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Jump-Diffusion Models for Valuing the Future: Discounting under Extreme Situations

Author

Listed:
  • Jaume Masoliver

    (Departament de Física de la Matèria Condensada, Universitat de Barcelona, 08028 Barcelona, Spain
    Universitat de Barcelona Institute of Complex Systems (UBICS), 08028 Barcelona, Spain)

  • Miquel Montero

    (Departament de Física de la Matèria Condensada, Universitat de Barcelona, 08028 Barcelona, Spain
    Universitat de Barcelona Institute of Complex Systems (UBICS), 08028 Barcelona, Spain)

  • Josep Perelló

    (Departament de Física de la Matèria Condensada, Universitat de Barcelona, 08028 Barcelona, Spain
    Universitat de Barcelona Institute of Complex Systems (UBICS), 08028 Barcelona, Spain)

Abstract

We develop the process of discounting when underlying rates follow a jump-diffusion process, that is, when, in addition to diffusive behavior, rates suffer a series of finite discontinuities located at random Poissonian times. Jump amplitudes are also random and governed by an arbitrary density. Such a model may describe the economic evolution, specially when extreme situations occur (pandemics, global wars, etc.). When, between jumps, the dynamical evolution is governed by an Ornstein–Uhlenbeck diffusion process, we obtain exact and explicit expressions for the discount function and the long-run discount rate and show that the presence of discontinuities may drastically reduce the discount rate, a fact that has significant consequences for environmental planning. We also discuss as a specific example the case when rates are described by the continuous time random walk.

Suggested Citation

  • Jaume Masoliver & Miquel Montero & Josep Perelló, 2021. "Jump-Diffusion Models for Valuing the Future: Discounting under Extreme Situations," Mathematics, MDPI, vol. 9(14), pages 1-26, July.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:14:p:1589-:d:589615
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    References listed on IDEAS

