Financial Mathematics - history notes

In Finance Stochastic Processes were introduced in 1900. and then forgotten. In the 1960's and 1970's they were investigated again. In that time the known Black-Scholed model of a financial market was developed. This model was based on the theory of and the continuus time analysis of the prices of the assets. Later was discovered that this Brownian Motion is not the perfect and the most appropriate solution for the changes in prices of the assets. Some ten years later the Browninan MotionGeometric Brwninan Motion was shown to be the best model solution.

These results made some mathematicians to look deeper into the financial market theory. The result of the later inhanced analysis was "The fundamental theorem of Asset Pricing" which gave some necesary boundaries for the solution of finding the best option price in funture betting. It was shown that these conditions were already satisfied both in Brownian Motion and in Geometrical Brownian Motion.

The necessary condition for the market was the "no-arbitrage" quality which is considered the most valuable quality for the market to be "fair". In short it states that there should not be possibilities for the non-risky profits in the market. Later this quality of the market was conected to the martingale representation of that particulat financial market. The fundamental Theorem of Asset Pricing states that if the market has the no-arbitrage quality then it exists another probability measure equivalent to the first one, in which the market is martingale.

Introduction

The various topics, ranging from Functional analysis, Stochastic Processes , to Hidden Markov Models and Statistical learning applications will be discussed here..........