Information transmission between bitcoin derivatives and spot markets: high-frequency causality analysis with Fourier approximation

This paper examines information transmission between Bitcoin derivatives and spot exchanges using 15-minutes interval data over May 2016 - September 2020. We employ a novel econometric framework with Fourier approximation, taking structural shifts in causal linkages, on the prices, returns, and volatilities of BitMEX, the derivatives market, and five other major spot exchanges, Coinbase, Bitstamp, Kraken, CEX.io, and Poloniex. Overall, the results provide robust evidence of information flow between the derivatives and spot exchanges, implying the markets react to new information simultaneously. The results are of importance for investors conducting portfolio allocation exercises and risk management strategies.

Metric Fourier Approximation of Set-Valued Functions of Bounded Variation

We introduce and investigate an adaptation of Fourier series to set-valued functions (multifunctions, SVFs) of bounded variation. In our approach we define an analogue of the partial sums of the Fourier series with the help of the Dirichlet kernel using the newly defined weighted metric integral. We derive error bounds for these approximants. As a consequence, we prove that the sequence of the partial sums converges pointwisely in the Hausdorff metric to the values of the approximated set-valued function at its points of continuity, or to a certain set described in terms of the metric selections of the approximated multifunction at a point of discontinuity. Our error bounds are obtained with the help of the new notions of one-sided local moduli and quasi-moduli of continuity which we discuss more generally for functions with values in metric spaces.

Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials

Fourier approximation plays a key role in qualitative theory of deterministic and random differential equations. In this paper, we will develop a new approximation tool. For an m m -order differentiable function f f on [ 0 , 1 0,1 ], we will construct an m m -degree algebraic polynomial P m {P}_{m} depending on values of f f and its derivatives at ends of [ 0 , 1 0,1 ] such that the Fourier coefficients of R m = f − P m {R}_{m}=f-{P}_{m} decay fast. Since the partial sum of Fourier series R m {R}_{m} is a trigonometric polynomial, we can reconstruct the function f f well by the combination of a polynomial and a trigonometric polynomial. Moreover, we will extend these results to the case of random processes.