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Is normal backwardation normal? Valuing financial futures with a local index-rate covariance

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  • Raimbourg, Philippe
  • Zimmermann, Paul

Abstract

Revisiting the two-factor valuation of futures contracts, we propose a new pricing model for financial futures and their derivatives. The linkage between the money market funding rate and the underlying asset price is stochastic and state-dependent, in compliance with investors’ arbitrage strategies. The model explicitly captures the impact of interest rate expectations in the marking-to-market feature of futures, as predicted by Cox, Ingersoll, and Ross (1981) theory. The backwardation vs. contango regime of financial futures depends on a new parameter, the contango factor, which paves the way for future empirical studies. Akin to the implied volatility of option contracts, the contango factor provides market participants with a universal gauge of futures contracts’ level of contango, consistent across futures markets and maturities. Our numerical simulations show significant deviations from the traditional cost-of-carry model of futures prices, with price deviations above 1% even for short-term futures contracts.

Suggested Citation

  • Raimbourg, Philippe & Zimmermann, Paul, 2022. "Is normal backwardation normal? Valuing financial futures with a local index-rate covariance," European Journal of Operational Research, Elsevier, vol. 298(1), pages 351-367.
  • Handle: RePEc:eee:ejores:v:298:y:2022:i:1:p:351-367
    DOI: 10.1016/j.ejor.2021.06.051
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    as
    1. Eugene F. Fama & Kenneth R. French, 2015. "Commodity Futures Prices: Some Evidence on Forecast Power, Premiums, and the Theory of Storage," World Scientific Book Chapters, in: Anastasios G Malliaris & William T Ziemba (ed.), THE WORLD SCIENTIFIC HANDBOOK OF FUTURES MARKETS, chapter 4, pages 79-102, World Scientific Publishing Co. Pte. Ltd..
    2. Paul H. Cootner, 1960. "Returns to Speculators: Telser versus Keynes," Journal of Political Economy, University of Chicago Press, vol. 68(4), pages 396-396.
    3. Tuomo Vuolteenaho, 2002. "What Drives Firm‐Level Stock Returns?," Journal of Finance, American Finance Association, vol. 57(1), pages 233-264, February.
    4. Robert A. JARROW & George S. OLDFIELD, 2008. "Forward Contracts And Futures Contracts," World Scientific Book Chapters, in: Financial Derivatives Pricing Selected Works of Robert Jarrow, chapter 11, pages 237-246, World Scientific Publishing Co. Pte. Ltd..
    5. Vasicek, Oldrich, 1977. "An equilibrium characterization of the term structure," Journal of Financial Economics, Elsevier, vol. 5(2), pages 177-188, November.
    6. 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.
    7. repec:bla:jfinan:v:43:y:1988:i:3:p:661-76 is not listed on IDEAS
    8. Bryan R. Routledge & Duane J. Seppi & Chester S. Spatt, 2000. "Equilibrium Forward Curves for Commodities," Journal of Finance, American Finance Association, vol. 55(3), pages 1297-1338, June.
    9. 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.
    10. Campbell, J.Y. & Shiller, R.J., 1988. "Stock Prices, Earnings And Expected Dividends," Papers 334, Princeton, Department of Economics - Econometric Research Program.
    11. François Longin & Bruno Solnik, 2001. "Extreme Correlation of International Equity Markets," Journal of Finance, American Finance Association, vol. 56(2), pages 649-676, April.
    12. Fishburn, Peter C, 1977. "Mean-Risk Analysis with Risk Associated with Below-Target Returns," American Economic Review, American Economic Association, vol. 67(2), pages 116-126, March.
    13. Vasicek, Oldrich Alfonso, 1977. "Abstract: An Equilibrium Characterization of the Term Structure," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 12(4), pages 627-627, November.
