Massachusetts Institute of Technology, Cambridge, USA
Introduction to Learning Theory
Humans, animals, and, increasingly - machines - all learn from experience. But what, precisely, does “learning” mean? Just as physics seeks to uncover the laws that govern the physical world, and to express those laws in a mathematical language - learning theory seeks to uncover the mathematical laws that govern systems that learn (be they biological or otherwise).
This course is an introduction to learning theory. We will present a few definitions of learning, and then proceed to investigate what can (and cannot) be learned according to each definition, and what amount of computational resources (such as data and compute) are necessary in the cases where learning is possible. Attention will be devoted to understanding definitions and proofs.
Institute of Mathematics, Justus-Liebig-University Gießen, Germany
Introduction and foundation of Mathematical Finance
Mathematical Finance builds the foundation of the mathematical formulation and understanding of financial markets. Since the work by Black, Scholes, and Merton (1973), presenting the nowadays ubiquitous Black-Scholes formula and the seminal paper by Harrison and Pliska (1981) discovering the strong connection to martingale theory and stochastic analysis, mathematical finance became a prominent topic in applied and pure mathematics. In this lecture we will first explain the basic principles in a model with one time period and two future states. These include the concepts of non-arbitrage, replication, and risk neutral pricing of options, like calls and puts, under an equivalent martingale measure Then we will extend these to the binomial model from Cox, Ross, and Rubinstein in discrete time and space, also indicating its computational application. This section will finish with the celebrated Black-Scholes Fomula. For continuous time markets we need to develop the basics of stochastic analysis, including martingale theory, Itô's formula, stochastic differential equation, change of measure. With this tools we will analyse absence of arbitrage, completeness of markets and the riks-neutral pricing formula. As an example we will again present the Black-Scholes formula from this more general point of view.
THM Business School, Gießen, Germany
Practical applications of Mathematical Finance
Mathematical Finance has spurred the practical application of financial derivatives. Notably, the seminal work of Black, Scholes, and Merton (1973) has given rigor and theoretical understanding to trading and hedging options. This has accelerated the development of products and techniques to transfer financial risk and to possibly generate arbitrage profits making use of small price discrepancies between different markets. In this lecture we will look at the practical side of financial market business and how concepts like stochastic analysis, no-arbitrage, replication, and risk neutral pricing can be applied. We will focus on differences between real markets and ideal markets as postulated in mathematical finance, for example discussing continuous-time hedging against discrete time hedging. Topics will include
Dynamical replication of options and gap risk
Implied volatility, smiles, and fat tails
Capital requirements for banks following the 2008 financial crisis
Risk measurement through value-at-risk and expected shortfall, capital allocation and covariance
Counterparty credit risk