  Identifying and quantifying co-dependence between financial instruments is a
key challenge for researchers and practitioners in the financial industry.
Linear measures such as the Pearson correlation are still widely used today,
although their limited explanatory power is well known. In this paper we
present a much more general framework for assessing co-dependencies by
identifying and interpreting linear and nonlinear causalities in the complex
system of financial markets. To do so, we use two different causal inference
methods, transfer entropy and convergent cross-mapping, and employ Fourier
transform surrogates to separate their linear and nonlinear contributions. We
find that stock indices in Germany and the U.S. exhibit a significant degree of
nonlinear causality and that correlation, while a very good proxy for linear
causality, disregards nonlinear effects and hence underestimates causality
itself. The presented framework enables the measurement of nonlinear causality,
the correlation-causality fallacy, and motivates how causality can be used for
inferring market signals, pair trading, and risk management of portfolios. Our
results suggest that linear and nonlinear causality can be used as early
warning indicators of abnormal market behavior, allowing for better trading
strategies and risk management.
