Modelling bursts and chaos regularization in credit risk with a deterministic nonlinear model

Financial markets’ deterministic aspects modeled by a low-dimensional equation

“…More generally, what we deal with here is similar to the one encountered in neuroscience, where initial ideas of simple chaotic or of power-law behavior characterizing the data, had to be replaced by more complicated and, in particular, scale-dependent characterizations 80 . This shifts the question towards determining exactly at what time-scales the two processes of low-dimensional chaos and of stochastic, respectively, are dominant, and to identify the horizon over which a deterministic prediction will yield optimal forecasting results 32 . In this way, our work may open a general avenue towards a better understanding and monitoring of the essential drivers in the data, not only in financial market dynamics, but also in similar processes beyond.…”

“…While this interplay between fast-slow dynamics is a general mechanism unterlying pattern formation, the Rulkov map offers through the involved parameters the possibility to adapt the general mechanism to specific features exhibited by a process modeled. In particular by its recursive nonlinear and mean reverting properties, Rulkov maps may be expected to be highly suitable for the modeling of financial time series, in particular regarding the occurrence of data clusters, heteroskedasticity, mutual synchronization, chaos and regularization of bursts of activity across the markets, after-shock asset classes, and more 32 . In our application, we will optimize the Rulkov family model to best reproduce the processes and events observed in our data.…”

“…This technique finds the optimal alignment between two time series by stretching or shrinking on time series along the time axis, and then evaluates the Euclidean distance between the two 41 , 42 . This commonly used technique in finance and econometrics 32 , 43 , 44 copes with slightly temporally varying responses in one of the time series 45 . This feature has made DTWD the predominant tool in speech recognition (for determining whether two waveforms represent the same spoken phrase) and one of the standard methods used for classification problems in data mining 46 , 47 .…”

Forecasting portfolio returns with skew‐geometric Brownian motions

“…As empirically investigated in Orlando and Bufalo, 3 returns, either standardized or not, do not seem to be unconditionally normally distributed. They often show a significant amount of skewness and extra‐kurtosis 4‐7 . Skew normal distributions were first introduced by Azzalini 8 and Henze 9 and have gained some momentum because of their suitability in modeling real data.…”

“…They often show a significant amount of skewness and extra-kurtosis. [4][5][6][7] Skew normal distributions were first introduced by Azzalini 8 and Henze 9 and have gained some momentum because of their suitability in modeling real data. In fact, a few years later, Kim 10 further enhanced the framework by introducing t-skew distribution as scale mixtures of skew-normal distributions.…”