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Singular Forward-Backward Stochastic Differential Equations and Emissions Derivatives

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  • Rene Carmona
  • Francois Delarue
  • Gilles-Edouard Espinosa
  • Nizar Touzi

Abstract

We introduce two simple models of forward-backward stochastic differential equations with a singular terminal condition and we explain how and why they appear naturally as models for the valuation of CO2 emission allowances. Single phase cap-and-trade schemes lead readily to terminal conditions given by indicator functions of the forward component, and using fine partial differential equations estimates, we show that the existence theory of these equations, as well as the properties of the candidates for solution, depend strongly upon the characteristics of the forward dynamics. Finally, we give a first order Taylor expansion and show how to numerically calibrate some of these models for the purpose of CO2 option pricing.

Suggested Citation

  • Rene Carmona & Francois Delarue & Gilles-Edouard Espinosa & Nizar Touzi, 2012. "Singular Forward-Backward Stochastic Differential Equations and Emissions Derivatives," Papers 1210.5773, arXiv.org.
  • Handle: RePEc:arx:papers:1210.5773
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    File URL: http://arxiv.org/pdf/1210.5773
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    References listed on IDEAS

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    1. Seifert, Jan & Uhrig-Homburg, Marliese & Wagner, Michael, 2008. "Dynamic behavior of CO2 spot prices," Journal of Environmental Economics and Management, Elsevier, vol. 56(2), pages 180-194, September.
    2. Marc Chesney & Luca Taschini, 2008. "The Endogenous Price Dynamics of the Emission Allowances: An Application to CO2 Option Pricing," Swiss Finance Institute Research Paper Series 08-01, Swiss Finance Institute, revised Jan 2008.
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