale Components1," Mathematical Finance, Wiley Blackwell, vol. 1(4), pages 39-55, October.
    6. Leland, Hayne E, 1985. "Option Pricing and Replication with Transactions Costs," Journal of Finance, American Finance Association, vol. 40(5), pages 1283-1301, December.
    7. repec:dau:papers:123456789/9300 is not listed on IDEAS
    8. Ernst Eberlein & Antonis Papapantoleon & Albert Shiryaev, 2008. "On the duality principle in option pricing: semimartingale setting," Finance and Stochastics, Springer, vol. 12(2), pages 265-292, April.
    9. Edirisinghe, Chanaka & Naik, Vasanttilak & Uppal, Raman, 1993. "Optimal Replication of Options with Transactions Costs and Trading Restrictions," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 28(1), pages 117-138, March.
    10. Jean-Guy Simonato, 2011. "Johnson binomial trees," Quantitative Finance, Taylor & Francis Journals, vol. 11(8), pages 1165-1176.
    11. Davide Lauria & W. Brent Lindquist & Stefan Mittnik & Svetlozar T. Rachev, 2022. "ESG-Valued Portfolio Optimization and Dynamic Asset Pricing," Papers 2206.02854, arXiv.org.
    12. Harrison, J. Michael & Pliska, Stanley R., 1983. "A stochastic calculus model of continuous trading: Complete markets," Stochastic Processes and their Applications, Elsevier, vol. 15(3), pages 313-316, August.
    13. Dilip B. Madan & Frank Milne, 1991. "Option Pricing With V. G. Martingale Components," Working Paper 1159, Economics Department, Queen's University.
    14. Boyle, Phelim P & Evnine, Jeremy & Gibbs, Stephen, 1989. "Numerical Evaluation of Multivariate Contingent Claims," The Review of Financial Studies, Society for Financial Studies, vol. 2(2), pages 241-250.
    15. Robert C. Merton, 2005. "Theory of rational option pricing," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 8, pages 229-288, World Scientific Publishing Co. Pte. Ltd..
    16. Svetlozar T. Rachev & Stoyan V. Stoyanov & Frank J. Fabozzi, 2017. "Financial Markets With No Riskless (Safe) Asset," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 20(08), pages 1-24, December.
    17. Alona Bock & Ralf Korn, 2016. "Improving Convergence of Binomial Schemes and the Edgeworth Expansion," Risks, MDPI, vol. 4(2), pages 1-22, May.
    18. Boyle, Phelim P., 1988. "A Lattice Framework for Option Pricing with Two State Variables," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 23(1), pages 1-12, March.
    19. Harrison, J. Michael & Pliska, Stanley R., 1981. "Martingales and stochastic integrals in the theory of continuous trading," Stochastic Processes and their Applications, Elsevier, vol. 11(3), pages 215-260, August.
    20. Nelson, Daniel B & Ramaswamy, Krishna, 1990. "Simple Binomial Processes as Diffusion Approximations in Financial Models," The Review of Financial Studies, Society for Financial Studies, vol. 3(3), pages 393-430.
    21. Yuan Hu & Abootaleb Shirvani & Stoyan Stoyanov & Young Shin Kim & Frank J. Fabozzi & Svetlozar T. Rachev, 2020. "Option Pricing In Markets With Informed Traders," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 23(06), pages 1-32, September.
    22. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    23. Audrino, Francesco & Huitema, Robert & Ludwig, Markus, 2014. "An Empirical Analysis of the Ross Recovery Theorem," Economics Working Paper Series 1411, University of St. Gallen, School of Economics and Political Science.
    24. Atar, Rami & Budhiraja, Amarjit, 2015. "On the multi-dimensional skew Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 125(5), pages 1911-1925.
    25. Yuan Hu & Abootaleb Shirvani & W. Brent Lindquist & Frank J. Fabozzi & Svetlozar T. Rachev, 2021. "Market Complete Option Valuation using a Jarrow-Rudd Pricing Tree with Skewness and Kurtosis," Papers 2106.09128, arXiv.org.
    26. Amin, Kaushik I., 1991. "On the Computation of Continuous Time Option Prices Using Discrete Approximations," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 26(4), pages 477-495, December.
    27. S. G. Kou, 2002. "A Jump-Diffusion Model for Option Pricing," Management Science, INFORMS, vol. 48(8), pages 1086-1101, August.
    28. Claudia Yeap & Simon S Kwok & S T Boris Choy, 2018. "A Flexible Generalized Hyperbolic Option Pricing Model and Its Special Cases," Journal of Financial Econometrics, Oxford University Press, vol. 16(3), pages 425-460.
    29. J. Michael Harrison & Stanley R. Pliska, 1981. "Martingales and Stochastic Integrals in the Theory of Continous Trading," Discussion Papers 454, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    30. Yuan Hu & Abootaleb Shirvani & Stoyan Stoyanov & Young Shin Kim & Frank J. Fabozzi & Svetlozar T. Rachev, 2020. "Option Pricing in Markets with Informed Traders," Papers 2006.02596, arXiv.org, revised Aug 2020.
    31. Steve Ross, 2015. "The Recovery Theorem," Journal of Finance, American Finance Association, vol. 70(2), pages 615-648, April.
