Miklós Csörgo - Random walking around financial mathematics

The 1997 Nobel Prize in Economic Sciences was awarded to Robert C. Merton and Myron S. Scholes who, in collaboration with the late Fischer Black, developed a pioneering formula for the valuation of stock options. In 1973, Black and Scholes published what has come to be known as the Black-Scholes formula. In order to derive and properly appreciate this formula, taking a historical route, we will first review some of the fundamental notions of Brownian motion-Wiener process, as well as some elements of Itô calculus. Consequently, we will summarize the derivation of the Black-Scholes formula for the European and American options. Though the solution is in terms of geometric Brownian motion, the latter will be highlighted also in terms of geometric fractional Brownian motion. Time permitting, we will also describe some long time path properties of various geometric processes of the call on average (Asian) option via those of the call on maximum option. The latter will be based on ongoing work with Endre Csáki, Antónia Földes and Pál Révész.