    14. 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.
    15. Laura L. Veldkamp, 2006. "Information Markets and the Comovement of Asset Prices," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 73(3), pages 823-845.
    16. 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.
    17. Donald Lien & Y. K. Tse, 2002. "Some Recent Developments in Futures Hedging," Journal of Economic Surveys, Wiley Blackwell, vol. 16(3), pages 357-396, July.
    18. repec:bla:jecsur:v:14:y:2000:i:2:p:215-53 is not listed on IDEAS
    19. Kwiatkowski, Denis & Phillips, Peter C. B. & Schmidt, Peter & Shin, Yongcheol, 1992. "Testing the null hypothesis of stationarity against the alternative of a unit root : How sure are we that economic time series have a unit root?," Journal of Econometrics, Elsevier, vol. 54(1-3), pages 159-178.
    20. Campbell, John Y, 1991. "A Variance Decomposition for Stock Returns," Economic Journal, Royal Economic Society, vol. 101(405), pages 157-179, March.
    21. 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.
    22. Holbrook Working, 1948. "Theory of the Inverse Carrying Charge in Futures Markets," American Journal of Agricultural Economics, Agricultural and Applied Economics Association, vol. 30(1), pages 1-28.
    23. Dickey, David A & Fuller, Wayne A, 1981. "Likelihood Ratio Statistics for Autoregressive Time Series with a Unit Root," Econometrica, Econometric Society, vol. 49(4), pages 1057-1072, June.
    24. French, Kenneth R., 1983. "A comparison of futures and forward prices," Journal of Financial Economics, Elsevier, vol. 12(3), pages 311-342, November.
    25. Lam, Kin & Yu, P.L.H. & Lee, P.H., 2010. "A margin scheme that advises on when to change required margin," European Journal of Operational Research, Elsevier, vol. 207(1), pages 524-530, November.
    26. Whitelaw, Robert F, 1994. "Time Variations and Covariations in the Expectation and Volatility of Stock Market Returns," Journal of Finance, American Finance Association, vol. 49(2), pages 515-541, June.
    27. Steven L. Heston & Mark Loewenstein & Gregory A. Willard, 2007. "Options and Bubbles," The Review of Financial Studies, Society for Financial Studies, vol. 20(2), pages 359-390.
    28. John C. Cox & Jonathan E. Ingersoll Jr. & Stephen A. Ross, 2005. "A Theory Of The Term Structure Of Interest Rates," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 5, pages 129-164, World Scientific Publishing Co. Pte. Ltd..
    29. Hull, John & White, Alan, 1990. "Valuing Derivative Securities Using the Explicit Finite Difference Method," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 25(1), pages 87-100, March.
    30. Ramaswamy, Krishna & Sundaresan, Suresh M, 1985. "The Valuation of Options on Futures Contracts," Journal of Finance, American Finance Association, vol. 40(5), pages 1319-1340, December.
    31. Hull, John & White, Alan, 1990. "Pricing Interest-Rate-Derivative Securities," The Review of Financial Studies, Society for Financial Studies, vol. 3(4), pages 573-592.
    32. Nicholas Kaldor, 1939. "Speculation and Economic Stability," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 7(1), pages 1-27.
    33. Andrea Buraschi & Paolo Porchia & Fabio Trojani, 2010. "Correlation Risk and Optimal Portfolio Choice," Journal of Finance, American Finance Association, vol. 65(1), pages 393-420, February.
    34. 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.
    35. Ying‐Foon Chow & Michael McAleer & John Sequeira, 2000. "Pricing of Forward and Futures Contracts," Journal of Economic Surveys, Wiley Blackwell, vol. 14(2), pages 215-253, April.
    36. John Y. Campbell, Robert J. Shiller, 1988. "The Dividend-Price Ratio and Expectations of Future Dividends and Discount Factors," The Review of Financial Studies, Society for Financial Studies, vol. 1(3), pages 195-228.
    37. repec:bla:jecsur:v:16:y:2002:i:3:p:357-96 is not listed on IDEAS
