Accuracy of the Approximation ln(St/S0) ≈ (St - S0) / S0 for Stock Market Analysis

Introduction

In the field of stock market analysis, it is often necessary to approximate the natural logarithm of the ratio of current stock price (St) to the initial stock price (S0). One common approximation used is:

(lnleft(frac{St}{S0}right) ≈ frac{St - S0}{S0})

This approximation is based on the Taylor series expansion of the natural logarithm function around (x 1) and is particularly useful when (St) is close to (S0). This article explores the accuracy of this approximation, its limitations, and the scenarios where it can be used effectively.

The Taylor Series Expansion and Its Limitations

The Taylor series expansion of the natural logarithm function (ln(1 x) approx x - frac{x^2}{2} frac{x^3}{3} - cdots) for (x) near 0 suggests the approximation:

(ln(1 x) ≈ x)

or equivalently,

(ln(x) ≈ x - 1) for (x) near 1.

The function (ln(x)) intersects the x-axis at (x 1) with a slope of 1, making the approximation valid near this point. Specifically, when (frac{St}{S0} ≈ 1), the approximation can be written as:

(lnleft(frac{St}{S0}right) ≈ frac{St - S0}{S0})

Plotting the Error

To illustrate the accuracy of this approximation, we can plot the function (y frac{St - S0}{S0}) and compare it to (lnleft(frac{St}{S0}right)). Below is the plot of (y frac{St - S0}{S0}) and (x - 1 - ln(x)) to highlight the error:

Plot of x - 1 - ln(x)

This plot demonstrates that the approximation (lnleft(frac{St}{S0}right) ≈ frac{St - S0}{S0}) is highly accurate for (frac{St}{S0} ≈ 1). However, as (frac{St}{S0}) deviates from 1, the error increases linearly.

Linear Deviation Example

For example, when (frac{St}{S0} 2), the absolute error in the approximation is approximately 44%, while for (frac{St}{S0} 1.5), it is about 23%, and for (frac{St}{S0} 1.25), it is around 12%. This indicates that the error grows almost linearly with the distance from 1.

Relative Error Analysis

In terms of relative error, the error is also significantly large for values of (frac{St}{S0}) significantly different from 1. The plot below shows the relative error in percent as a function of (frac{St}{S0}). The linear nature of the relative error indicates that the approximation is not very accurate for large deviations from 1.

Plot of Relative Error in Percent

Technically, the Taylor series expansion of the natural logarithm (ln(1 x)) is only valid for (-1 x 1). In the context of stock market analysis, if (St) diverges far from (S0), the approximation may become invalid, leading to significant errors in the analysis. Therefore, it is crucial to use the approximation with caution, especially when (St) is no longer close to (S0).

Conclusion

In summary, the approximation (lnleft(frac{St}{S0}right) ≈ frac{St - S0}{S0}) is highly accurate when (St) is close to (S0). However, it should be used with caution when (St) significantly deviates from (S0). In such cases, the approximation may lead to substantial errors in stock market analysis. The key takeaway is that while this approximation can be useful, it is important to understand its limitations and use it appropriately.

Keywords: logarithmic approximation, stock market, Taylor series, natural logarithm