Inverted anhamonic oscillator model for distribution of financial returns

We construct a quantum-mechanical model to explain the distribution of financial returns in a stock market when it exhibits an upward trend. By combining a critical phenomenon effect in the form of a power law and the Schrodinger equation, we show that an appropriate potential of the financial returns is given by a time-dependent inverted anharmonic oscillator, whose coefficients depend on the critical time and exponent, which are empirically obtained from the Stock Exchange of Thailand (SET) from 1992 to 1994, during the critical phase of the Asian financial crisis. With the derived potential, we simulate the dynamics of returns as a function of time by employing the time-dependent variational method and the fourth-order Runge-Kutta method. Then we compute key characteristics of the return distribution such as mean, variance, skewness, and kurtosis and compare them with real financial data from SET. The results are found that the mean, skewness and kurtosis show good agreement with actual data computed from SET, but the variance is higher than that from the SET data.