Normal and Log-Normal Distributions in Finance

1. Why We Use Log Returns for Modeling

Before we dive into distributions, we need to clarify what exactly we're modeling. While we often think in terms of simple returns ($\frac{P_t-P_{t-1}}{P_{t-1}}$), for statistical modeling, we almost always use logarithmic returns (log returns).

A log return is calculated as:

There are several reasons for this, but two are paramount for our purposes. The following reading explains them well.

The key takeaway is time-additivity. To get the total return over a long period, you can simply sum the individual log returns. This is an incredibly convenient mathematical property that makes modeling much easier. For instance, the total log return from time 0 to $T$ is just $\ln (P_T)-\ln (P_0)$, regardless of the path it took in between. This is not possible with simple returns, which require compounding (multiplication).

This additivity is a property you're likely familiar with from other domains, like calculating total signal attenuation over a series of components in electronics, where you sum the decibel losses.

2. The Normal Distribution: A Starting Point for Returns

The Normal distribution, or "bell curve," is ubiquitous in statistics. Given your math background, you've certainly encountered it. It's defined by two parameters we already know: the mean ($\mu$) and the standard deviation ($\sigma$).

A common assumption in many financial models is that log returns are approximately normally distributed over short time intervals. Why? The Central Limit Theorem provides some justification: if a return is the result of many small, independent random shocks, their sum will tend to be normally distributed.

However, a critical problem arises if we try to model asset prices directly with a normal distribution:

• A normal distribution has infinite support, from $-\infty$ to $+\infty$.
• This means it assigns a non-zero (though perhaps tiny) probability to asset prices being negative, which is impossible.

This fundamental flaw leads us to a more appropriate model for prices.

3. The Log-Normal Distribution: A Better Model for Prices

If we stick with the powerful assumption that log returns, $\ln (P_t/P_{t-1})$, are normally distributed, what does this imply about the prices themselves?

It implies that asset prices, $P_t$, follow a Log-Normal Distribution.

A random variable $X$ is log-normally distributed if its natural logarithm, $\ln (X)$, is normally distributed. This elegant connection solves the negative price problem and introduces other realistic features.

The next video clearly explains this concept in the context of the famous Black-Scholes option pricing model, which is built on this very assumption.

Key Properties of a Log-Normal Price Distribution

As the video highlights, the log-normal distribution has characteristics that make it a much better fit for asset prices than the normal distribution:

1. Non-Negative Support: The distribution is defined only for values greater than zero. The probability of a price being less than or equal to zero is exactly zero.
2. Positive Skew (Right-Skewed): Unlike the symmetric bell curve, the log-normal distribution has a long tail to the right. This implies that while the most likely outcomes are clustered, there's a possibility of extremely large positive price movements, while losses are bounded by the price falling to zero. This often reflects market reality better than a symmetric model.

4. Under the Hood: The Mathematical Origins

Your preference for understanding the underlying mechanics is important here. The log-normal model isn't just a convenient choice; it can be derived from a set of foundational assumptions about how markets work.

This involves modeling the asset price as a stochastic process—a process that evolves randomly over time. You might recall from your differential equations coursework that such processes can be described by stochastic differential equations (SDEs).

The following video provides a more rigorous mathematical derivation. It uses concepts like Brownian motion and Itô's formula. While you don't need to master these concepts right now (they are part of advanced quantitative finance), seeing how the model is built will give you a deeper appreciation for its foundations.

This derivation solidifies the idea that the log-normal distribution is a natural consequence of modeling asset prices as a geometric random walk, where percentage changes (not absolute changes) are random.

5. A Critical View: How Well Does the Model Fit Reality?

As a developer experienced in building robust systems, you know that models are simplifications. The log-normal model is a powerful first approximation, but it's not perfect. Your skepticism about common strategies is healthy and should extend to the models themselves.

The biggest discrepancy between the theory (normally distributed log returns) and reality is the presence of "fat tails". Real-world financial returns experience extreme events—market crashes and explosive rallies—far more frequently than a normal distribution would predict.

The following resource provides a practical look at this limitation using Python.

This is a crucial insight. While we start with the normal/log-normal framework, a sophisticated quant must be aware of its shortcomings, particularly its tendency to underestimate the risk of extreme events ("tail risk"). In later modules, we will explore techniques and other distributions (like the Student's t-distribution you just saw) that better account for these fat tails.

Conclusion

In this lesson, we moved from describing data with moments to modeling it with distributions. We established why the Normal/Log-Normal framework is a cornerstone of quantitative finance.

Key Takeaways:

• Log returns ($\ln (P_t/P_{t-1})$) are used for modeling because they are time-additive and more symmetric than simple returns.
• A foundational assumption in finance is that log returns are normally distributed.
• This directly implies that asset prices follow a log-normal distribution.
• The key properties of a log-normal distribution—being non-negative and positively skewed—make it a much more realistic model for prices than a normal distribution.
• While powerful, this model has limitations. Real-world returns exhibit "fat tails," meaning extreme events are more common than the normal distribution predicts.

Preview of the Next Lesson:

So far, we've only analyzed assets in isolation. But in trading, the relationships between assets are just as important. How does Bitcoin's price movement relate to Ethereum's? In our next lesson, we will learn how to quantify these relationships by calculating correlation and covariance, opening the door to strategies based on diversification and relative value.