    38. Bawa, Vijay S. & Lindenberg, Eric B., 1977. "Capital market equilibrium in a mean-lower partial moment framework," Journal of Financial Economics, Elsevier, vol. 5(2), pages 189-200, November.
    39. Peter Hecht & Tuomo Vuolteenaho, 2006. "Explaining Returns with Cash-Flow Proxies," The Review of Financial Studies, Society for Financial Studies, vol. 19(1), pages 159-194.
    40. 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.
    41. Jaime Casassus & Peng Liu & Ke Tang, 2013. "Economic Linkages, Relative Scarcity, and Commodity Futures Returns," The Review of Financial Studies, Society for Financial Studies, vol. 26(5), pages 1324-1362.
    42. David Heath & Robert Jarrow & Andrew Morton, 2008. "Bond Pricing And The Term Structure Of Interest Rates: A New Methodology For Contingent Claims Valuation," World Scientific Book Chapters, in: Financial Derivatives Pricing Selected Works of Robert Jarrow, chapter 13, pages 277-305, World Scientific Publishing Co. Pte. Ltd..
    43. Robert W. Kolb, 1992. "Is normal backwardation normal?," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 12(1), pages 75-91, February.
    44. Michael W. Brandt & Francis X. Diebold, 2006. "A No-Arbitrage Approach to Range-Based Estimation of Return Covariances and Correlations," The Journal of Business, University of Chicago Press, vol. 79(1), pages 61-74, January.
    45. Chiu, Mei Choi & Wong, Hoi Ying & Zhao, Jing, 2015. "Commodity derivatives pricing with cointegration and stochastic covariances," European Journal of Operational Research, Elsevier, vol. 246(2), pages 476-486.
    46. Hemler, Michael L. & Longstaff, Francis A., 1991. "General Equilibrium Stock Index Futures Prices: Theory and Empirical Evidence," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 26(3), pages 287-308, September.
    47. Cox, John C. & Ingersoll, Jonathan Jr. & Ross, Stephen A., 1981. "The relation between forward prices and futures prices," Journal of Financial Economics, Elsevier, vol. 9(4), pages 321-346, December.
    48. Long Chen & Xinlei Zhao, 2009. "Return Decomposition," The Review of Financial Studies, Society for Financial Studies, vol. 22(12), pages 5213-5249, December.
    49. 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.
    50. Joost Driessen & Pascal J. Maenhout & Grigory Vilkov, 2009. "The Price of Correlation Risk: Evidence from Equity Options," Journal of Finance, American Finance Association, vol. 64(3), pages 1377-1406, June.
    51. Black, Fischer, 1976. "The pricing of commodity contracts," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 167-179.
    52. 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.
    53. Miltersen, Kristian R. & Schwartz, Eduardo S., 1998. "Pricing of Options on Commodity Futures with Stochastic Term Structures of Convenience Yields and Interest Rates," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 33(1), pages 33-59, March.
    54. Gourieroux, C. & Jasiak, J. & Sufana, R., 2009. "The Wishart Autoregressive process of multivariate stochastic volatility," Journal of Econometrics, Elsevier, vol. 150(2), pages 167-181, June.
    55. Laws, Jason & Thompson, John, 2004. "The efficiency of financial futures markets: Tests of prediction accuracy," European Journal of Operational Research, Elsevier, vol. 155(2), pages 284-298, June.
    56. Chen, Sheng-Syan & Lee, Cheng-few & Shrestha, Keshab, 2003. "Futures hedge ratios: a review," The Quarterly Review of Economics and Finance, Elsevier, vol. 43(3), pages 433-465.
    57. Donald Lien & Yiu Kuen Tse, 1998. "Hedging time‐varying downside risk," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 18(6), pages 705-722, September.
    58. Alexander, Carol & Kaeck, Andreas & Sumawong, Anannit, 2019. "A parsimonious parametric model for generating margin requirements for futures," European Journal of Operational Research, Elsevier, vol. 273(1), pages 31-43.
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