Stochastic Processes in Finance

The theory of stochastic integration and martingale theory is applied to pricing and hedging of derivative securities and other problems arising in finance. This field, which descends from the seminal work of F. Black, R. Merton and B. Scholes (the "Black-Scholes- formula`` for option pricing which was honoured by the Nobel prize in economics in 1997) has grown very rapidly over the past 25 years. The research group of Professor Schachermayer investigates the foundational as well as several applied aspects of this theory.

The application of probabilistic methods had a strong impact on the finance industry, as reflected, e.g., in the accords Basel I and II. The underlying mathematical theory goes back to the pioneering work of L. Bachelier (1900) and gained seminal importance through the work of F. Black, R. Merton, and M. Scholes (1973). The basic economic idea (the principle of "no arbitrage") is remarkably simple: a mathematical model of a financial market should be designed in such a way that it does not allow for gains without risk. This convincing principle ("there is no such thing as a free lunch") allows for surprisingly far-reaching conclusions. Modelling the price- evolution of stocks on the basis of Brownian motion with a fixed volatility, one may, e.g., deduce from the principle of no arbitrage unique prices for options on these stocks. In addition, the theory also yields a recipe, how to replicate these options by dynamic trading strategies. This approach, due to Black-Scholes-Merton (1973) and honoured by the Nobel prize in economics in 1997, has been further developed in practice as well as in its theoretical foundations on a rapid path. W. Schachermayer and his research group at TU Wien have substantially contributed to this development. In particular, the principle of no arbitrage was characterized on the basis of a general mathematical theorem and put into relation with probabilistic theories. This had impact, e.g., on the theory of portfolio optimisation of risk-averse investors, as well as the extension of the theory to include transaction costs as well as the stochastic evolution of the yield curve. This was shown in numerous papers published in international journals by the group around W. Schachermayer.