Monitoring the volatility in a process which reflects trading in the financial market

Abstract: Recently, the financial market has become an area of increased research interest for mathematician and statisticians. The Black and Scholes breakthrough in this area triggered a lot of new research activity. Commonly the research concerns the log returns of assets (shares, bond, foreign exchange, option). The variation in the log returns is called volatility and it is widely studied and because of its relevance for applications in the financial world. The volatility is mostly used for measuring the risk and also for forecasting future prices. In this research work a process of trading activities is considered. It is assumed that at a random time-point a parameter change in the laws of the trading occurs, indicating changed trading behaviour. For inferential matters about the process it is of vital importance to be able to state that such change has occurred quickly and accurately. The methods used to this end are called stopping rules which signal alarm as soon as some statistics based on-line observations goes beyond some boundary. The model considered for this process of log returns is the family of Autoregressive Conditional Heteroskedastic (ARCH) model. It is widely accepted that this well describes a lot of phenomena in the financial market. In this work statements about this process will be derived, the stopping rule will be defined, evaluated and their properties discussed.