## Info about the course

**Lectures**: Tuesdays, 12:15-13:45, Delta - 2039**Labs**: Wednesdays, 12:15-13:45, Delta - 2030**Lecturer**: Raul Kangro (Assoc. prof., Institute of Mathematics and statistics), raul.kangro@ut.ee**Amount of credits**: 6 EAP

Course homepage in Moodle

**Goals**: After successful completion of the course the students derive and implement numerical methods for pricing various financial options. More precisely, after successful completion of the course the students

- know the basic properties and derivation procedures of the partial differential equations arising in mathematical finance and can apply the procedures in the case of new market models,
- can derive numerical methods for computing the prices of financial options and are able to implement them in Python programming language,
- obtain practical skills for computing the prices of common options,
- know and can use some popular methods of estimating the parameters of various market models and for computing the quantities that are necessary for constructing hedging portfolios.

**Topics of the course**: Definition and types of financial options. Arbitrage principle. Black-Scholes market model. Methods for estimating market parameters: maximal likelihood method, least squares minimization. Monte-Carlo method for pricing options. Derivation of Black-Scholes Partial Differential Equation (PDE) for European options. Alternative approaches for deriving PDE. The idea of finite difference methods. Explicit method, implicit method, Crank-Nicolson method. Stability of numerical methods. Computing option prices with given accuracy. Derivation of PDE for Asian options. A finite difference method for pricing Asian options. PDE and an explicit numerical method for options depending on two stocks. The idea of finite element methods. Derivation of the method for pricing European options.

**Independent works**: There are 3 theoretical homework assignments (specified in electronic course materials) and 7 practical homework assignments (one in every second week starting from the third 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 Python files, the solutions of the theoretical homework assignments can be submitted either via Moodle or in the lecture. Maximal score for each homework assignment is 4 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 consists of a theoretical part (derivation of a numerical method, derivation of PDE for option pricing) and a computational part (pricing of a concrete option). 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 labs of the 3rd, 7th, 11th,15th week are completely structured for self-study by using appropriate e-learning activities. The lecture notes and lab handouts contain all of the theoretical materials that is required for this course. The solutions of the 7 practical homework assignments have to be submitted through Moodle. The solutions of the theoretical homework can be handed in on paper or submitted electronically through Moodle. The final examination has to be taken in person.