A long-term alternative formula for a stochastic stock price model

TL;DR

This paper introduces a new long-term stock price variation formula based on median instead of mean, improving prediction accuracy in heavy-tailed, skewed distributions by reducing outlier influence.

Contribution

It proposes a median-based alternative formula for geometric Brownian motion, addressing limitations of expected value in skewed, heavy-tailed distributions for long-term stock modeling.

Findings

Median-based formula offers more realistic long-term predictions.

Improves modeling accuracy for high-volatility, heavy-tailed distributions.

Reduces outlier sensitivity compared to traditional expected value methods.

Abstract

This study presents a long-term alternative formula for stock price variation described by a geometric Brownian motion on the basis of median instead of mean or expected values. The proposed method is motivated by the observation made in remote fields, where optimality of bet-hedging or diversification strategies is explained based on a measure different from expected value, like geometric mean. When the probability distribution of possible outcomes is significantly skewed, it is generally known that expected value leads to an erroneous picture owing to its sensitivity to outliers, extreme values of rare occurrence. Since geometric mean, or its counterpart median for the log-normal distribution, does not suffer from this drawback, it provides us with a more appropriate measure especially for evaluating long-term outcomes dominated by outliers. Thus, the present formula makes a more realistic prediction for long-term outcomes of a large volatility, for which the probability distribution becomes conspicuously heavy-tailed.