Wavelet-Based Methods for High-Frequency Lead-Lag Analysis

Abstract

We propose a novel framework to investigate lead-lag relationships between two financial assets. Our framework bridges a gap between continuous-time modeling based on Brownian motion and the existing wavelet methods for lead-lag analysis based on discrete-time models and enables us to analyze the multi-scale structure of lead-lag effects. We also present a statistical methodology for the scale-by-scale analysis of lead-lag effects in the proposed framework and develop an asymptotic theory applicable to a situation including stochastic volatilities and irregular sampling. Finally, we report several numerical experiments to demonstrate how our framework works in practice. J for some universal constant c 4 > 0, hence we obtainWe rewrite the target quantity as, we obtain (35) by the dominated convergence theorem once we prove Ξ J (i 1 , i 2 ) → 0 for any fixed i 1 , i 2 . By Lemma 7 we haveWe can take sufficiently large r such that 2/r < 1 − κ. Then we haveJ )|X]| r ] = O τ 1−2/r J L 2 log 2 τ J r/2 = o(1) by Lemma 12. This yields the desired result. N τ J by the triangle and Schwarz inequalities. Therefore, the Markov inequality yields P max l∈L + J |I J (l)| > ε ≤ ε −1 l∈L + J I J (l) 2 2 N 2 τ J for any ε > 0, hence max l∈L + J |I J (l)| → p 0.Noting that L 2 τ J → 0, we can prove max l∈L + J |II J (l)| → p 0 in an analogous manner to the above.Next we prove max l∈L + J |III J (l)| → p 0. For any k ∈ I m,N (i) we haveWe can prove max l∈L + J |IV J (l)| → p 0 in an analogous manner.Now we prove max l∈L + J |V J (l)| → p 0. By Lemma 14 and the boundedness of σ 1 and σ 2 , we haveψ 1 → 0 by assumptions. This especially implies that max l∈L + J |V J (l)| → p 0. Finally, by Lemmas 4(b) and 8 we have max l∈L + J |VI J (l)| = o p (M J · τ −1 J τ m · τ J ). Since M J = O(τ −1 m ), we obtain max l∈L + J |VI J (l)| → p 0. This completes the proof. Completion of the proof of Theorem 2We need the following auxiliary result:Lemma 15. Λ −j D(λ)Π(λ)e

From Glosten-Milgrom to the Whole Limit Order Book and Applications to Financial Regulation

“…• One informed trader: by this term, we mean a trader who undergoes low latency and is able to access market data and assess efficient price jumps faster than other participants, creating asymmetric information in the market. For instance, he can analyze external information or use lead-lag relationships between assets or platforms to evaluate the efficient price (for details about lead-lag see [13,15,20]). Therefore, we assume that the informed trader receives the value of the price jump size B (and the efficient price P (t)) just before it happens.…”

Inference for time-varying lead–lag relationships from ultra-high-frequency data

“…A related model has also been studied in Robert and Rosenbaum (2010) by utilizing the random matrix theory and Ito and Sakemoto (2020) by multinomial dynamic time warping. Other approaches to investigate lead-lag relationships in a continuous-time framework include Hawkes process-based models (Bacry et al 2013; Da Fonseca and Zaatour 2015), a wavelet-based method (Hayashi and Koike 2018), and a multi-asset lagged adjustment model (Buccheri et al 2020). Several empirical approaches have been proposed, as well; see Pomponio and Abergel (2013) and Dobrev and Schaumburg (2016), for example.…”

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