Quantifying horizon dependence of asset prices: a cluster entropy approach

Market dynamic is quantified in terms of the entropy $S(\tau,n)$ of the clusters formed by the intersections between the series of the prices $p_t$ and the moving average $\widetilde{p}_{t,n}$. The entropy $S(\tau,n)$ is defined according to Shannon as $\sum P(\tau,n)\log P(\tau,n),$ with $P(\tau,n)$ the probability for the cluster to occur with duration $\tau$. \par The investigation is performed on high-frequency data of the Nasdaq Composite, Dow Jones Industrial Avg and Standard \& Poor 500 indexes downloaded from the Bloomberg terminal. The cluster entropy $S(\tau,n)$ is analysed in raw and sampled data over a broad range of temporal horizons $M$ varying from one to twelve months over the year 2018. The cluster entropy $S(\tau,n)$ is integrated over the cluster duration $\tau$ to yield the Market Dynamic Index $I(M,n)$, a synthetic figure of price dynamics. A systematic dependence of the cluster entropy $S(\tau,n)$ and the Market Dynamic Index $I(M,n)$ on the temporal horizon $M$ is evidenced. \par Finally, the Market Horizon Dependence}, defined as $H(M,n)=I(M,n)-I(1,n)$, is compared with the horizon dependence of the pricing kernel with different representative agents obtained via a Kullback-Leibler entropy approach. The Market Horizon Dependence $H(M,n)$ of the three assets is compared against the values obtained by implementing the cluster entropy $S(\tau,n)$ approach on artificially generated series (Fractional Brownian Motion).