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

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.