ÿþBayesowska analiza skoków w stopach zwrotu z kontraktów forward na gaz
i kontraktów futures na pozwolenia do emisji CO2
[Bayesian analysis of jumps in returns on gas forwards and CO2 allowance futures]
Maciej Kostrzewski (Katedra Ekonometrii i BadaD Operacyjnych, UE Kraków)
In the study, three Bayesian models are applied to model returns on gas forward
prices and rates of return on CO2 allowance futures prices and employed to detect
jumps therein:
a) the double exponential jump-diffusion model,
b) the stochastic volatility model with leverage effect, normal errors and a jump
component with a double exponential distribution of a jump value,
c) the stochastic volatility model with leverage effect, normal errors and a jump
component with a normal distribution of a jump value.
The results of Bayesian estimation are presented and discussed. The Bayesian
framework, founded upon the idea of latent variables is computationally facilitated
with Markov Chain Monte Carlo methods, which enables one to detect jumps and to
infer upon their frequency. The series of waiting times between two consecutive
jumps is also of interest in the paper. Periods of no jumps alternating with the
ones of frequent jumps could be indicative of existence of the jump clustering
phenomenon. Moreover, the asymmetry of the jump-size distribution and the frequency
of negative jumps versus the positive ones is explored.
Financial time series of daily logarithmic rates of return are often modelled by
the SV processes. Extending basic SV structures to SV models featuring a jump
component may often appear empirically superfluous, because of only a rather small
number of jumps to be identified in a given dataset, which is believed to be
a consequence of the SV component s capability of modelling sharp movements in the
data. The time series considered in the present research, are also analysed to
verify the impact of the jump component as an addition to the pure SV structure.
Preliminary empirical results suggest some important role of the jump component.
A test of whether the jump activity may vary over time is also conducted, with the
procedure aiming at either accepting or rejecting the hypotheses of a constant
jump intensity. The results presented in the paper indicate the existence of jump
clustering phenomenon under the jump-diffusion structure.