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A Multivariate Stochastic Unit Root Model with an Application to Derivative Pricing

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Abstract

This paper extends recent findings of Lieberman and Phillips (2014) on stochastic unit root (SUR) models to a multivariate case including a comprehensive asymptotic theory for estimation of the model's parameters. The extensions are useful because they lead to a generalization of the Black-Scholes formula for derivative pricing. In place of the standard assumption that the price process follows a geometric Brownian motion, we derive a new form of the Black-Scholes equation that allows for a multivariate time varying coefficient element in the price equation. The corresponding formula for the value of a European-type call option is obtained and shown to extend the existing option price formula in a manner that embodies the effect of a stochastic departure from a unit root. An empirical application reveals that the new model is consistent with excess skewness and kurtosis in the price distribution relative to a lognormal distribution.

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  • Offer Lieberman & Peter C.B. Phillips, 2014. "A Multivariate Stochastic Unit Root Model with an Application to Derivative Pricing," Cowles Foundation Discussion Papers 1964, Cowles Foundation for Research in Economics, Yale University.
    • Handle: RePEc:cwl:cwldpp:1964
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    References listed on IDEAS

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    1. Bakshi, Gurdip & Cao, Charles & Chen, Zhiwu, 1997. "Empirical Performance of Alternative Option Pricing Models," Journal of Finance, American Finance Association, vol. 52(5), pages 2003-2049, December.
    2. Peter C. B. Phillips & Jun Yu, 2011. "Dating the timeline of financial bubbles during the subprime crisis," Quantitative Economics, Econometric Society, vol. 2(3), pages 455-491, November.
    3. Peter C. B. Phillips, 2015. "Edmond Malinvaud: a tribute to his contributions in econometrics," Econometrics Journal, Royal Economic Society, vol. 18(2), pages 1-13, June.
    4. Phillips, P C B, 1987. "Time Series Regression with a Unit Root," Econometrica, Econometric Society, vol. 55(2), pages 277-301, March.
    5. Itzhak Gilboa & Offer Lieberman & David Schmeidler, 2012. "Empirical Similarity," World Scientific Book Chapters, in: Case-Based Predictions An Axiomatic Approach to Prediction, Classification and Statistical Learning, chapter 9, pages 211-243, World Scientific Publishing Co. Pte. Ltd..
    6. Peter Christoffersen & Kris Jacobs, 2004. "Which GARCH Model for Option Valuation?," Management Science, INFORMS, vol. 50(9), pages 1204-1221, September.
    7. 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.
    8. Phillips, Peter C.B. & Magdalinos, Tassos, 2007. "Limit theory for moderate deviations from a unit root," Journal of Econometrics, Elsevier, vol. 136(1), pages 115-130, January.
    9. Offer Lieberman & Peter C. B. Phillips, 2014. "Norming Rates And Limit Theory For Some Time-Varying Coefficient Autoregressions," Journal of Time Series Analysis, Wiley Blackwell, vol. 35(6), pages 592-623, November.
    10. Offer Lieberman, 2012. "A similarity‐based approach to time‐varying coefficient non‐stationary autoregression," Journal of Time Series Analysis, Wiley Blackwell, vol. 33(3), pages 484-502, May.
    11. Ai[diaeresis]t-Sahalia, Yacine & Kimmel, Robert, 2007. "Maximum likelihood estimation of stochastic volatility models," Journal of Financial Economics, Elsevier, vol. 83(2), pages 413-452, February.
    12. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    13. Lieberman, Offer, 2010. "Asymptotic Theory For Empirical Similarity Models," Econometric Theory, Cambridge University Press, vol. 26(4), pages 1032-1059, August.
    14. Phillips, P C B, 1987. "Time Series Regression with a Unit Root," Econometrica, Econometric Society, vol. 55(2), pages 277-301, March.
    15. 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.
    16. Eric Renault & Nizar Touzi, 1996. "Option Hedging And Implied Volatilities In A Stochastic Volatility Model1," Mathematical Finance, Wiley Blackwell, vol. 6(3), pages 279-302, July.
    17. 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.
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    Cited by:

    1. Lieberman, Offer & Phillips, Peter C.B., 2022. "Understanding temporal aggregation effects on kurtosis in financial indices," Journal of Econometrics, Elsevier, vol. 227(1), pages 25-46.
    2. Chaoyi Chen & Thanasis Stengos, 2022. "Estimation and Inference for the Threshold Model with Hybrid Stochastic Local Unit Root Regressors," JRFM, MDPI, vol. 15(6), pages 1-15, May.
    3. Bykhovskaya, Anna & Phillips, Peter C.B., 2020. "Point optimal testing with roots that are functionally local to unity," Journal of Econometrics, Elsevier, vol. 219(2), pages 231-259.
    4. Liyu Dou & Ulrich K. Müller, 2021. "Generalized Local‐to‐Unity Models," Econometrica, Econometric Society, vol. 89(4), pages 1825-1854, July.
    5. Tao, Yubo & Phillips, Peter C.B. & Yu, Jun, 2019. "Random coefficient continuous systems: Testing for extreme sample path behavior," Journal of Econometrics, Elsevier, vol. 209(2), pages 208-237.
    6. Samuel Brien & Michael Jansson & Morten Ørregaard Nielsen, 2022. "Nearly Efficient Likelihood Ratio Tests of a Unit Root in an Autoregressive Model of Arbitrary Order," Working Paper 1429, Economics Department, Queen's University.
    7. Farzad Sabzikar & Piotr Kokoszka, 2023. "Tempered functional time series," Journal of Time Series Analysis, Wiley Blackwell, vol. 44(3), pages 280-293, May.
    8. Lieberman, Offer & Phillips, Peter C.B., 2020. "Hybrid stochastic local unit roots," Journal of Econometrics, Elsevier, vol. 215(1), pages 257-285.
    9. Christis Katsouris, 2023. "Estimation and Inference in Threshold Predictive Regression Models with Locally Explosive Regressors," Papers 2305.00860, arXiv.org, revised May 2023.
    10. Muriel, Nelson & González-Farías, Graciela, 2018. "Testing the null of difference stationarity against the alternative of a stochastic unit root: A new test based on multivariate STUR," Econometrics and Statistics, Elsevier, vol. 7(C), pages 46-62.
    11. Horváth, Lajos & Trapani, Lorenzo, 2019. "Testing for randomness in a random coefficient autoregression model," Journal of Econometrics, Elsevier, vol. 209(2), pages 338-352.
    12. Liu, Yanbo & Phillips, Peter C.B., 2023. "Robust inference with stochastic local unit root regressors in predictive regressions," Journal of Econometrics, Elsevier, vol. 235(2), pages 563-591.
    13. Lingjie Du & Tianxiao Pang, 2021. "Asymptotic Theory for a Stochastic Unit Root Model with Intercept and Under Mis-Specification of Intercept," Methodology and Computing in Applied Probability, Springer, vol. 23(3), pages 767-799, September.
    14. Andreas Hetland, 2018. "The Stochastic Stationary Root Model," Econometrics, MDPI, vol. 6(3), pages 1-33, August.

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    More about this item

    Keywords

    Autoregression; Derivative; Diffusion; Options; Similarity; Stochastic unit root; Time-varying coefficients;
    All these keywords.

    JEL classification:

    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes

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