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Computational Finance and Physics,
Oslo, March 22-23, 2004
ABSTRACTS (chronologically ordered):
Kevin Schmidt
Quantum Monte Carlo methods have been used and developed
to calculate a variety of physical properties. The methods have
been applied to many electrons in atoms, molecules and solids; nucleons
in the atomic nucleus; and for interacting atoms and molecules.
I will describe how to formulate calculations of some of the desired
properties, and will then give an overview of some methods as well
as results for example systems in atomic and molecular physics and
nuclear physics using variational Monte Carlo and the diffusion,
reptation and path integral ground state methods for implementing
path integral calculations with Monte Carlo techniques.
Erik Bølviken
Finance and insurance are really social sciences. The mechanisms
there are influenced by cultural, political and other factors that
change in a way that is hard to describe mathematically. Surely
this has repercussions for academic research and practical risk
analysis. Major applications deal with financial risk up to half
a century ahead! But there are also time scales much smaller than
a second.
Monte Carlo is an invaluable tool for all of this, but I believe
the issues mentioned influence how it should be put to use. The
lecture is an attempt of a discussion through some general examples.
Much can be achieved by bread-and butter Monte Carlo, but there
are some areas where high speed and advanced Monte Carlo may be
important. See additional article here.
Jordi Boronat
In this talk I introduce nowadays standard techniques for
tackling N-body quantum problems with the aid of the Monte Carlo
method. Among the different alternatives, I will focus on the diffusion
Monte Carlo method which allows for an exact solution of the Schr\"odinger
equation for bosons and a nearly-exact solution for fermions. The
high accuracy of the method is dramatically proved in the study
of quantum fluids at zero temperature. The results obtained in our
group in the last years concerning the physics of liquid $^4$He
and liquid $^3$He will be presented in comparison with experimental
data. As it will be shown, the agreement between theory
and experiment is excellent in spite of the difficulties of these
calculations due to the strong interatomic correlations.
Agnes Sulem
See separate pdf-file here.
Eirik Flekkøy
A brief historical review of the particle methods for hydrodynamics
that evolved from the lattice gas cellular
automata of the 1980's is given, along with some random examples
of how these methods perform. At the end point of this evolution
we find models that rely directly on Focker-Planck equations, not
unlike the Black-Scholes equation.
Espen Jakobsen
I will discuss resent results on error bounds for monotone approximation
schemes for Hamilton-Jacobi-Bellman equations. These are second
order degenerate elliptic and fully non-linear equations having
non-smooth solutions. They appear in optimal stochastic control
theory which has many applications e.g. in finance. For more than
a decade, nobody was able to obtain error bounds for numerical schemes
for such equations. The breakthrough was made by Krylov in 1997
and 2000, and more recently these results have been improved and
generalized by Barles and the speaker.
I will try to explain the ideas and some of the techniques involved
and state the best result available. Then I will use this result
to obtain the rate of convergence for a monotone finite difference
method and give a brief discussion about the different existing
results.
Anders Szepessy
Optimal control theory is used to solve many inverse problems
for differential equations. Optimal control in general leads to
non smooth control where viscosity solutions of the corresponding
Hamilton-Jacobi-Bellman equation provide good theoretical foundation,
but poor computational efficiency in high dimensions. The alternative
to this PDE approach is to mimimize directly and use the differential
equation as a constraint, which is computational feasible even in
very high dimension (e.g discretized PDE). Both approaches are well
established. I will focus on the relation between the two; In particular
I will present optimal order error estimates and construct regularizations
for Euler approximations to optimally controlled ordinary differential
equations in $\mathbb{R}^d$, with non smooth control. Controls can
be discontinuous due to a lack of regularity in the Hamiltonian
or due to colliding backward paths, i.e. shocks. I will describe
why it is necessary to regularize, how to do this and how to obtain
optimal convergence rates using consistency of the Hamilton-Jacobi-Bellman
equation, in the viscosity solution sense, and a discrete Pontryagin
principle, but not on solving the costly Hamilton-Jacobi-Bellman
equations. The error analysis leads to estimates useful also in
high dimensions since the bounds depend on the Lipschitz norm of
the flux and the running cost but not on $d$ explicitly: I will
show applications in topology optimization and implicit volatility
estimation.
Evgeny Zabrodhin
Xeni Dimakos
We present a new and spatial method for pricing insurance policies
based on Markov Chain Monte Carlo. The method provides estimates
of the risk premium of a policy and is intended for claim types
for which the underlying risk is believed to depend on the geographical
location of the policy.
The method includes and extends classical approaches based on generalized
linear models.
Information on the regions and individuals is included via covariates
that influence the expectation of the claim frequency and claim
size. Furthermore, we introduce latent variables with a spatial
prior. These latent variables capture effects that are not explained
by the measured and available covariates, and thus replace important
missing information. The spatial structure induce a geographical
smoothing since regions defined as neighbours are assumed to exhibit
similar risk patterns. The approach permits to quantify the risk
of policies in regions where little historical information is available,
since such regions may borrow strength from nearby regions. Estimation
is done using Markov Chain Monte Carlo (MCMC) simulation.
This is joint work with Arnoldo Frigessi.
David Dean
Lars O. Dahl
The talk will cover use of qmc in finance applications.
Many central problems in finance is in essence integration problems,
in which the use of qmc can give substantial improvements in accuracy
on a given computational budget. The use of qmc techniques usually
go hand in hand with some tailoring of the original problem to get
it in a form that comply with qmc, and gives the best gain in performance.
As examples of applications I will show the use in various option
pricing problems, show use of Malliavin calculus to formulate derivatives
of such option prices with regards to underlying parameters so that
these problems also comply with qmc, and as a last application show
how the use of qmc can be utilized to get better results in optimization
of portfolios.
Claudia La Chioma
The aim of this lecture is to present some analytical and numerical
results, recently obtained in the framework of my Ph.D. Thesis,
which concern viscosity solutions to integro differential problems
arising in Mathematical Finance when derivatives are valuated in
a market driven by general jump-diffusion processes.
When analyzing financial data it comes out that the prices process
can jump: in particular jumps become more visible as one samples
the path more frequently, making the assumption of high or infinite
jump frequencies plausible. This approach is based upon a Levy modeling
of the prices of the asset and gives a better fit to real-life data.
In view of these considerations, the prices of the stocks are modeled
in terms of exponential Levy models, the choice of a particular
Levy process standing in the choice of its distribution. Using Ito's
calculus, we can derive a nonlinear integro--partial differential
problem to get the price of a prescribed financial product.
In this lecture we shall discuss a new comparison principle for
unbounded semicontinuous viscosity sub- and supersolutions for this
kind of equations, in the case of geometric Levy process.
As a consequence of the "geometric form" of the underlying
processes, the comparison principle holds without assigning spatial
boundary data. Applications of this result will be presented for:
- backward stochastic differential equations,
- Merton problem
- pricing of European and American derivatives via backward stochastic
differential equations.
Despite presenting a great resemblance to real markets, which is
appealing for practitioner, this problem is nonlinear and does not
have a closed form solution. To overcome this difficulty a useful
tool is given by numerical approximations, which makes possible
to deal with more complicated nonlinear problems.
Starting from the fundamental result by Barles and Souganidis, we
shall show convergence for monotone, stable, consistent schemes
approximating integro-differential parabolic problems with bounded
and unbounded Levy
measures.
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