# The Geometric Random Walk in Finance

The concept of the Geometric Random Walk is foundational in understanding how financial markets behave. At its core, this theory suggests that stock prices evolve unpredictably, with their movements akin to a particle undergoing a random walk. This randomness is intrinsic to the market’s nature, reflecting a continuum of unforeseen events and information.

When we look at stock prices and try to predict their future, we often use the concept of a random walk. There are two types: the normal random walk and the geometric random walk. Both are used to model how stock prices move, but they differ in important ways.

## Normal Random Walk

In a normal random walk, we say that the next price of a stock is the current price plus some random change. This change could be up or down and is entirely unpredictable. The formula is:

$$P_t = P_{t-1} + \epsilon_t$$

where:

• $$P_t$$ is the future stock price,
• $$P_{t-1}$$ is the current stock price,
• $$\epsilon_t$$ is the random change.

This model is simple but has a big limitation: it suggests that stock prices could potentially go below zero, which isn’t possible in the real world.

## Geometric Random Walk

The geometric random walk fixes this by looking at percentage changes rather than absolute changes.

Here, we multiply the current price by a factor to get the next price. This factor represents the expected rate of return, which is random.

The formula is: $$P_t = P_{t-1} \times e^{\epsilon_t}$$

• $$e^{\epsilon_t}$$ is the factor by which the price changes, and it’s based on the rate of return, which is assumed to have a normal distribution.
• The “geometric” part of the name comes from the way we multiply to get the new price.
• This approach is more realistic because it ensures prices can’t go negative and fits better with how stock prices actually behave.

## Key Differentiators

Here’s how the geometric random walk stands out:
• Positive Prices: It inherently ensures that stock prices do not go negative, aligning with the reality that prices cannot be less than \$0.
• Volatility Clustering: This model accounts for volatility clustering—a phenomenon where large price changes tend to be followed by more large changes, and small changes tend to follow small changes.
• Compounded Returns: It mirrors the compound nature of asset returns over time, which is a cornerstone of investment analysis.
While the normal random walk might serve as a good model for price differences, the geometric random walk is more suited for modeling compounded returns and for pricing derivatives, where the log-normal distribution of returns plays a critical role.

## In Summary

The main difference between the two models is how they approach changes in stock prices.

The normal random walk adds or subtracts random amounts to the price, while the geometric random walk multiplies the price by a random factor. For financial professionals, knowing these differences is key to choosing the right model for the right situation.

## Fitting into the Model -

From Arithmetic to Geometric:
• Instead of fixed increases or decreases, we consider proportional changes, as stock prices change in percentages.
• This change turns our model from a straightforward arithmetic random walk into a geometric random walk.
Modeling Stock Price Changes:
• For a heads outcome, we increase the stock price by 1%.
• For tails, we decrease it by 1%.
• This process is repeated through ‘N’ iterations, simulating realistic stock price movements.

Let’s fit the said model to our current equation –

To fit your model of proportional changes due to coin toss outcomes into the geometric random walk equation, we would modify the model as follows:

In the geometric random walk model, the stock price at time $$t$$ is a product of the stock price at time $$t-1$$ and an exponential factor that represents the rate of return, which in this case is determined by the coin toss. The equation is:

$$P_t = P_{t-1} \times e^{\epsilon_t}$$

• If the outcome is heads, the stock price increases by 1%, which can be expressed as a rate of return $$\epsilon_t$$ of $$\log(1.01)$$.
• If the outcome is tails, the stock price decreases by 1%, which can be expressed as a rate of return $$\epsilon_t$$ of $$\log(0.99)$$.

So, the modified equation for each coin toss would be:

• For heads: $$P_t = P_{t-1} \times e^{\log(1.01)}$$ which simplifies to $$P_t = P_{t-1} \times 1.01$$
• For tails: $$P_t = P_{t-1} \times e^{\log(0.99)}$$ which simplifies to $$P_t = P_{t-1} \times 0.99$$

This process would be repeated through ‘N’ iterations, with the rate of return $$\epsilon_t$$ being determined by the outcome of each coin toss.

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