Abstract
This article presents a dual investigation of stock price returns: one theoretical and one empirical. It explores how returns should behave if they are entirely driven by information and compares these theoretical expectations to actual market data. The central question is whether the statistical features of price returns can be reconciled with the notion that prices reflect only the arrival and processing of new information.
In the first, epistemic component of the study, a probability distribution is constructed to describe the impact of small units of information (“news particles”) on asset prices. The model assumes that stock prices evolve exclusively in response to these informational events, with each news particle contributing incrementally to the total information state. Since information is not directly measurable and has no inherent numerical value, its value is defined operationally as the degree to which it moves prices. The expected distribution of returns over different time intervals is then derived by convolving the distribution of individual news particles. These convolutions simulate how the accumulation of many small informational updates might affect return distributions over longer periods.
Particular attention is paid to kurtosis, which captures the relative weight of observations in the tails of a distribution, is sensitive to extreme events, and therefore provides a natural summary of the heavy-tailed behaviour and extreme price movements emphasised in the empirical finance and risk-management literature. The theoretical model predicts that, as time progresses and more informational updates are convolved, kurtosis should decay steadily and ultimately converge to the kurtosis of a normal distribution, in accordance with the central limit theorem. This behaviour is taken as a benchmark: if returns are purely information-driven, then kurtosis should follow a predictable decay path.
The empirical portion of the study analyses return data from the FTSE/JSE Top 40 Total Return Index (J200T) across a range of time intervals, from one second to one year. The returns are carefully prepared to exclude distortions due to weekend gaps and non-trading hours. For each time interval, key statistical moments are calculated, with a particular focus on kurtosis. The analysis shows that empirical kurtosis, while it declines as time intervals increase (as expected), remains significantly higher than the theoretical model predicts. Most notably, it does not appear to converge to the normal level of kurtosis (i.e. a value of 3). Even at longer intervals (e.g. daily or weekly), kurtosis remains elevated, suggesting that something other than pure information is affecting return distributions.
It is proposed that the persistent excess kurtosis observed in real return data is not the result of the information process itself, but rather of a transformation applied to information as it is reflected in prices. This transformation is argued not to be a random artefact, but to follow a consistent pattern: it is the result of a structural heuristic that emerges when market participants infer measurable values (such as returns) from unmeasurable causes (such as the significance of news). This process is referred to as hyperbolic transformation, a concept developed in earlier work. It arises from the geometric observation that when the domain of a function is not directly measurable, estimators tend to interpolate between known output values (on the range or value axis), rather than inputs (on the domain axis). Under certain assumptions, interpolation on the value axis is proportionally equivalent to interpolation on the functional curve, resulting in a distortion of the underlying distribution, often captured by an increase in kurtosis.
To support this hypothesis, the rate of kurtosis decay in the theoretical model (where kurtosis evolves as a function of the number of convolutions) is compared to the empirical decay of kurtosis (as a function of the return interval, measured in seconds). A functional transformation is sought that aligns the two decay trajectories in their first and higher-order derivatives. A simple exponential function is found to provide an excellent match. Using numerical optimisation, parameter values are identified that minimise the difference between the theoretical and empirical kurtosis derivatives across intervals. The match is sufficiently strong to suggest that both distributions are in the same phase of decay but displaced in absolute magnitude. This displacement is attributed to hyperbolic transformation.
Furthermore, it is tested whether this excess kurtosis can be attributed to conditional heteroskedasticity (volatility clustering), a well-known feature of financial time series. Using GARCH(1,1) models and Ljung–Box tests on squared returns, it is shown that, although volatility clustering is indeed present, it cannot on its own account for the magnitude of the kurtosis observed – especially at very short intervals. It is concluded that volatility clustering contributes to excess kurtosis but is not the dominant cause.
To model returns incorporating the hypothesised transformation, a process of the form:
is used. Here, 𝑍 is a standardised version of sinh(𝜅𝑋 + λ), where 𝑋 is standard normally distributed variable and λ and 𝜅 are shape parameters. The sinh function introduces hyperbolic curvature, and the 𝑍-transformation ensures zero mean and unit variance. The volatility term σ(𝑡) follows a GARCH(1,1) process, calibrated to capture volatility dynamics. All model parameters are calibrated by maximising the log-likelihood of the observed return series. The residuals from the model (the inferred 𝑋-values) are tested for normality, autocorrelation and volatility clustering. The results are consistent with the model’s assumptions: residuals approximate standard normality and exhibit no significant autocorrelation or ARCH effects, provided the data is pre-processed to remove multi-day returns and calendar artefacts.
The findings suggest that hyperbolic transformation is a viable explanation for the persistent excess kurtosis seen in return distributions, especially at intraday and daily frequencies. The model performs well in matching both the shape and dynamic structure of returns. However, the scope of the study is limited to an equity index under favourable conditions (high liquidity, active trading). Further research is needed to assess the generalisability of the findings to other assets, market regimes and levels of market efficiency. Overall, the study proposes a synthesis of information-based pricing theory and heuristic modelling, offering a coherent framework for explaining the shape and dynamics of stock return distributions and inviting further exploration of the geometric and epistemic underpinnings of market behaviour.
Keywords: central limit theorem; GARCH; hyperbolic transformation; information-driven process; kurtosis; nonlinear transformations; probability distribution; stock price return; volatility clustering
- This article’s featured image was created by Mikhail Nilov and obtained from Pexels.
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