12,214 research outputs found
Fast Estimation of True Bounds on Bermudan Option Prices under Jump-diffusion Processes
Fast pricing of American-style options has been a difficult problem since it
was first introduced to financial markets in 1970s, especially when the
underlying stocks' prices follow some jump-diffusion processes. In this paper,
we propose a new algorithm to generate tight upper bounds on the Bermudan
option price without nested simulation, under the jump-diffusion setting. By
exploiting the martingale representation theorem for jump processes on the dual
martingale, we are able to explore the unique structure of the optimal dual
martingale and construct an approximation that preserves the martingale
property. The resulting upper bound estimator avoids the nested Monte Carlo
simulation suffered by the original primal-dual algorithm, therefore
significantly improves the computational efficiency. Theoretical analysis is
provided to guarantee the quality of the martingale approximation. Numerical
experiments are conducted to verify the efficiency of our proposed algorithm
Optimal dual martingales, their analysis and application to new algorithms for Bermudan products
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 It\^o-L\'evy
environment we outline a particular regression based backward algorithm which
allows for computing dual upper bounds without 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 the backward dual regression algorithm in a Wiener environment
at well known benchmark examples. It turns out that the method is at least
comparable to the one in Belomestny et. al. (2009) regarding accuracy, but
regarding computational robustness there are even several advantages.Comment: This paper is an extended version of Schoenmakers and Huang, "Optimal
dual martingales and their stability; fast evaluation of Bermudan products
via dual backward regression", WIAS Preprint 157
Optimal Transport and Skorokhod Embedding
The Skorokhod embedding problem is to represent a given probability as the
distribution of Brownian motion at a chosen stopping time. Over the last 50
years this has become one of the important classical problems in probability
theory and a number of authors have constructed solutions with particular
optimality properties. These constructions employ a variety of techniques
ranging from excursion theory to potential and PDE theory and have been used in
many different branches of pure and applied probability.
We develop a new approach to Skorokhod embedding based on ideas and concepts
from optimal mass transport. In analogy to the celebrated article of Gangbo and
McCann on the geometry of optimal transport, we establish a geometric
characterization of Skorokhod embeddings with desired optimality properties.
This leads to a systematic method to construct optimal embeddings. It allows
us, for the first time, to derive all known optimal Skorokhod embeddings as
special cases of one unified construction and leads to a variety of new
embeddings. While previous constructions typically used particular properties
of Brownian motion, our approach applies to all sufficiently regular Markov
processes.Comment: Substantial revision to improve the readability of the pape
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