A model for stock returns and volatility

• We show that historic volatility is best described by the generalized inverse gamma distribution.
• We show that historic stock returns are best described by the generalized Student’s distribution.
• We discuss stochastic stock and volatility models that produce these distributions.
• We obtain the mean and the variance of relaxation times on approach to steady state distributions.
• We examine 1/f1/f noise in volatility and stock returns.

We prove that Student’s t-distribution provides one of the better fits to returns of S&P component stocks and the generalized inverse gamma distribution best fits VIX and VXO volatility data. We further prove that stock returns are best fit by the product distribution of the generalized inverse gamma and normal distributions. We find Brown noise in VIX and VXO time series and explain the mean and the variance of the relaxation times on approach to the steady-state distribution.