19 research outputs found
Multilevel dual approach for pricing American style derivatives
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 different 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 significantly 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
Simulation based policy iteration for American style derivatives --- A multilevel approach
This paper presents a novel approach to reduce the complexity of simulation based policy iteration methods for pricing American options. Typically, Monte Carlo construction of an improved policy gives rise to a nested simulation algorithm for the price of the American product. In this respect our new approach uses the multilevel idea in the context of the inner simulations required, where each level corresponds to a specific number of inner simulations. A thorough analysis of the crucial convergence rates in the respective multilevel policy improvement algorithm is presented. A detailed complexity analysis shows that a significant reduction in computational effort can be achieved in comparison to standard Monte Carlo based policy iteration
Multilevel Monte Carlo methods
The author's presentation of multilevel Monte Carlo path simulation at the
MCQMC 2006 conference stimulated a lot of research into multilevel Monte Carlo
methods. This paper reviews the progress since then, emphasising the
simplicity, flexibility and generality of the multilevel Monte Carlo approach.
It also offers a few original ideas and suggests areas for future research
Deep optimal stopping
In this paper we develop a deep learning method for optimal stopping problems
which directly learns the optimal stopping rule from Monte Carlo samples. As
such, it is broadly applicable in situations where the underlying randomness
can efficiently be simulated. We test the approach on three problems: the
pricing of a Bermudan max-call option, the pricing of a callable multi barrier
reverse convertible and the problem of optimally stopping a fractional Brownian
motion. In all three cases it produces very accurate results in
high-dimensional situations with short computing times
From rough path estimates to multilevel Monte Carlo
New classes of stochastic differential equations can now be studied using
rough path theory (e.g. Lyons et al. [LCL07] or Friz--Hairer [FH14]). In this
paper we investigate, from a numerical analysis point of view, stochastic
differential equations driven by Gaussian noise in the aforementioned sense.
Our focus lies on numerical implementations, and more specifically on the
saving possible via multilevel methods. Our analysis relies on a subtle
combination of pathwise estimates, Gaussian concentration, and multilevel
ideas. Numerical examples are given which both illustrate and confirm our
findings.Comment: 34 page
Multilevel Monte Carlo methods for applications in finance
Since Giles introduced the multilevel Monte Carlo path simulation method
[18], there has been rapid development of the technique for a variety of
applications in computational finance. This paper surveys the progress so far,
highlights the key features in achieving a high rate of multilevel variance
convergence, and suggests directions for future research.Comment: arXiv admin note: text overlap with arXiv:1202.6283; and with
arXiv:1106.4730 by other author
Multilevel Richardson-Romberg extrapolation
We propose and analyze a Multilevel Richardson-Romberg (MLRR) estimator which
combines the higher order bias cancellation of the Multistep Richardson-Romberg
method introduced in [Pa07] and the variance control resulting from the
stratification introduced in the Multilevel Monte Carlo (MLMC) method (see
[Hei01, Gi08]). Thus, in standard frameworks like discretization schemes of
diffusion processes, the root mean squared error (RMSE) can
be achieved with our MLRR estimator with a global complexity of
instead of with the standard MLMC method, at least when the weak
error of the biased implemented estimator
can be expanded at any order in and . The MLRR estimator is then halfway between a regular MLMC
and a virtual unbiased Monte Carlo. When the strong error , , the gain of MLRR over MLMC becomes even
more striking. We carry out numerical simulations to compare these estimators
in two settings: vanilla and path-dependent option pricing by Monte Carlo
simulation and the less classical Nested Monte Carlo simulation.Comment: 38 page