2024
Advanced Financial Calculus
Name: Advanced Financial Calculus
Code: MAT10217M
6 ECTS
Duration: 15 weeks/156 hours
Scientific Area:
Mathematics
Teaching languages: Portuguese
Languages of tutoring support: Portuguese
Sustainable Development Goals
Learning Goals
Learning outcomes:
Obtain a solid knowledge of stochastic processes, in particular about martingalas,.
Obtain a solid knowledge of stochastic differential equations and discretizations. Know how to apply the theoretical knowledge acquired to
Get a solid knowledge of financial applications.
Skills to be developed by the student:
At the end of this course students will be able to apply the concepts of stochastic differential equations for application to real data modeling. They should also be able to apply acquired financial concepts, namely the application of the Black-Scholes formula for the calculation of european and american call and put options and other financial assets.
Obtain a solid knowledge of stochastic processes, in particular about martingalas,.
Obtain a solid knowledge of stochastic differential equations and discretizations. Know how to apply the theoretical knowledge acquired to
Get a solid knowledge of financial applications.
Skills to be developed by the student:
At the end of this course students will be able to apply the concepts of stochastic differential equations for application to real data modeling. They should also be able to apply acquired financial concepts, namely the application of the Black-Scholes formula for the calculation of european and american call and put options and other financial assets.
Contents
Module1: Introduction to Stochastic Differential Equations and applications:
Wiener Process and diffusions.
Martingales, adapted processes.
Stochastic integrals, sketch of the construction of the Itô integral, and use of Itôs Theorem.
Existence and Uniqueness theorem for Stochastic Differential Equations.
Strong and weak solutions
Formula of Feynman-Kac.
Module2: Financial Applications of Stochastic Differential Equations
Model of Cox-Ross-Rubinstein.
European e american options of buying and selling. Generalization of the methodology to other financial assets.
Statement and interpretation of Girsanovs theorem, transition to the risk-neutral probability.
Derivation of the Black-Scholes formulas.
The model of Black-Scholes at the stock exchange, implicit volatility.
Wiener Process and diffusions.
Martingales, adapted processes.
Stochastic integrals, sketch of the construction of the Itô integral, and use of Itôs Theorem.
Existence and Uniqueness theorem for Stochastic Differential Equations.
Strong and weak solutions
Formula of Feynman-Kac.
Module2: Financial Applications of Stochastic Differential Equations
Model of Cox-Ross-Rubinstein.
European e american options of buying and selling. Generalization of the methodology to other financial assets.
Statement and interpretation of Girsanovs theorem, transition to the risk-neutral probability.
Derivation of the Black-Scholes formulas.
The model of Black-Scholes at the stock exchange, implicit volatility.
Teaching Methods
Teaching methodology:
Structured presentation of stochastic analysis, guided by models.
Illustration of concepts and arguments of the theory by means of concrete examples.
Processing by exercises and applications in other areas of mathematics.
Verification of the level of acquisition of the subject-matter during the period of lecturing by means of classroom questions.
Evaluation:
Continous evaluation - Assessment by list of exercises (50%), to be solved individually and application project (50%) using real data or simulation study using, for example, R software.
Evaluation by exam - The student may opt for the exam evaluation (70%) and project (30%).
Structured presentation of stochastic analysis, guided by models.
Illustration of concepts and arguments of the theory by means of concrete examples.
Processing by exercises and applications in other areas of mathematics.
Verification of the level of acquisition of the subject-matter during the period of lecturing by means of classroom questions.
Evaluation:
Continous evaluation - Assessment by list of exercises (50%), to be solved individually and application project (50%) using real data or simulation study using, for example, R software.
Evaluation by exam - The student may opt for the exam evaluation (70%) and project (30%).
