Abstract
We ask whether empirical finance market data (Financial Stress Index, swap and equity, emerging and developed, corporate and government, short and long maturity), with their recently observed alternations between calm periods and financial turmoil, could be described by a low-dimensional deterministic model, or whether this requests a stochastic approach. We find that a deterministic model performs at least as well as one of the best stochastic models, but may offer additional insight into the essential mechanisms that drive financial markets.
Highlights
Substantial mathematically minded economic research has dealt with the question whether financial data contain in a substantial manner low-dimensional chaotic features, or whether their nature requires the stochastic approach predominantly used by the practitioners dealing with the analysis of financial markets
An obvious asset of the deterministic model is that it operates in a low-dimensional subspace of models defined by means of the form of the deterministic equation, whereas the stochastic approach lacks such a limiting element
Whereas the ARIMA-GARCH approach deals separately with mean and volatility, in our two-dimensional deterministic approach, the first vector component represents an instantaneous activity that is functionally linked to an underlying trend represented by the second vector component
Summary
We ask whether empirical finance market data (Financial Stress Index, swap and equity, emerging and developed, corporate and government, short and long maturity), with their recently observed alternations between calm periods and financial turmoil, could be described by a low-dimensional deterministic model, or whether this requests a stochastic approach. To provide a first general insight into the nature of Rulkov maps, we start from a two-dimensional non-linear recurrence of the form α xt+1 = fα(xt ) + γ yt + δ, with fα(xt ) = 1 + xn , n ∈ N, α, γ , δ, x ∈ R Such a setting is widely used for the modelling short-time effects like pulses and shocks in time series. Combining Eq (1) with Eq (2) leads to xt+1 =fα(xt ) + γ yt + δ, yt+1 =β yt − μ xt + η This version, that differs slightly from the original Rulkov version, is the map that we will use for our financial time series modeling; its two-dimensional form takes care of the short-term (x) and the average prediction (y) horizon (the two main subjects of interest in financial modeling), which provides the basis for an optimal modeling framework.
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