## Info about the course

**Labs**: Wednesdays, 14:15-15:45, J. Liivi 2 - 003**Lecturer**: Raul Kangro (Institute of Mathematics and Statistics), raul.kangro@ut.ee**Course Home Page in Moodle**: https://moodle.ut.ee/course/view.php?id=1435**Amount of credits**: 3 EAP

**Goals**: After successful completion of the course the student

- Knows the principles and practical error estimates of the Monte-Carlo method; is able to generate the trajectories of the solutions of stochastic differential equations (especially stock price trajectories) and knows how to use the skill in constructing simulation methods for pricing financial options
- Knows several variance reductions methods for speeding up the Monte-Carlo simulation (antithetic variates, control variables, importance sampling, stratified sampling) and is able to use them in option pricing
- Knows the concept of the quasi-random variables and is able to use them in option pricing.

**Topics of the course**: Introduction to the R package. Generating random numbers. MC method for numerical computation of expected values of random variables. Numerical evaluation of integrals using MC method. Implementing Black-Scholes formulas in R. Simulating the trajectories of stock prices. Implementing Euler's method. The analysis of the convergence rate of the method. Euler's method in the case of a general market model. Milstein's method and a weakly second order method for generating stock prices. Variance reduction methods: antithetic variates and control variates, importance sampling and stratified sampling. Additional points about stratified sampling, using stratified sampling for generating the trajectories of the stock price. Pricing Asian options by MC. Using stratified sampling for pricing Asian options. Quasi-Monte-Carlo methods, Halton points, Sobol points. Computing the option price sensitivities with MC. Using MC for pricing American options.

**Independent works**: There are 8 practical homework assignments (one in every second week starting from the first one). The homework solutions are due one week after they were handed out. The late submissions are allowed but the score for such submissions is reduced by 50%. The solutions to the practical homework assignments have to be submitted through the Moodle web page of the course as R scripts (.R files) or R notebook files (.Rmd files). Maximal score for each homework assignment is 5 points. The maximal total score for homework assignments is 40 points.

**Requirement to be met for final assessment**: At least 20 points (50%) for homework assignments is required for qualifying for the final examination.

**Composition of the final grade**: 60% of the total score is given by the final exam, 40% comes from the homework assignments. The final exam takes place in a computer lab and consists of 4 computational problems related to option pricing. The final grade is determined by the total score as follows: the score less than 50 gives F, from 50 to 59.9 gives E, from 60 to 69.9 gives D, from 70 to 79.9 gives C, from 80 to 89.9 gives B and a score of 90 or more gives A

**E-learning activities**: The course materials are divided between 16 study weeks and can be used for independent study or as supporting materials for the lectures and the computer labs. The lecture notes and lab handouts contain all of the theoretical materials that is required for this course. The solutions of the 8 practical homework assignments have to be submitted through Moodle. The final examination has to be taken in person.