Modelling, Mathematical Analysis and Numerical Simulation to Value Derivatives Related to Renewable Energy Certificates

Programa Oficial de Doutoramento en Métodos Matemáticos e Simulación Numérica en Enxeñaría e Ciencias Aplicadas. 551V01[Resumo] O obxectivo principal desta tese céntrase na modelaxe, análise matemática e resolución numérica de problemas de ecuacións en derivadas parciais (EDPs) para a fixación de prezos de certificados de enerxía renovable (RECs, polas súas siglas en inglés) e produtos derivados asociados. Na modelaxe, o prezo do REC xoga un papel importante. Proponse un modelo de EDP non lineal con dous factores estocásticos. Os factores estocásticos son os certificados verdes acumulados e a taxa de xeración de enerxía renovable. Unha novidade desta tese é o tratamento numérico do término convectivo non lineal na EDP. Para resolver o problema linealizado obtido, propóñense esquemas de semi-Lagrange en tempo conxugados con diferenzas finitas, ou m´etodos alternativos de Lagrange-Galerkin. Utilizouse unha metodoloxía semellante para a valoración dos derivados de REC para obter un modelo de EDP lineal unha vez coñecido o prezo do REC. A existencia de solución obtense neste escenario. Abórdase a fixación de prezos de opcións europeas e futuros sobre RECs. Finalmente, móstranse os resultados do comportamento dos modelos e dos métodos numéricos implementados.[Resumen] El objetivo principal de esta tesis se centra en el modelado, análisis matemático y resolución numérica de problemas de ecuaciones en derivadas parciales (EDPs) para la fijación de precios de certificados de energía renovable (RECs, por sus siglas en inglés) y productos derivados asociados. En el modelado, el precio del REC juega un papel relevante. Se propone un modelo de EDP no lineal con dos factores estocásticos. Los factores estocásticos son los certificados verdes acumulados y la tasa de generación de energía renovable. Una novedad de esta tesis es el tratamiento numérico del término convectivo no lineal en la EDP. Para resolver el problema linealizado obtenido, se proponen esquemas de semi-Lagrange en tiempo combinados con diferencias finitas, o métodos alternativos de Lagrange-Galerkin. Se ha utilizado una metodología equivalente para la valoración de los derivados de REC para obtener un modelo de EDP lineal una vez conocido el precio del REC. La existencia de solución se obtiene en este escenario. Se aborda la fijación de precios de opciones europeas y futuros sobre RECs. Finalmente, se muestran resultados del comportamiento de los modelos y de los métodos numéricos implementados.[Abstract] The main objective of this thesis concerns to the modelling, mathematical analysis and numerical solution of partial differential equations (PDEs) models for pricing renewable energy certificates (RECs) and associated derivatives products. In the modelling, the price of the REC plays a relevant role. A non-linear PDE model with two stochastic factors is proposed. The stochastic factors are the accumulated green certificates and the renewable electricity generation rate. One novelty of this thesis comes from the numerical treatment of the non-linear convective term in the PDE. In order to solve the obtained linearized problem, semi-Lagrangian schemes in time combined with finite differences discretizations, or alternative Lagrange-Galerkin methods are proposed. An equivalent methodology has been used for the valuation of the REC derivatives to obtain a linear PDE model once the REC price is known. Existence of solution is obtained in this setting. The application to the pricing of European options and futures on RECs is addressed. Finally, we show illustrative results of the performance of the models and numerical methods that have been implemented.Xunta de Galicia; ED431C 2018/033Xunta de Galicia; ED431G 2019/01This research has been partially funded by the following projects: • Project PID2019-108584RB-I00 from Ministerio de Ciencia e Innovavi_on. • Project MTM2016-76497-R from Ministerio de Economía y Competitividad. • Project ED431C 2018/033 from Xunta de Galicia. • Project ED431G 2019/01 from Xunta de Galicia. All previous projects include FEDER funding

