Compound distributions for financial returns

Abstract

In this paper, we propose six Student's t based compound distributions where the scale parameter is randomized using functional forms of the half normal, Fréchet, Lomax, Burr III, inverse gamma and generalized gamma distributions. For each of the proposed distribution, we give expressions for the probability density function, cumulative distribution function, moments and characteristic function. GARCH models with innovations taken to follow the compound distributions are fitted to the data using the method of maximum likelihood. For the sample data considered, we see that all but two of the proposed distributions perform better than two popular distributions. Finally, we perform a simulation study to examine the accuracy of the best performing model.

Fig 1. Histogram of standard deviations computed over non-overlapping windows of length 50 days for the specified daily log-returns (S&P500, DJI, Diesel, Propane, BTC and LTC).

Fig 2. Kurtosis values corresponding to (11) (top left), (13) (top right), (15) (middle left), (17) (middle right), (19) (bottom left) and (21) (bottom right) versus ν and selected values of other parameters.

Fig 3. Time series plots of the daily log-returns of S&P500, DJI, Diesel, Propane, BTC and LTC with their histograms and kernel based density estimates.

Fig 4. P-P plots of the standardized residuals of the GARCH(1, 1) model with innovations given by (14).

Fig 5. Q-Q plots of the standardized residuals of the GARCH(1, 1) model with innovations given by (14).

Fig 6. Biases of the parameter estimates of the GARCH(1, 1) model with innovations given by (14) based on the simulation study of Section 5.

Fig 7. Mean squared errors of the parameter estimates of the GARCH(1, 1) model with innovations given by (14) based on the simulation study of Section 5.