8:00 PM (1 hour ago)
Ladies and Gentlemen,
The Zulf model is a seven parameter model: five parameters from the Heston (1993) model
kappa – mean reversion rate for volatility
v0,vT – initial and long run volatility
rho – correlation between return and volatility
sigma – vol-of-vol
alpha – long memory parameter
delta – time-scale parameter
So how well does this model explain return distributions?
See attached graphs for an example. These are for 2,5,10,30,60 day returns. In these cases the KL divergence of empirical returns versus model is below 0.1 for 2,5,10 and around 0.9 for 30 and 2.5 for 60. So it’s not a perfect fit and seems to need some investigation for the longer period returns but for the short term it seems extremely good at telling us the exact return distribution. We already know that it improves upon Heston fits for volatility surfaces (statistically significantly and in over 70% of the cases, which should be the standard benchmark anyway since it’s totally meaningless in 2016 to benchmark against BSM models because it tells us nothing useful; all valid models trivially beat BSM). The real question now is whether Zulf model explains return distributions better than some of the fancier generalized hyperbolic models which will be the next phase.