Stochastic Processes Applied to Financial Models

Aim

The course gives an introduction to the theory of stochastic processes and the Black-Scholes model. After a completed course the student is expected to be able to:

• handle advanced items and theorems within the theory of stochastic processes, such as the Kolmogorov extension theorem, ergodicity of time discrete Markov chains, the Kolmogorov differential equations for time continuous Markov chains, Wiener process, Ornstein-Uhlenbeck process, stochastic Itô-integral, Itô-formula, Martingales in discrete and continuous time Doleans measure and stopping
• construct solutions to stochastic differential equations
• understand the two different approaches to the Black-Scholes formula, on the one hand by means of the geometric Brownian motion and related partial differential equations, on the other hand by means of the fundamental theorem of pricing
• calculate the fair price of certain financial assets Course content Martingales, Markov processes, stochastic integrals, stochastic differential equations, Brownian motion, Itô's formula, Girsanov's theorem, diffusion processes, random walk, Ising model, Black-Scholes formula, risk-neutal valuation, volatility, geometric Brownian motion and statistical analysis of stock prices, Teaching and working methods Lectures and tutorials. Examination TEN1Written Examination6 creditsU, 3, 4, 5Grades for examination modules are decided in accordance with the assessment criteria presented at the start of the course. Grades Four-grade scale, LiU, U, 3, 4, 5 Other information About teaching and examination languageThe teaching language is presented in the Overview tab for each course. The examination language relates to the teaching language as follows: If teaching language is “Swedish”, the course as a whole could be given in Swedish, or partly in English. Examination language is Swedish, but parts of the examination can be in English.If teaching language is “English”, the course as a whole is taught in English. Examination language is English.If teaching language is “Swedish/English”, the course as a whole will be taught in English if students without prior knowledge of the Swedish language participate. Examination language is Swedish or English depending on teaching language.OtherThe course is conducted in such a way that there are equal opportunities with regard to sex, transgender identity or expression, ethnicity, religion or other belief, disability, sexual orientation and age.The planning and implementation of a course should correspond to the course syllabus. The course evaluation should therefore be conducted with the course syllabus as a starting point. The course is campus-based at the location specified for the course, unless otherwise stated under “Teaching and working methods”. Please note, in a campus-based course occasional remote sessions could be included. Department Matematiska institutionen Course literature BooksPaul, Wolfgang, Baschnagel, Jörg, Stochastic Processes: From Physics to Finance Springer ISBN: 978-3-642-08582-6

Content

Martingales, Markov processes, stochastic integrals, stochastic differential equations, Brownian motion, Itô's formula, Girsanov's theorem, diffusion processes, random walk, Ising model, Black-Scholes formula, risk-neutal valuation, volatility, geometric Brownian motion and statistical analysis of stock prices,