Unit #1 Introduction to pricing of assets in complete and incomplete markets, implications of the no-arbitrage principle

Unit#2 Fundamental theorem of asset pricing, review of the binomial option pricing model

Unit#3 Stochastic processes in continuous time, Ito’s lemma

Unit#4 Hedging in continuous time, Girsanov’s theorem, market price of risk

Unit#5 Black-Scholes formula, pricing and hedging of derivatives, implied volatility and limitations of the Black-Scholes framework

Unit#6 Introduction to Monte-Carlo simulation, application to terminal and pathwise payoffs for Geometric Brownian Motions

After completing the course, students will be able to

- understand the no-arbitrage principle and its implications for asset pricing in complete and incomplete markets
- know the statistical properties of important continuous-time stochastic processes and are able to interpret and estimate the model parameters
- understand the concept of hedging in continuous time
- differentiate between the physical (empirical) and the equivalent risk-neutral (martingale) probability measure
- understand the general relation of asset prices to conditional expected payoffs under the risk-neutral measure
- understand and apply the Black-Scholes formula as an important example for an analytical model
- understand and apply the all-purpose concept of Monte-Carlo simulation to numerically compute expected pay-offs of arbitrary complexity under the risk-neutral measure

Although this course touches some very involved mathematical concepts (e.g., measure theory, stochastic calculus), the design of the course will only require prior knowledge in mathematics and statistics which could be expected from good students in economics and business administration showing some extra interest and motivation with respect to quantitative methods. In particular, students should be familiar with the concepts of partial derivatives, definite integrals, and conditional probabilities which could be refreshed by using any undergraduate textbook. As a consequence, students who might be interested in a rigorous treatment of the mathematical fundamentals are referred to graduate programs in this field or to suitable electives offered by the Institute for Statistics and Mathematics for undergraduate students.

One of the main course outcomes is the ability of students to compute fair values of financial instruments numerically with Monte-Carlo simulation. Although all cases and applications are designed in a way that MS Excel can still be used, a good command of R might be helpful.

Participation is compulsory. Students are not allowed to miss more than one unit.

The course will be delivered by in-class presentations in six units of 3:15 hours. Each unit will provide enough space for Q/A sessions. Two case studies have to be solved by students in take home assigments where feedback will be given individually.

- Midterm exam (written, in-class open book): 25%
- Endterm exam (written, in-class open book): 35%
- Home assignment 1: 15%
- Home assignment 2: 25%

Students will pass the course **if **more than 50% of all credits are earned **and** more than 50% of the credits of the endterm exam are earned **and** more than 50% of the credits of the home assignment 2 are earned. The pass marks will be distributed as follows: 3 if total credits > 60%, 2 if total credits > 70%, 1 if total credits > 80%.

Positive completion of Course I and Course II

Registration via LPIS