Feeds:
Posts

## STATISTICALLY SIGNIFICANT TEST OF FRACTIONAL POISSON SHOCKS TO VOLATILITY

ComteRenault1998 studied long memory stochastic volatility models in 1998.  Their model for $Y_t = \log \sigma(t)$ is $dY_t = k (\theta - Y_t)dt + \gamma dw^H_t$ where $H>1/2$ is the Hurst exponent of the fractional Brownian motion.  They generally found on real data that $H ~ 0.8$ for $|\sigma_{t+1}-\sigma_t|$.  So empirical evidence for non-Markovian volatilities is not new.

Thanks to the method-of-moment estimators for fractional Poisson process (http://www2.latech.edu/~dcahoy/fPp.pdf) which depend on two parameters (mu,nu) where nu=1 corresponds to Markovian Poisson intensity.  I calculated their estimators of mu and nu for 1900 volatilities and found (sample size ~1900)

(‘mean mu=’, 0.25356579252485467, ‘ std(mu)=’, 0.058293974647685647)
(‘mean nu=’, 1.180714603354504, ‘ std(nu)=’, 0.10268867530749047)
I counted shocks to volatility by a threshold of 0.1 (arbitrary) and calculated noisy daily volatility using the stochastic-volatility type model log(return^2+eps).  The code is attached.  This is on daily data, so this is fairly significant in the debate in finance regarding non-Markovian character of the markets: often people look at TICK data and assume that non-Markovian character persists for 15 minutes but here at least for volatilities, daily data contains quantifiable evidence for a non-Markovian process, and indeed fractional Poisson process is more appropriate to model waiting times for volatility shocks than Markovian models.

2 Attachments