# Stochastic Calculus and Applications to Finance

## Objectives

-Main objectives related to knowledge:

Knowledge of Itô calculus. Knowledge of the classical theorem of existence and uniqueness of solution for a stochastic differential equation. Knowledge of the Markov property for the solution of a stochastic differential equation. Knowledge of the relationship between stochastic differential equations and certain deterministic partial differential equations. Knowledge of the Black-Scholes model, and the method of evaluating the price of options through replication.

-Main competencies:

know the Itô calculus. know how to solve linear stochastic differential equations. know how to use the Markov property to manipulate relationships between stochastic differential equations and equations with partial derivatives. Know how to determine the price of assets and derivatives in the Black-Scholes model.

## General characterization

##### Code

11579

##### Credits

6.0

##### Responsible teacher

Maria Fernanda de Almeida Cipriano Salvador Marques

##### Hours

Weekly - 4

Total - 56

##### Teaching language

Português

### Prerequisites

Knowledge of measure theory and Lebesgue integration. Knowledge of ordinary differential equations.

### Bibliography

1 -- Hui-Hsiung Kuo: *Introduction to Stochastic Integration*. Springer. 2006

2 -- Bernt Oksendal: *Stochastic Differential Equations*. Sringer. 1998

### Teaching method

*The professor gives the course by lectures, where he explains all topics referred to in the syllabus.*

### Evaluation method

The evaluation is performed by Continuous Evaluation or Exam.

1- Continuous Evaluation

The Continuous Evaluation is performed through two Tests and a Work. Each Test has the classification of 20 values and the Work has the classification of 20 values.

The classification of Continuous Evaluation is obtained by doing the arithmetic mean of the 3 classifications obtained.

If the Continuous Evaluation classification is higher than or equal to 9.5, the student is approved with this rounded classification to the units.

2 - Exam

All students enrolled in the course (and not yet approved) can realize the Exam.

The final classification will be equal to the classification of the Exam.

If the final classification is higher than or equal to 9.5 the student is approved with this rounded classification to the units.

If the final classification is less than or equal to 9.4 the student is not approved.

## Subject matter

1- Stochastic Integral

2-Itô Formula

3 - Stochastic differential equations (existence and uniqueness theorem)

4- Solving linear equations.

5- Markov''''s property of the solution of a stochastic differential equation.

6- Forward and backward Kolmogorov''''s equations

7- Girsanov''''s Theorem

8- Martingale representation theorem.

9- Black-Sholes Model

9.1- martingale measure

9.2- Replication strategy.

9.3- Price of European options

9.4- Black-Scholes Equation

9.5- Price of barrier options

9.6 Price of American options

9.7- Price of an America option as a solution to a free boundary problem.