Annals of Applied Probability
- Ann. Appl. Probab.
- Volume 23, Number 3 (2013), 1086-1128.
Singular forward–backward stochastic differential equations and emissions derivatives
René Carmona, François Delarue, Gilles-Edouard Espinosa, and 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 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