    32. Carr, Peter & Wu, Liuren, 2004. "Time-changed Levy processes and option pricing," Journal of Financial Economics, Elsevier, vol. 71(1), pages 113-141, January.
    33. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    34. Broadie, Mark & Cvitanic, Jaksa & Soner, H Mete, 1998. "Optimal Replication of Contingent Claims under Portfolio Constraints," The Review of Financial Studies, Society for Financial Studies, vol. 11(1), pages 59-79.
    35. Kim, Young Shin & Stoyanov, Stoyan & Rachev, Svetlozar & Fabozzi, Frank J., 2019. "Enhancing binomial and trinomial equity option pricing models," Finance Research Letters, Elsevier, vol. 28(C), pages 185-190.
    36. Lo-Bin Chang & Ken Palmer, 2007. "Smooth convergence in the binomial model," Finance and Stochastics, Springer, vol. 11(1), pages 91-105, January.
    37. Yisong “Sam” Tian, 1999. "A flexible binomial option pricing model," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 19(7), pages 817-843, October.
    38. He, Hua, 1990. "Convergence from Discrete- to Continuous-Time Contingent Claims Prices," The Review of Financial Studies, Society for Financial Studies, vol. 3(4), pages 523-546.
    39. L.C.G. Rogers & E.J. Stapleton, 1997. "Fast accurate binomial pricing," Finance and Stochastics, Springer, vol. 2(1), pages 3-17.
    40. Omberg, Edward, 1988. "Efficient Discrete Time Jump Process Models in Option Pricing," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 23(2), pages 161-174, June.
    41. Adelchi Azzalini & Antonella Capitanio, 2003. "Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t‐distribution," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 65(2), pages 367-389, May.
    42. Harrison, J. Michael & Kreps, David M., 1979. "Martingales and arbitrage in multiperiod securities markets," Journal of Economic Theory, Elsevier, vol. 20(3), pages 381-408, June.
    43. T. R. A. Corns & S. E. Satchell, 2007. "Skew Brownian Motion and Pricing European Options," The European Journal of Finance, Taylor & Francis Journals, vol. 13(6), pages 523-544.
    44. Omberg, Edward, 1987. "A Note on the Convergence of Binomial-Pricing and Compound-Option," Journal of Finance, American Finance Association, vol. 42(2), pages 463-469, June.
    45. Abootaleb Shirvani & Yuan Hu & Svetlozar T. Rachev & Frank J. Fabozzi, 2019. "Option Pricing with Mixed Levy Subordinated Price Process and Implied Probability Weighting Function," Papers 1910.05902, arXiv.org, revised Apr 2020.
    46. Paolo Guasoni & Emmanuel Lépinette & Miklós Rásonyi, 2012. "The fundamental theorem of asset pricing under transaction costs," Finance and Stochastics, Springer, vol. 16(4), pages 741-777, October.
    47. 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.
    48. Lai, Tze Leung & Lim, Tiong Wee, 2009. "Option hedging theory under transaction costs," Journal of Economic Dynamics and Control, Elsevier, vol. 33(12), pages 1945-1961, December.
    49. Rendleman, Richard J, Jr & Bartter, Brit J, 1979. "Two-State Option Pricing," Journal of Finance, American Finance Association, vol. 34(5), pages 1093-1110, December.
    50. Bardia Kamrad & Peter Ritchken, 1991. "Multinomial Approximating Models for Options with k State Variables," Management Science, INFORMS, vol. 37(12), pages 1640-1652, December.
    51. Madan, Dilip B & Milne, Frank & Shefrin, Hersh, 1989. "The Multinomial Option Pricing Model and Its Brownian and Poisson Limits," The Review of Financial Studies, Society for Financial Studies, vol. 2(2), pages 251-265.
    52. Hull, John C & White, Alan D, 1987. "The Pricing of Options on Assets with Stochastic Volatilities," Journal of Finance, American Finance Association, vol. 42(2), pages 281-300, June.
    53. Cox, John C. & Ross, Stephen A. & Rubinstein, Mark, 1979. "Option pricing: A simplified approach," Journal of Financial Economics, Elsevier, vol. 7(3), pages 229-263, September.
    54. Brennan, Michael J. & Schwartz, Eduardo S., 1978. "Finite Difference Methods and Jump Processes Arising in the Pricing of Contingent Claims: A Synthesis," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 13(3), pages 461-474, September.
    55. Robert Jarrow, 2012. "The Third Fundamental Theorem Of Asset Pricing," Annals of Financial Economics (AFE), World Scientific Publishing Co. Pte. Ltd., vol. 7(02), pages 1-11.
    56. Black, Fischer, 1972. "Capital Market Equilibrium with Restricted Borrowing," The Journal of Business, University of Chicago Press, vol. 45(3), pages 444-455, July.
    57. Arturo Leccadito & Pietro Toscano & Radu S. Tunaru, 2012. "Hermite Binomial Trees: A Novel Technique For Derivatives Pricing," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 15(08), pages 1-36.
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    1. Svetlozar Rachev & Nancy Asare Nyarko & Blessing Omotade & Peter Yegon, 2023. "Bachelier's Market Model for ESG Asset Pricing," Papers 2306.04158, arXiv.org.

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