Modeling and Risk Analysis Using Parametric Distributions with an Application in Equity-Linked Securities

Table 5

The estimated parameters of the eight distributions and maximum log-likelihood for the filtered residuals from S&P 500’s log return.

Model	No.	Parameters					Log-L.
Cauchy	2	μ	σ				−3452.9
		0.0761	0.5122
Laplace	2	μ	σ				−3207.6
		0.0606	0.3003
Normal	2	μ	σ				−3263.2
		0.0063	0.9985
Student’s t	3	μ	σ	ν			−3203.9
		0.0436	0.7991	5.3023
Skew normal	3	μ	σ	α			−3248.5
		−0.0074	0.9955	0.8763
Skew Cauchy	3	μ	σ	α			−3447.6
		0.1819	0.5175	−0.2155
Skew Laplace	3	μ	σ	α			−3200.6
		0.1332	0.3665	0.5422
Skew Student’s t	4	μ	σ	α	ν		−3193.9
		0.2421	0.9888	−0.2373	5.8101
Hyperbolic	4	μ	σ	α	β		−3186.4
		0.2062	0.9914	−0.1999	0.8485
NIG	4	μ	λ	σ	β		−3189.1
		0.2259	0.9869	−0.2195	1.4165
Variance gamma	4	μ	σ	α	λ		−3185.1
		0.2043	0.9886	−0.1979	1.6782
G. hyperbolic	5	μ	σ	α	β	λ	−3185.1
		0.2042	0.9893	−0.1982	0.0781	1.6772