Quantitative analysis of financial markets often assumes that asset prices or returns follow a normal distribution. This approximation is not perfect, but it is useful in many contexts. Below, we explain why the normal distribution has become one of the basic models for understanding market movements.
1. The Central Limit Theorem and the Normality of Returns
One of the main reasons for using the normal distribution is the Central Limit Theorem (CLT). This theorem states that the sum or average of a sufficiently large number of independent random variables tends to follow a normal distribution, regardless of the original distribution of the variables.
This implies that, if asset prices are influenced by many small and independent factors (market information, investor decisions, random noise, etc.), returns (percentage changes in price) can approximate a normal distribution in the short term.
2. Mathematical facilities and modeling
The normal distribution has mathematical properties that facilitate statistical calculations and financial decision-making:
- It is completely defined by two variables: the mean (μ) i la standard deviation (σ).
- It allows simple calculations of probabilities and risks using the standard normal table.
- Analytical models such as Value at Risk (VaR) or the Black-Scholes model for options can be constructed.
3. 68-95-99.7% rule and risk analysis
When modeling financial returns with a normal distribution, general rules can be established to quantify risks:
- 68% of the values are within one standard deviation of the mean (low volatility).
- 95% within two deviations (more extreme, but relatively common events).
- 99.7% within three deviations (rare events, such as severe financial crises).
This property is key in risk management and in predicting the probability of severe losses.
4. Geometric Brownian Motion and financial models
Many financial models assume that prices follow a geometric Brownian motion (GBM):
$$dS_t = \mu S_t dt + \sigma S_t dW_t$$
This model assumes that percentage price increases follow a normal distribution, which allows elegant solutions to be derived for the pricing of financial derivatives.
5. Key concepts in the study of the normal distribution in markets
5.1 Mathematical expectancy (μ) i Variance (σ2)
- Mathematical expectancy (μ): It represents the average of the returns of a financial asset. It indicates the central tendency of the distribution of prices or returns.
- Variance (σ2): It measures the dispersion of returns around the average, indicating the volatility of the asset.
5.2 Standard Deviation (σ)
- The standard deviation is the square root of the variance and is directly interpreted as volatility.
- EIn financial models, we often work with annualized volatility, which is the standard deviation of daily returns multiplied by \(\sqrt{252}\) (per trading day in a year).
5.3 Standard Normal Distribution (N(0,1))
- The standard version of the normal has a mean of zero and a standard deviation of one. It is key to working with statistics like Z-score: $$Z = \frac{X – \mu}{\sigma}$$ This allows any normal variable to be transformed into a standard normal for probability calculations.
5.4 Value at Risk (VaR)
- Based on the normal distribution, VaR estimates the maximum expected loss over a given time horizon for a given confidence level.
- Example: a daily VaR of 5% indicates the loss that, with a 95% probability, will not be exceeded in one trading day.
5.5 Normality Test
- Financial returns do not always follow a perfect normal because they can have:
- Skewness: Whether the distribution is more likely to be skewed to the right or left.
- Kurtosis: If it has heavier tails (example: more extreme events).
- To test if the returns follow a normal pattern, you can do:
- Jarque-Bera test.
- Shapiro-Wilk test.
- QQ-Plot to see deviations from normal.
6. Limitations of normality in the markets
Although the normal distribution is a useful first approximation, it does not always correctly capture the reality of markets:
- Markets often have thicker tails than normal predicts (more extreme events).
- Volatility is not constant, which violates one of the normality assumptions.
- There are dynamic correlations and time dependence effects between price movements.
For these reasons, more sophisticated models are used, such as Lévy distributions, extreme value theory (EVT) and heteroscedastic models (GARCH).
Conclusion
The appeal of the normal distribution in the study of financial markets is due to its mathematical simplicity and the theoretical support of the Central Limit Theorem. Although it is not a perfect model, it remains an essential tool for understanding volatility and risk. To better capture extreme phenomena and market irregularities, more complex models are often used.
