Annals of Applied Probability

Singular forward–backward stochastic differential equations and emissions derivatives

René Carmona, François Delarue, Gilles-Edouard Espinosa, and Nizar Touzi

Full-text: Open access

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 CO${}_{2}$ 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 CO${}_{2}$ option pricing.

Article information

Source
Ann. Appl. Probab., Volume 23, Number 3 (2013), 1086-1128.

Dates
First available in Project Euclid: 7 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1362684855

Digital Object Identifier
doi:10.1214/12-AAP865

Mathematical Reviews number (MathSciNet)
MR3076679

Zentralblatt MATH identifier
1276.60070

Subjects
Primary: 60H30: Applications of stochastic analysis (to PDE, etc.) 91G80: Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems)

Keywords
Stochastic analysis forward–backward stochastic differential equations emissions derivatives

Citation

Carmona, René; Delarue, François; Espinosa, Gilles-Edouard; Touzi, Nizar. Singular forward–backward stochastic differential equations and emissions derivatives. Ann. Appl. Probab. 23 (2013), no. 3, 1086--1128. doi:10.1214/12-AAP865. https://projecteuclid.org/euclid.aoap/1362684855


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