27 research outputs found
Inversion of analytic characteristic functions and infinite convolutions of exponential and Laplace densities
We prove that certain quotients of entire functions are characteristic
functions. Under some conditions, the probability measure corresponding to a
characteristic function of that type has a density which can be expressed as a
generalized Dirichlet series, which in turn is an infinite linear combination
of exponential or Laplace densities. These results are applied to several
examples
A new look at the Heston characteristic function
A new expression for the characteristic function of log-spot in Heston model
is presented. This expression more clearly exhibits its properties as an
analytic characteristic function and allows us to compute the exact domain of
the moment generating function. This result is then applied to the volatility
smile at extreme strikes and to the control of the moments of spot. We also
give a factorization of the moment generating function as product of Bessel
type factors, and an approximating sequence to the law of log-spot is deduced
Multilevel Monte Carlo simulation for Levy processes based on the Wiener-Hopf factorisation
In Kuznetsov et al. (2011) a new Monte Carlo simulation technique was
introduced for a large family of Levy processes that is based on the
Wiener-Hopf decomposition. We pursue this idea further by combining their
technique with the recently introduced multilevel Monte Carlo methodology.
Moreover, we provide here for the first time a theoretical analysis of the new
Monte Carlo simulation technique in Kuznetsov et al. (2011) and of its
multilevel variant for computing expectations of functions depending on the
historical trajectory of a Levy process. We derive rates of convergence for
both methods and show that they are uniform with respect to the "jump activity"
(e.g. characterised by the Blumenthal-Getoor index). We also present a modified
version of the algorithm in Kuznetsov et al. (2011) which combined with the
multilevel methodology obtains the optimal rate of convergence for general Levy
processes and Lipschitz functionals. This final result is only a theoretical
one at present, since it requires independent sampling from a triple of
distributions which is currently only possible for a limited number of
processes
The β-Meixner model
AbstractWe propose to approximate the Meixner model by a member of the β-family introduced by Kuznetsov (2010) in [2]. The advantage of the approximation is the semi-explicit formulae for the running extrema under the β-family processes which enables us to produce more efficient algorithms for pricing path dependent options through the Wiener–Hopf factors. We will explore the performance of the approximation both in an equity framework and in the credit risk setting, where we use the approximation to calibrate a surface of credit default swaps. The paper follows the approach of the study made by Schoutens and Damme (2010) in [1], where the aim was to approximate the variance gamma. We will contextualize the results by Schoutens and Damme (2010) in [1] and the ones here with respect to the approach taken by Jeannin and Pistorius (2010) in [15]. An asymptotic expression for the rate of convergence of the approximation is derived
The β-Meixner model
We propose to approximate the Meixner model by a member of the B-family introduced in [Kuz10a]. The advantage of such approximations are the semi-explicit formulas for the running extrema under the B-family processes which enables us to produce more efficient algorithms for certain path dependent options
A new look at the Heston characteristic function
A new expression for the characteristic function of log-spot in Heston model is presented. This expression more clearly exhibits its properties as an analytic characteristic function and allows us to compute the exact domain of the moment generating function. This result is then applied to the volatility smile at extreme strikes and to the control of the moments of spot. We also give a factorization of the moment generating function as product of Bessel type factors, and an approximating sequence to the law of log-spot is deduced