46,723 research outputs found
Levy process simulation by stochastic step functions
We study a Monte Carlo algorithm for simulation of probability distributions
based on stochastic step functions, and compare to the traditional
Metropolis/Hastings method. Unlike the latter, the step function algorithm can
produce an uncorrelated Markov chain. We apply this method to the simulation of
Levy processes, for which simulation of uncorrelated jumps are essential.
We perform numerical tests consisting of simulation from probability
distributions, as well as simulation of Levy process paths. The Levy processes
include a jump-diffusion with a Gaussian Levy measure, as well as
jump-diffusion approximations of the infinite activity NIG and CGMY processes.
To increase efficiency of the step function method, and to decrease
correlations in the Metropolis/Hastings method, we introduce adaptive hybrid
algorithms which employ uncorrelated draws from an adaptive discrete
distribution defined on a space of subdivisions of the Levy measure space.
The nonzero correlations in Metropolis/Hastings simulations result in heavy
tails for the Levy process distribution at any fixed time. This problem is
eliminated in the step function approach. In each case of the Gaussian, NIG and
CGMY processes, we compare the distribution at t=1 with exact results and note
the superiority of the step function approach.Comment: 20 pages, 18 figure
Diffusion covariation and co-jumps in bidimensional asset price processes with stochastic volatility and infinite activity Levy jumps
In this paper we consider two processes driven by diffusions and jumps. The
jump components are Levy processes and they can both have finite activity and
infinite activity. Given discrete observations we estimate the covariation
between the two diffusion parts and the co-jumps. The detection of the co-jumps
allows to gain insight in the dependence structure of the jump components and
has important applications in finance. Our estimators are based on a threshold
principle allowing to isolate the jumps. This work follows Gobbi and Mancini
(2006) where the asymptotic normality for the estimator of the covariation,
with convergence speed given by the squared root of h, was obtained when the
jump components have finite activity. Here we show that the speed is the
squared root of h only when the activity of the jump components is moderate
Continuous-time VIX dynamics: on the role of stochastic volatility of volatility
This paper examines the ability of several different continuous-time one- and two-factor jump-diffusion models to capture the dynamics of the VIX volatility index for the period between 1990 and 2010. For the one-factor models we study affine and non-affine specifications, possibly augmented with jumps. Jumps in one-factor models occur frequently, but add surprisingly little to the ability of the models to explain the dynamic of the VIX. We present a stochastic volatility of volatility model that can explain all the time-series characteristics of the VIX studied in this paper. Extensions demonstrate that sudden jumps in the VIX are more likely during tranquil periods and the days when jumps occur coincide with major political or economic events. Using several statistical and operational metrics we find that non-affine one-factor models outperform their affine counterparts and modeling the log of the index is superior to modeling the VIX level directly
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