Abstract
We empirically quantify the relation between trading activity — measured by the number of transactions N — and the price change G(t) for a given stock, over a time interval [t, t + Δt]. We relate the time-dependent standard deviation of price changes—volatility—to two microscopic quantities: the number of transactions N(t) in Δt and the variance W 2(t) of the price changes for all transactions in Δt. We find that the long-ranged volatility correlations are largely due to those of N. We then argue that the tail-exponent of the distribution of N is insufficient to account for the tail-exponent of P{G > x}. Our results suggest that the fat tails of the distribution P{G > x} arises from W, which has a power-law distribution with an exponent consistent with that of G.
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The hypothesis that the conditional distribution has the same form for all W and N might strike the reader as surprising since one expects the conditional distribution to be “closer” to a Gaussian for increasing N. If W and N are independent, then the hypothesis is exact only for a stable distribution for δpi such as a Gaussian (consistent with our findings later in the text).
The ≃ sign is used because although the tails of the conditional distribution are consistent with Gaussian, the central part is affected by discreteness of price changes in units of 1/16 or 1/32 of a dollar.
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Gopikrishnan, P., Plerou, V., Gabaix, X., Amaral, L.A.N., Stanley, H.E. (2002). Price fluctuations and Market Activity. In: Takayasu, H. (eds) Empirical Science of Financial Fluctuations. Springer, Tokyo. https://doi.org/10.1007/978-4-431-66993-7_2
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DOI: https://doi.org/10.1007/978-4-431-66993-7_2
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