922 research outputs found
Derivatives pricing in energy markets: an infinite dimensional approach
Based on forward curves modelled as Hilbert-space valued processes, we
analyse the pricing of various options relevant in energy markets. In
particular, we connect empirical evidence about energy forward prices known
from the literature to propose stochastic models. Forward prices can be
represented as linear functions on a Hilbert space, and options can thus be
viewed as derivatives on the whole curve. The value of these options are
computed under various specifications, in addition to their deltas. In a second
part, cross-commodity models are investigated, leading to a study of square
integrable random variables with values in a "two-dimensional" Hilbert space.
We analyse the covariance operator and representations of such variables, as
well as presenting applications to pricing of spread and energy quanto options
Integration theory for infinite dimensional volatility modulated Volterra processes
We treat a stochastic integration theory for a class of Hilbert-valued,
volatility-modulated, conditionally Gaussian Volterra processes. We apply
techniques from Malliavin calculus to define this stochastic integration as a
sum of a Skorohod integral, where the integrand is obtained by applying an
operator to the original integrand, and a correction term involving the
Malliavin derivative of the same altered integrand, integrated against the
Lebesgue measure. The resulting integral satisfies many of the expected
properties of a stochastic integral, including an It\^{o} formula. Moreover, we
derive an alternative definition using a random-field approach and relate both
concepts. We present examples related to fundamental solutions to partial
differential equations.Comment: Published at http://dx.doi.org/10.3150/15-BEJ696 in the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
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
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