# Statistical Properties and Dynamics of Stock Prices

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## Chapter 1: Understanding Stock Price Dynamics

This article delves into the statistical characteristics of stock prices. Contrary to popular belief, the distribution of returns is not normally distributed. Instead, Lévy distributions, which possess broad tails, prove more effective in forecasting potential market crashes and other infrequent events. Additionally, the topic of correlations will be examined. Contrary to the common assumption that correlations are zero initially, real financial data only becomes uncorrelated after a certain time interval Δt*. Prior to this interval, correlations behave as a decaying exponential function.

### Statistical Analysis of Stock Prices

The dynamic behavior of stock prices generally relies on a few key assumptions:

Continuous trading, meaning that price quotes occur at non-zero intervals.

Prices evolve as a stochastic process influenced by fundamental random variables.

label: random_variables text{Random variables that describe the dynamics of stock prices are independent and identically distributed (i.i.d.) with finite mean } mu text{ and variance } sigma.

The price dynamics of ln S(t) follow a diffusive process. Diffusion refers to the distribution of an object from a region of higher concentration to areas of lower concentration.

The random variables are assumed to have Gaussian increments. This model is referred to as geometric Brownian motion (GBM), which is governed by the following stochastic differential equation (SDE):

label: GBM_SDE dS(t) = mu S(t) dt + sigma S(t) dW(t)

where W(t) represents a Wiener process or Brownian motion, with μ indicating drift and σ denoting volatility. The solution to this SDE is:

label: GBM_solution S(t) = S(0) e^{(mu - frac{sigma^2}{2})t + sigma W(t)}

The exponent of the standard deviation σ(t) hovers around 0.5, suggesting that price changes are independent of one another. Consequently, as stated in the initial assumption, the total number of quotations within a finite interval diverges. Therefore, according to the central limit theorem, the distribution approaches a Gaussian form.

Assumption 1 is a simplification. A more accurate model would involve S(t) as a discrete-time stochastic process with price quotes occurring at intervals of Δt. The characteristics of these discrete processes have been meticulously analyzed by physicists.

### Stock Returns

Physical studies typically concentrate on price increments rather than the prices themselves. When ΔS(n) is much smaller than S(n) and Δt is brief, indicating slow price fluctuations (for instance, in the absence of crashes), we can approximate:

label: price_increment Delta S(n) approx text{some function of } S(n)

By comparing a Monte Carlo simulation of geometric Brownian motion:

with another simulation for Brownian motion, using an annual drift of μ=10% and annual volatility of σ=20%, we observe a strong agreement.

Thus, additive models (as opposed to multiplicative models) can be applied to stock prices:

label: additive_model R(n) = Delta S(n) + text{previous returns}

In additive models, price increments accumulate, whereas multiplicative models rely on successive ratios.

### A Mathematical Interlude

The Lévy distribution is a probability distribution for non-negative random variables, introduced by French mathematician P. Lévy and Soviet mathematician A. Khintchine. Lévy distributions account for leptokurtosis (the tendency for extreme values).

This distribution is stable, meaning that a linear combination of independent variables following this distribution retains the same distribution characteristics. This leads to a scaling property:

label: stable_distributions X_1 + X_2 sim text{Lévy distribution}

However, Lévy distributions may have divergent standard deviations, with their maxima being larger and narrower than expected in real financial series. This can be addressed by truncating the tails using a cutoff parameter.

label: truncated_levy P(X) = text{some truncated function}

Benoit Mandelbrot was among the first to observe that asset prices exhibit larger fluctuations than predicted by Gaussian models, indicating the presence of fat tails.

In a significant paper, H. E. Stanley and R. N. Mantegna analyzed data from the S&P 500 between 1984 and 1989 to understand the distribution of stock variations.

The findings revealed that the Lévy distribution fits well for certain ranges of price variations, while showing an exponential decay beyond a defined threshold.

label: levy_fit P(l) sim text{Lévy distribution for } l/sigma leq 6

This distribution possesses two key traits: stability (self-similarity) and attraction in probabilistic space.

Truncated Lévy distributions maintain self-similarity over extended periods before truncation takes effect.

### Correlations

The Geometric Brownian motion model assumes zero correlations between price changes. To validate this assumption, we utilize the following correlation function:

label: correlation_function C(Delta S) = text{some correlation function}

This correlation can range from -1 to 1, with specific cases being particularly noteworthy. If assumptions 1 and 2 were true, we would have:

label: price_correlation C(Δt) = e^{-lambda Δt}

In reality, the correlation behaves as an exponential decay from 1 to 0, typically within a time frame of around Δt* ≈ 15 minutes, showcasing strong correlations at shorter intervals.

Thus, we arrive at a model for price distribution after Δt* > 15 minutes, where price variations can be considered independent.

label: price_distribution P(ΔS) = text{some distribution function}

The cumulative distribution, when convolved with N* factors, shows interesting characteristics:

label: cumulative_distribution C(N) = text{cumulative probability}

Key observations include: - The distribution is well represented by a truncated Lévy distribution with specific parameters. - The convolution closely approximates probabilities at longer time intervals, gradually conforming to a Gaussian distribution. - Real financial data diverges from the cumulative distribution, particularly at the extremes.

This convergence to a Gaussian distribution can take several days to weeks, depending on the specific market analyzed.

Following the work of Stanley and Mantegna, I will briefly explore the dynamics of the DJIA (Dow Jones Industrial Average). They discovered the maximum and minimum correlation coefficients among the 30 stocks within the DJIA. The highest correlation, 0.73, occurs between Coca-Cola and Procter & Gamble.

Furthermore, they measured the characteristic time scale over which strong correlations persist. Their analysis from 1990 to 1994 indicated correlations ranging from 0.73 to 0.51, suggesting a high level of synchronization among the stocks.

For additional insights on finance, physics, machine learning, deep learning, and mathematics, feel free to visit my personal website at www.marcotavora.me.

## Chapter 2: Understanding Stock Risk through Statistics

In this section, we explore the fundamental concepts of statistical risk associated with stock trading.

The first video, "Statistics 101: Normal Distribution and Stock Risk," provides insights into how statistical principles apply to stock market risks.

The second video, "Ep 112: Trading Stocks by the Numbers (Statistical Probabilities + Options)," discusses the application of statistical probabilities in stock trading and options.