65C05 Monte Carlo methods
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- Monte Carlo simulation (3)
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- Bermudan options (1)
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- Financial derivatives (1)
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- Libor market model (1)
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- Monte-Carlo methods (1)
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In this article we show, that the binding kinetics of a molecular system can be
identied by a projection of a continuous process onto a nite number of macro states. We thus
interpret binding kinetics as a projection. When projecting onto non-overlapping macro states the
Markovianity is spoiled. As a consequence, the description of e.g. a receptor-ligand system by a two
state kinetics is not accurate. By assigning a degree of membership to each state, we abandon the
non-overlapping approach. This overlap is crucial for a correct mapping of binding eects by Markov
State Models with regard to their long time behavior. It enables us to describe the highly discussed
rebinding eect, where the spatial arrangement of the system has the be included. By introducing
a \degree of fuzziness" we have an indicator for the strength of the rebinding eect, such that the
minimal rebinding eect can be derived from an optimization problem. The fuzziness also includes
some new paradigms for molecular kinetics. These new model paradigms show good agreement with
experimental data.
Markov State Models (MSMs) are widely used to represent molecular
conformational changes as jump-like transitions between subsets of the conformational
state space. However, the simulation of peptide folding in explicit water is
usually said to be unsuitable for the MSM framework. In this article, we summarize
the theoretical background of MSMs and indicate that explicit water simulations do
not contradict these principles. The algorithmic framework of a meshless conformational
space discretization is applied to an explicit water system and the sampling
results are compared to a long-term molecular dynamics trajectory. The meshless
discretization approach is based on spectral clustering of stochastic matrices (MSMs)
and allows for a parallelization of MD simulations. In our example of Trialanine we
were able to compute the same distribution of a long term simulation in less computing
time.
In this article we propose a novel approach to reduce the computational complexity
of the dual method for pricing American options. We consider a sequence of
martingales that converges to a given target martingale and decompose the original
dual representation into a sum of representations that correspond to dierent levels
of approximation to the target martingale. By next replacing in each representation
true conditional expectations with their Monte Carlo estimates, we arrive at what
one may call a multilevel dual Monte Carlo algorithm. The analysis of this algorithm
reveals that the computational complexity of getting the corresponding target upper
bound, due to the target martingale, can be signicantly reduced. In particular, it
turns out that using our new approach, we may construct a multilevel version of the
well-known nested Monte Carlo algorithm of Andersen and Broadie (2004) that is,
regarding complexity, virtually equivalent to a non-nested algorithm. The performance
of this multilevel algorithm is illustrated by a numerical example.
Optimal dual martingales, their analysis and application to new algorithms for Bermudan products
(2012)
In this paper we introduce and study the concept of optimal and surely
optimal dual martingales in the context of dual valuation of Bermudan
options, and outline the development of new algorithms in this context.
We provide a characterization theorem, a theorem which gives conditions
for a martingale to be surely optimal, and a stability theorem concerning martingales which are near to be surely optimal in a sense. Guided
by these results we develop a framework of backward algorithms for constructing such a martingale. In turn this martingale may then be utilized
for computing an upper bound of the Bermudan product. The methodology is pure dual in the sense that it doesn't require certain (input)
approximations to the Snell envelope.
In an Ito-Levy environment we outline a particular regression based
backward algorithm which allows for computing dual upper bounds with-
out nested Monte Carlo simulation. Moreover, as a by-product this algorithm also provides approximations to the continuation values of the
product, which in turn determine a stopping policy. Hence, we may obtain lower bounds at the same time.
In a first numerical study we demonstrate a backward dual regression algorithm in a Wiener environment that is easy to implement and
is regarding accuracy comparable with the method of Belomestny et. al.
(2009).
In order to compute the thermodynamic weights of the different metastable conformations
of a molecule, we want to approximate the molecule’s Boltzmann distribution in a reasonable
time. This is an essential issue in computational drug design. The energy landscape of active
biomolecules is generally very rough with a lot of high barriers and low regions. Many of the
algorithms that perform such samplings (e.g. the hybrid Monte Carlo method) have difficulties
with such landscapes. They are trapped in low-energy regions for a very long time and cannot
overcome high barriers. Moving from one low-energy region to another is a very rare event. For
these reasons, the distribution of the generated sampling points converges very slowly against
the thermodynamically correct distribution of the molecule.
The idea of ConfJump is to use a priori knowledge of the localization of low-energy regions
to enhance the sampling with artificial jumps between these low-energy regions. The artificial
jumps are combined with the hybrid Monte Carlo method. This allows the computation of
some dynamical properties of the molecule. In ConfJump, the detailed balance condition is
satisfied and the mathematically correct molecular distribution is sampled.
In this paper we introduce efficient Monte Carlo estimators for the valuation
of high-dimensional derivatives and their sensitivities (”Greeks”).
These estimators are based on an analytical, usually approximative representation
of the underlying density. We study approximative densities
obtained by the WKB method. The results are applied in the context of
a Libor market model.
In this paper we lay the foundation for a numerical algorithm to
simulate high-dimensional coupled FBSDEs under weak coupling or
monotonicity conditions. In particular we prove convergence of a
time discretization and a Markovian iteration. The iteration
differs from standard Picard iterations for FBSDEs in that the
dimension of the underlying Markovian process does not increase
with the number of iterations. This feature seems to be
indispensable for an efficient iterative scheme from a numerical
point of view. We finally suggest a fully explicit numerical
algorithm and present some numerical examples with up to
10-dimensional state space.
We introduce a forward scheme to simulate backward SDEs. Compared
to existing schemes, we avoid high order nestings of conditional
expectations backwards in time. In this way the error, when
approximating the conditional expectation, in dependence of the
time partition is significantly reduced. Besides this generic
result, we present an implementable algorithm and provide an error
analysis for it. Finally, we demonstrate the strength of the new
algorithm by solving some financial problems numerically.
We propose a valuation method for callable structures in a multi-factor Libor model which are path-dependent in the sense that, after calling, one receives a sequence of cash-flows in the future, instead of a well specified cash-flow at the calling date. The method is based on a Monte Carlo procedure for standard Bermudans recently developed in Kolodko & Schoenmakers (2004), and is applied to the cancelable snowball interest rate swap. The proposed procedure is quite generic, straightforward to implement, and can be easily adapted to other related path-dependent products.