Jonathan Khedair, Reimer Kühn
The standard models of risk in the mathematical finance community typically connect to the well known Geometric Brownian motion (GBM). However, they are faced with the usual shortcomings that they do not reproduce the so called ‘stylised facts’ of financial markets. Several other alternatives have since been proposed where most of these non-trivial features are forced in by hand – leading to ever more complicated approaches. In the contrary however, those in the Econophysics community have recognised that such universal features are in fact a signature of collective behaviour and instead propose a ‘bottom-up’ agent based approach. In most of such examples, one typically considers the actions of individual agents and their subsequent interactions making little attempt to connect with more descriptive approach described above. Therefore, whilst substantial and insightful, the modelling capabilities in market risk setting are already naturally limited.
In this work we present an alternative approach which can be thought of as a middle ground between the two. We propose a model in which an interacting non-linear feedback mechanism is added to the standard GBM. The rationale behind this starts with a hypothetical complete knowledge of all degrees of freedom in a market setting. If one were to integrate all of such except the price we would be left with resulting effective interactions between prices and memory. Here, we take the minimalist Markovian approach with pairwise interactions only. Interestingly, the resulting dynamics is equivalent to that of a graded response neural network which have been well studied and are known to exhibit glassy dynamics being largely dominated by a landscape with many meta-stable states. Using such methods, analytical tractability can be made and we find closed form expressions for return distributions as well as other quantities of interest. Another promising feature is that a natural study of the market state detection arises since the dynamics exhibits significantly different behaviour when transitioning across metastable states as opposed to fluctuations within one. It is of surprise and great interest that even in this first approximation approach, essentially all non-trivial effects one hopes to observe are recovered in full detail – suggesting that interactions must be acknowledged if accurate models of risk and portfolio management are to be constructed.