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Sequential Monte Carlo Methods for Option Pricing
In the following paper we provide a review and development of sequential
Monte Carlo (SMC) methods for option pricing. SMC are a class of Monte
Carlo-based algorithms, that are designed to approximate expectations w.r.t a
sequence of related probability measures. These approaches have been used,
successfully, for a wide class of applications in engineering, statistics,
physics and operations research. SMC methods are highly suited to many option
pricing problems and sensitivity/Greek calculations due to the nature of the
sequential simulation. However, it is seldom the case that such ideas are
explicitly used in the option pricing literature. This article provides an
up-to date review of SMC methods, which are appropriate for option pricing. In
addition, it is illustrated how a number of existing approaches for option
pricing can be enhanced via SMC. Specifically, when pricing the arithmetic
Asian option w.r.t a complex stochastic volatility model, it is shown that SMC
methods provide additional strategies to improve estimation.Comment: 37 Pages, 2 Figure
Convergence of numerical methods for stochastic differential equations in mathematical finance
Many stochastic differential equations that occur in financial modelling do
not satisfy the standard assumptions made in convergence proofs of numerical
schemes that are given in textbooks, i.e., their coefficients and the
corresponding derivatives appearing in the proofs are not uniformly bounded and
hence, in particular, not globally Lipschitz. Specific examples are the Heston
and Cox-Ingersoll-Ross models with square root coefficients and the Ait-Sahalia
model with rational coefficient functions. Simple examples show that, for
example, the Euler-Maruyama scheme may not converge either in the strong or
weak sense when the standard assumptions do not hold. Nevertheless, new
convergence results have been obtained recently for many such models in
financial mathematics. These are reviewed here. Although weak convergence is of
traditional importance in financial mathematics with its emphasis on
expectations of functionals of the solutions, strong convergence plays a
crucial role in Multi Level Monte Carlo methods, so it and also pathwise
convergence will be considered along with methods which preserve the positivity
of the solutions.Comment: Review Pape
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