2,937 research outputs found
Pricing path-dependent Bermudan options using Wiener chaos expansion: an embarrassingly parallel approach
In this work, we propose a new policy iteration algorithm for pricing
Bermudan options when the payoff process cannot be written as a function of a
lifted Markov process. Our approach is based on a modification of the
well-known Longstaff Schwartz algorithm, in which we basically replace the
standard least square regression by a Wiener chaos expansion. Not only does it
allow us to deal with a non Markovian setting, but it also breaks the
bottleneck induced by the least square regression as the coefficients of the
chaos expansion are given by scalar products on the L^2 space and can therefore
be approximated by independent Monte Carlo computations. This key feature
enables us to provide an embarrassingly parallel algorithm.Comment: The Journal of Computational Finance, Incisive Media, In pres
A Parallel Algorithm for solving BSDEs - Application to the pricing and hedging of American options
We present a parallel algorithm for solving backward stochastic differential
equations (BSDEs in short) which are very useful theoretic tools to deal with
many financial problems ranging from option pricing option to risk management.
Our algorithm based on Gobet and Labart (2010) exploits the link between BSDEs
and non linear partial differential equations (PDEs in short) and hence enables
to solve high dimensional non linear PDEs. In this work, we apply it to the
pricing and hedging of American options in high dimensional local volatility
models, which remains very computationally demanding. We have tested our
algorithm up to dimension 10 on a cluster of 512 CPUs and we obtained linear
speedups which proves the scalability of our implementationComment: 25 page
Local Indicators for Plurisubharmonic Functions
A notion of local indicator for a plurisubharmonic function is introduced.
The indicator is a certain plurisubharmonic function in the unit polydisc,
which controls the behavior of the considered function near a fixed point of
its singularity, as well as the residual Monge-Ampere mass of the function at
this point. A pluricomplex Green function with respect to given indicators is
constructed and applied to the Dirichlet problem for the Monge- Ampere operator
in the class of plurisubharmonic functions with isolated singularities.Comment: 18 pages, LaTeX; to appear in J. Math. Pures App
Long time behaviour of a stochastic nano particle
In this article, we are interested in the behaviour of a single ferromagnetic
mono-domain particle submitted to an external field with a stochastic
perturbation. This model is the first step toward the mathematical
understanding of thermal effects on a ferromagnet. In a first part, we present
the stochastic model and prove that the associated stochastic differential
equation is well defined. The second part is dedicated to the study of the long
time behaviour of the magnetic moment and in the third part we prove that the
stochastic perturbation induces a non reversibility phenomenon. Last, we
illustrate these results through numerical simulations of our stochastic model.
The main results presented in this article are the rate of convergence of the
magnetization toward the unique stable equilibrium of the deterministic model.
The second result is a sharp estimate of the hysteresis phenomenon induced by
the stochastic perturbation (remember that with no perturbation, the magnetic
moment remains constant)
A Parallel Algorithm for solving BSDEs - Application to the pricing and hedging of American options
We present a parallel algorithm for solving backward stochastic differential equations (BSDEs in short) which are very useful theoretic tools to deal with many financial problems ranging from option pricing option to risk management. Our algorithm based on Gobet and Labart (2010) exploits the link between BSDEs and non linear partial differential equations (PDEs in short) and hence enables to solve high dimensional non linear PDEs. In this work, we apply it to the pricing and hedging of American options in high dimensional local volatility models, which remains very computationally demanding. We have tested our algorithm up to dimension 10 on a cluster of 512 CPUs and we obtained linear speedups which proves the scalability of our implementationbackward stochastic differential equations, parallel computing, Monte- Carlo methods, non linear PDE, American options, local volatility model.
Green vs. Lempert functions: a minimal example
The Lempert function for a set of poles in a domain of at a
point is obtained by taking a certain infimum over all analytic disks going
through the poles and the point , and majorizes the corresponding multi-pole
pluricomplex Green function. Coman proved that both coincide in the case of
sets of two poles in the unit ball. We give an example of a set of three poles
in the unit ball where this equality fails.Comment: v3: proof of the upper estimate for the Green function added;
accepted in Pacific Journal of Mathematic
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