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    1. Christian Gollier & Phoebe Koundouri & Theologos Pantelidis, 2008. "Declining discount rates: Economic justifications and implications for long-run policy [‘Regime switches in interest rates’]," Economic Policy, CEPR, CESifo, Sciences Po;CES;MSH, vol. 23(56), pages 758-795.
    2. S. G. Kou, 2002. "A Jump-Diffusion Model for Option Pricing," Management Science, INFORMS, vol. 48(8), pages 1086-1101, August.
    3. Cox, John C & Ingersoll, Jonathan E, Jr & Ross, Stephen A, 1981. "A Re-examination of Traditional Hypotheses about the Term Structure of Interest Rates," Journal of Finance, American Finance Association, vol. 36(4), pages 769-799, September.
    4. Stern,Nicholas, 2007. "The Economics of Climate Change," Cambridge Books, Cambridge University Press, number 9780521700801, November.
    5. M. F. M. Osborne, 1959. "Brownian Motion in the Stock Market," Operations Research, INFORMS, vol. 7(2), pages 145-173, April.
    6. Masoliver, Jaume & Weiss, George H., 1988. "First passage time statistics for some stochastic processes with superimposed shot noise," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 149(3), pages 395-405.
    7. 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..
    8. J. Doyne Farmer & John Geanakoplos & Matteo G. Richiardi & Miquel Montero & Josep Perelló & Jaume Masoliver, 2024. "Discounting the Distant Future: What Do Historical Bond Prices Imply about the Long-Term Discount Rate?," Mathematics, MDPI, vol. 12(5), pages 1-25, February.
    9. J. Doyne Farmer & John Geanakoplos, 2009. "Hyperbolic discounting is rational: Valuing the far future with uncertain discount rates," Levine's Working Paper Archive 814577000000000356, David K. Levine.
    10. 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..
    11. Stern, Nicholas, 2014. "Ethics, Equity And The Economics Of Climate Change Paper 1: Science And Philosophy," Economics and Philosophy, Cambridge University Press, vol. 30(3), pages 397-444, November.
    12. Masoliver, Jaume & Montero, Miquel & Perello, Josep & Weiss, George H., 2006. "The continuous time random walk formalism in financial markets," Journal of Economic Behavior & Organization, Elsevier, vol. 61(4), pages 577-598, December.
    13. Newell, Richard G. & Pizer, William A., 2003. "Discounting the distant future: how much do uncertain rates increase valuations?," Journal of Environmental Economics and Management, Elsevier, vol. 46(1), pages 52-71, July.
    14. Martin L. Weitzman, 2009. "On Modeling and Interpreting the Economics of Catastrophic Climate Change," The Review of Economics and Statistics, MIT Press, vol. 91(1), pages 1-19, February.
    15. Josep Perell'o & Miquel Montero & Jaume Masoliver & J. Doyne Farmer & John Geanakoplos, 2019. "Statistical analysis and stochastic interest rate modelling for valuing the future with implications in climate change mitigation," Papers 1910.01928, arXiv.org, revised Feb 2020.
    16. Ryszard Kutner & Jaume Masoliver, 2017. "The continuous time random walk, still trendy: fifty-year history, state of art and outlook," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 90(3), pages 1-13, March.
    17. Arrow, K. & Cropper, M. & Gollier, C. & Groom, B. & Heal, G. & Newell, R. & Nordhaus, W. & Pindyck, R. & Pizer, W. & Portney, P. & Sterner, T. & Tol, R. S. J. & Weitzman, Martin L., 2013. "Determining Benefits and Costs for Future Generations," Scholarly Articles 12841963, Harvard University Department of Economics.
    18. Dothan, L. Uri, 1978. "On the term structure of interest rates," Journal of Financial Economics, Elsevier, vol. 6(1), pages 59-69, March.
    19. Madan, Dilip B & Seneta, Eugene, 1990. "The Variance Gamma (V.G.) Model for Share Market Returns," The Journal of Business, University of Chicago Press, vol. 63(4), pages 511-524, October.
    20. Martin L. Weitzman, 2011. "Fat-Tailed Uncertainty in the Economics of Catastrophic Climate Change," Review of Environmental Economics and Policy, Association of Environmental and Resource Economists, vol. 5(2), pages 275-292, Summer.
    21. Vasicek, Oldrich, 1977. "An equilibrium characterization of the term structure," Journal of Financial Economics, Elsevier, vol. 5(2), pages 177-188, November.
    22. Gilles, Christian & Leroy, Stephen F, 1986. "A Note on the Local Expectations Hypothesis: A Discrete-Time Exposition," Journal of Finance, American Finance Association, vol. 41(4), pages 975-979, September.
    23. repec:bla:jfinan:v:43:y:1988:i:1:p:155-74 is not listed on IDEAS
    24. Stern, Nicholas, 2014. "Ethics, Equity And The Economics Of Climate Change Paper 2: Economics And Politics," Economics and Philosophy, Cambridge University Press, vol. 30(3), pages 445-501, November.
    25. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    26. Weitzman, Martin L., 1998. "Why the Far-Distant Future Should Be Discounted at Its Lowest Possible Rate," Journal of Environmental Economics and Management, Elsevier, vol. 36(3), pages 201-208, November.
    27. J. Doyne Farmer & John Geanakoplos & Jaume Masoliver & Miquel Montero & Josep Perello, 2014. "Discounting the Distant Future," Cowles Foundation Discussion Papers 1951, Cowles Foundation for Research in Economics, Yale University.
    28. William D. Nordhaus, 2007. "A Review of the Stern Review on the Economics of Climate Change," Journal of Economic Literature, American Economic Association, vol. 45(3), pages 686-702, September.
    29. Stern, Nicholas, 2014. "Ethics, equity and the economics of climate change paper 2: economics and politics," LSE Research Online Documents on Economics 62704, London School of Economics and Political Science, LSE Library.
    30. -, 2009. "The economics of climate change," Sede Subregional de la CEPAL para el Caribe (Estudios e Investigaciones) 38679, Naciones Unidas Comisión Económica para América Latina y el Caribe (CEPAL).
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