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Time-Changed Ornstein-Uhlenbeck Processes And Their Applications In Commodity Derivative Models

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  • Lingfei Li
  • Vadim Linetsky

Abstract

This paper studies subordinate Ornstein-Uhlenbeck (OU) processes, i.e., OU diffusions time changed by L\'{e}vy subordinators. We construct their sample path decomposition, show that they possess mean-reverting jumps, study their equivalent measure transformations, and the spectral representation of their transition semigroups in terms of Hermite expansions. As an application, we propose a new class of commodity models with mean-reverting jumps based on subordinate OU process. Further time changing by the integral of a CIR process plus a deterministic function of time, we induce stochastic volatility and time inhomogeneity, such as seasonality, in the models. We obtain analytical solutions for commodity futures options in terms of Hermite expansions. The models are consistent with the initial futures curve, exhibit Samuelson's maturity effect, and are flexible enough to capture a variety of implied volatility smile patterns observed in commodities futures options.

Suggested Citation

  • Lingfei Li & Vadim Linetsky, 2012. "Time-Changed Ornstein-Uhlenbeck Processes And Their Applications In Commodity Derivative Models," Papers 1204.3679, arXiv.org.
    • Handle: RePEc:arx:papers:1204.3679
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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. Bessembinder, Hendrik, et al, 1995. "Mean Reversion in Equilibrium Asset Prices: Evidence from the Futures Term Structure," Journal of Finance, American Finance Association, vol. 50(1), pages 361-375, March.
    3. Schwartz, Eduardo S, 1997. "The Stochastic Behavior of Commodity Prices: Implications for Valuation and Hedging," Journal of Finance, American Finance Association, vol. 52(3), pages 923-973, July.
    4. repec:dau:papers:123456789/607 is not listed on IDEAS
    5. Peter Carr & Hélyette Geman & Dilip B. Madan & Marc Yor, 2003. "Stochastic Volatility for Lévy Processes," Mathematical Finance, Wiley Blackwell, vol. 13(3), pages 345-382, July.
    6. Jaime Casassus & Pierre Collin‐Dufresne, 2005. "Stochastic Convenience Yield Implied from Commodity Futures and Interest Rates," Journal of Finance, American Finance Association, vol. 60(5), pages 2283-2331, October.
    7. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    8. Hull, John & White, Alan, 1993. "One-Factor Interest-Rate Models and the Valuation of Interest-Rate Derivative Securities," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 28(2), pages 235-254, June.
    9. Nina Boyarchenko & Sergei Levendorskiǐ, 2007. "The Eigenfunction Expansion Method In Multi‐Factor Quadratic Term Structure Models," Mathematical Finance, Wiley Blackwell, vol. 17(4), pages 503-539, October.
    10. Robert S. Pindyck, 2001. "The Dynamics of Commodity Spot and Futures Markets: A Primer," The Energy Journal, , vol. 22(3), pages 1-29, July.
    11. Vadim Linetsky, 2004. "The Spectral Decomposition Of The Option Value," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 7(03), pages 337-384.
    12. Jimmy E. Hilliard & Jorge A. Reis, 1999. "Jump Processes in Commodity Futures Prices and Options Pricing," American Journal of Agricultural Economics, Agricultural and Applied Economics Association, vol. 81(2), pages 273-286.
    13. Les Clewlow & Chris Strickland, 1999. "Valuing Energy Options in a One Factor Model Fitted to Forward Prices," Research Paper Series 10, Quantitative Finance Research Centre, University of Technology, Sydney.
    14. Carr, Peter & Wu, Liuren, 2004. "Time-changed Levy processes and option pricing," Journal of Financial Economics, Elsevier, vol. 71(1), pages 113-141, January.
    15. Liming Feng & Vadim Linetsky, 2008. "Pricing Discretely Monitored Barrier Options And Defaultable Bonds In Lévy Process Models: A Fast Hilbert Transform Approach," Mathematical Finance, Wiley Blackwell, vol. 18(3), pages 337-384, July.
    16. Helyette Geman, 2005. "Commodities and Commodity Derivatives. Modeling and Pricing for Agriculturals, Metals and Energy," Post-Print halshs-00144182, HAL.
    17. Liming Feng & Vadim Linetsky, 2009. "Computing exponential moments of the discrete maximum of a Lévy process and lookback options," Finance and Stochastics, Springer, vol. 13(4), pages 501-529, September.
    18. Hilliard, Jimmy E. & Reis, Jorge, 1998. "Valuation of Commodity Futures and Options under Stochastic Convenience Yields, Interest Rates, and Jump Diffusions in the Spot," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 33(1), pages 61-86, March.
    19. repec:dau:papers:123456789/1392 is not listed on IDEAS
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    Cited by:

    1. Lingfei Li & Vadim Linetsky, 2013. "Optimal Stopping and Early Exercise: An Eigenfunction Expansion Approach," Operations Research, INFORMS, vol. 61(3), pages 625-643, June.
    2. Orcan Ogetbil & Bernhard Hientzsch, 2022. "A Flexible Commodity Skew Model with Maturity Effects," Papers 2212.07972, arXiv.org.

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