The study of the Black-Scholes type formula plays an important role in the analysis of financial markets and instruments, for example in option pricing. It is known that if the underlying asset price is a strict local martingale, then pricing the Black-Scholes type formula becomes delicate and difficult to analyze. For instance, if the underlying asset price is a "true" martingale, then the European call price is a submartingale, hence the function is increasing. On the other hand, if the underlying asset price is a strict local martingale, then the European call price is not in general increasing. The project develops techniques from probability theory to build a bridge between mathematical models and financial markets. A first main direction of the project uses the interpretation of the generalized Black-Scholes quantities in terms of last passage times in the framework of Bessel processes in order to derive fine properties of the call option process, as a function of maturity, written for the strict local Bessel martingales. Concretely, the work aims to provide a rigorous mathematical framework for understanding market bubbles. A second main focus of the project investigates the enlargement filtration of the last passage time in the Black-Scholes type formula, in particular, its "distance" from the stopping time. In a financial context, an example of the improper use of stopping time may be interpreted as using signals coming from available inside information. The project will develop mathematical theory and tools for detecting the presence of arbitrage trading in this setting which is also related to market bubble phenomena.
Recently, there has been a series of market crashes and bubbles where mispricing takes place. In economic terms, a bubble is defined as a deviation between the price of an asset and its underlying values. Mathematically, bubbles are phenomena for which the discounted asset prices are strictly local martingale but not martingale under the risk-free measure. In other words, if the discounted asset price processes are strict local martingales, option prices no longer increase in maturity and Merton's no early exercise theorem does not hold. Intuitively, if a bubble exists, the asset price will increase initially, but a crash is likely to take place in the future. It is well-known that many sophisticated financial derivatives have been built on the foundation of mathematical theories since the 1970s. As a consequence, such a rapid increase of derivative markets necessitates proper communication between financial analysts and mathematicians. A general goal of the project is to develop a rigorous mathematical understanding of market bubbles and related phenomena. The project is interdisciplinary in nature and will be integrated into practical applications.