Model and numerical methods for pricing renewable energy certificate derivatives

[Abstract]: Assuming that the price of the renewable energy certificate (REC) is known, we formulate the valuation problem of a European REC derivative in terms of a linear PDE model where the underlying stochastic factors are the accumulated green certificates sold by an authorized producer and the natural logarithm of the renewable generation rate. Next, a result of existence of solution is obtained for this PDE problem. In order to solve numerically the PDE problem, we propose two different methods: a semi-Lagrangian scheme in time combined with finite differences discretizations in the spatial-like variables, or alternative Lagrange–Galerkin methods. The application to the pricing of European options and futures on renewable energy certificates is addressed. Finally, we show illustrative results about the performance of the models and numerical methods that have been implemented.Xunta de Galicia; ED431C 2018/033Xunta de Galicia; ED431C 2022/047Xunta de Galicia; ED431G 2019/01This work has been funded by MICINN from Spain (Grant PID2019-10858RB-I00), and by Xunta of Galicia (Grant ED431C 2018/033 and ED431C 2022/047), all including FEDER financial co-funding. All authors are members of CITIC, so that they acknowledge grant ED431G 2019/01, funded by Consellería de Educación, Universidade e Formación Profesional of Xunta de Galicia through FEDER, Spain funds with 80%, from FEDER Galicia 2014–2020 Program and 20% from Secretaría Xeral de Universidades. Funding for open access charge: Universidade da Coruña/CISUG. Authors are also grateful to two anonymous reviewers whose remarks helped to improve the article

Pricing renewable energy certificates with a Crank–Nicolson Lagrange–Galerkin numerical method

Financiado para publicación en acceso aberto: Universidade da Coruña/CISUG[Abstract]: The valuation problem of renewable energy certificates can be formulated in terms of a nonlinear PDE model where the underlying stochastic factors are the accumulated green certificates sold by an authorized producer and the natural logarithm of the renewable generation rate. In the present paper, the nonlinear convective term is treated with the Bermúdez–Moreno duality method for maximal monotone operators as in Baamonde-Seoane et al. (2021). The main novelty of this article comes from the proposed techniques for the numerical solution of the resulting linear problem. In this case, we propose a Lagrange–Galerkin method which mainly consists of Crank–Nicolson characteristics for time discretization combined with finite elements for the discretization in the accumulated green certificates and the natural logarithm of the renewable generation rate directions. Finally, several numerical examples are presented to illustrate the good performance of the method and model, and its comparison with other numerical schemes employed to solve the same problem.Xunta de Galicia; ED431C 2018/033Xunta de Galicia; ED431C 2022/47Xunta de Galicia; ED431G 2019/01The authors acknowledge the funding by Spanish MINECO with the grants MTM2016-76497-R and PID2019-10858RB-I00, and by Galician Government, Spain with the grants ED431C 2018/033 and ED431C 2022/47, both including FEDER financial support. As members of CITIC, they also acknowledge the grant ED431G 2019/01, funded by Consellería de Educación, Spain , Universidade e Formación Profesional of Xunta de Galicia through FEDER funds with 80%, from FEDER Galicia 2014–2020 Program and 20% from Secretaría Xeral de Universidades

Method for Pricing Renewable Energy Certificates

Cursos e Congresos, C-155[Abstract] In this work we present one valuation method for Renewable Energy Certificates (RECs). Starting from a system of FBSDEs and using Ito lemma, we propose a mathematical model based on a semilinear PDE with two stochastic factors. The main novelty comes from the use of the Bermúdez-Moreno algorithm to deal the non-linear convective term in the PDE. This duality algorithm is based on the Yosida regularization of non-linear maximal monotone operators. The resulting linear problem is discretized by using a characteristics method combined with a second order implicit finite differences scheme. We showillustrative results of the performance of the proposed model and the numerical methodCITIC is funded by the Xunta de Galicia through the collaboration agreement between the Consellería de Cultura, Educación, Formación Profesional e Universidades and the Galician universities for the reinforcement of the research centres of the Galician University System (CIGUS