The Actuarial and Financial Engineering

What determines the premiums of a life-insurance contract? Clearly the insurance company has to take into account the age, occupation, sex and even habits of the customer and use a model of lifetime. In that sense each contract should be unique. However cash reserves are also required by certain regulations and thus the insurance company must also view its clients as a whole. The big picture is presented in this course.

• Computational Finance

In order to use mathematics for making sound financial decisions, one has to be able to select a suitable market model, to calibrate the model and to use it for computing various numerical quantities (prices of options, hedging parameters etc). Since new and better models are introduced very often, it is not enough to learn to use a fixed set of methods for some concrete applications. The goal of this course is to familiarize the students with the process of deriving appropriate equations from various market models and with the basic techniques of deriving numerical methods for those equations together with giving hands-on experience in implementing the methods effectively in a high level language; the questions of model calibration are also addressed.

• Models of Financial Mathematics

Modelling financial market requires several assumptions which are discussed in the first part of the course. The second part introduces scenario based models with discrete time, binomial models and Black-Scholes models, all applicable for option pricing.

• Non-Life Insurance Mathematics

Most people have bought or will buy at least one non-life insurance policy (home insurance, car insurance, etc) in their life, which means they are involved with the non-life insurance business as customers. This course gives another point of view to insurance business: through the eyes of an insurance company. We will study what is the reason of several rules and regulations, by what principles the insurance premiums and compensations are calculated and many more topics involving the internal life of an insurance company.

• Martingales

Martingales are a fundamental tool in many fields of applied probability. The course covers the fundamentals of martingales with examples arising in the financial context. Starting with discrete time martingales, it also covers some basics of continuous time martingales. Ito calculus as well as applications to option pricing thery are discussed.

• Risk Theory

Risk is a possibility that an unfavorable event occurs. However, risk is also an opportunity! In order to earn more than by just keeping money on the bank account, one has to take a risk, smaller or larger. Insurance companies take risks by selling insurance policies, which can sometimes lead to huge payouts or even bankruptcy. A classical problem in risk theory what is the ruin probability of an insurance company is the first topic of the course. Risk measures in modern portfolio theory are also covered, including Markowitz model and capital asset pricing model (CAPM). Ubiquitous Value at Risk (VaR) methodology for measuring financial risks, together with the concept of coherent risk measures, are considered.

• Simulation Methods in Financial Mathematics

Very often one is interested in computing values of quantities (like option prices) than can be expressed as expected value of some random variable. A very popular method that can be applied in such cases is the Monte-Carlo method that corresponds to simulating the behavior of the random variable (generating independent samples of the variable) and computing the average of the result. This course discusses the possibilities of using the method for solving problems in mathematical finance and, more importantly, some methods and tricks that can be used to speed up the convergence of the method.