Applied theoretical research, financial mathematics

IIa. Methods of quantum measurements and pricing on financial markets

For a long time physicists tried to numerically simulate the behavior of nonlinear stochastic Schrödinger equation, used in describing the processes of measurement of the state of quantum system. In 2002 analytical solution of this tasks was received. It turned out, that mathematical methods of non-linear filtration, used for receiving this solution, can be widely used in other fields of knowledge for describing of the observed phenomenon (unlike prediction or smoothing, which are traditional ways of usingthe theory of filtration). For example, they can be used for describing human behaviour: people behave in accordance with their "optimal estimation" or "optimal assumption" about the future - actually, everything we do is defined by our estimation about the future, conscious or unconscious.

In application to finance, this research field led to emerging of so called information-based asset pricing theory on the stock market. In the framework of this theory mathematical methods of filtration may be used for different observed phenomena, e.g. for describing the behaviour of traders on financial markets.

The traditional framework for asset pricing and risk management relies on the approach whereby the price process (which itself is an emergent concept resulting from the flow of information) is modelled exogenously, and market information is supposed to be generated by the random movement of prices. In the approach, introduced by Brody et al., however, this point of view is turned around: market information flow is modelled from the outset, from which price processes are derived. A number of areas have been investigated in this framework, however, there remain numerous important and open issues. In the laboratory we aim to develop a modelling framework within the information-based approach for:

1) the energy market (where traditional approach seems limited in explaining sharp peaks seen, for example, in the electricity markets);

2) the real-estate market (for which relatively little work has been done from the viewpoint of modern financial mathematics);

3) insurance / reinsurance markets (build on the earlier work in and extended into other Lévy processes depending on the type of insurance contract);

4) credit risk modelling, where are various issues that arose more recently in relation to credit valuation adjustments.

IIb. Social discounting and sustainable society

The well-known theorem of Dybvig, Ingersoll and Ross shows that the long zero-coupon rate can never fall. This result, which—although undoubtedly correct—has been regarded by many as counterintuitive and even pathological, stems from the implicit assumption that the long-term discount function has an exponential tail. It has recently been shown, however, that if the long "simple" interest rate (or Libor rate) is finite, then this rate (unlike the zero-coupon rate) acts viably as a state variable, the value of which can fluctuate randomly in line with other economic indicators.
The conditions necessary for the existence of such "hyperbolic" long rates turn out to be those of so-called social discounting, which allow for long-term cash flows to be treated as broadly  "just as important" as those of the short or medium term.