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A multivariate stochastic unit root model with an application to derivative pricing

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  • Lieberman, Offer
  • Phillips, Peter C.B.

Abstract

This paper extends recent findings of Lieberman and Phillips (2014) on stochastic unit root (STUR) models to a multivariate case including asymptotic theory for estimation of the model’s parameters. The extensions are useful for applications of STUR modeling and 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 substantially reduces the average percentage pricing error of the Black–Scholes and Heston’s (1993) stochastic volatility (with zero volatility risk premium) pricing schemes in most moneyness-maturity categories considered.

Suggested Citation

  • Lieberman, Offer & Phillips, Peter C.B., 2017. "A multivariate stochastic unit root model with an application to derivative pricing," Journal of Econometrics, Elsevier, vol. 196(1), pages 99-110.
    • Handle: RePEc:eee:econom:v:196:y:2017:i:1:p:99-110
    DOI: 10.1016/j.jeconom.2016.05.019
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    1. Peter Christoffersen & Kris Jacobs, 2004. "Which GARCH Model for Option Valuation?," Management Science, INFORMS, vol. 50(9), pages 1204-1221, September.
    2. 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..
    3. 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.
    4. 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.
    5. 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.
    6. 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.
    7. 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.
    8. 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.
    9. Phillips, P C B, 1987. "Time Series Regression with a Unit Root," Econometrica, Econometric Society, vol. 55(2), pages 277-301, March.
    10. 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.
    11. 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.
    12. Lieberman, Offer, 2010. "Asymptotic Theory For Empirical Similarity Models," Econometric Theory, Cambridge University Press, vol. 26(4), pages 1032-1059, August.
    13. 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.
    14. 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.
    15. Phillips, P C B, 1987. "Time Series Regression with a Unit Root," Econometrica, Econometric Society, vol. 55(2), pages 277-301, March.
    16. 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.
    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. 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.
    4. Farzad Sabzikar & Piotr Kokoszka, 2023. "Tempered functional time series," Journal of Time Series Analysis, Wiley Blackwell, vol. 44(3), pages 280-293, May.
    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. 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.
    7. 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.
    8. 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.
    9. Donald W. K. Andrews & Ming Li, 2024. "Inference in a Stationary/Nonstationary Autoregressive Time-Varying-Parameter Model," Cowles Foundation Discussion Papers 2389, Cowles Foundation for Research in Economics, Yale University.
    10. 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.
    11. Lieberman, Offer & Phillips, Peter C.B., 2020. "Hybrid stochastic local unit roots," Journal of Econometrics, Elsevier, vol. 215(1), pages 257-285.
    12. Liyu Dou & Ulrich K. Müller, 2021. "Generalized Local‐to‐Unity Models," Econometrica, Econometric Society, vol. 89(4), pages 1825-1854, July.
    13. Christis Katsouris, 2023. "Estimation and Inference in Threshold Predictive Regression Models with Locally Explosive Regressors," Papers 2305.00860, arXiv.org, revised May 2023.
    14. Horváth, Lajos & Trapani, Lorenzo, 2019. "Testing for randomness in a random coefficient autoregression model," Journal of Econometrics, Elsevier, vol. 209(2), pages 338-352.
    15. Donald W. K. Andrews & Ming Li, 2024. "Inference in a Stationary/Nonstationary Autoregressive Time-Varying-Parameter Model," Papers 2411.00358, arXiv.org.
    16. 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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