A record is an entry in a time series that is larger (upper record) or smaller (lower record) than all
previous entries. In financial mathematics record events play a big role for example when we think
of changes in stock prices. Obviously not only the probability that a record occurs is of interest but
also the correlations between record events. How likely is that one record is followed by another one
immediately? The financial crisis is a good example to be considered, since consecutive lower records
may occur. Realizing such an extreme event in its early stages can prevent heavy loss by reacting
appropriately. What you should do is drop the affected financial products.
In this work we consider four stochastic processes: the Wiener process with drift, the geometric
Brownian motion, the Ornstein-Uhlenbeck process and the Cox-Ingersoll-Ross model. We calculate
the probability Pn that all n entries in a time series are records, the probability pn that the nth entry
is a record and the joint probability pn,n?1 that the (n ? 1)th entry is a record as well as the nth
entry, enabling us to consider correlations between record events. Finally we divide pn,n?1 by the
probabilities pn and pn?1 to get the normalized joint probability ln,n?1. We perform a distributionfree method based on the record correlations to detect heavy-tail behavior. We show that the method
works well for small dataset. The maximum likelihood method for estimating the parameters of the
processes and the model validation and selection are part of this work, too
Forschungs-