1,808 research outputs found
Smoothing the payoff for efficient computation of Basket option prices
We consider the problem of pricing basket options in a multivariate Black
Scholes or Variance Gamma model. From a numerical point of view, pricing such
options corresponds to moderate and high dimensional numerical integration
problems with non-smooth integrands. Due to this lack of regularity, higher
order numerical integration techniques may not be directly available, requiring
the use of methods like Monte Carlo specifically designed to work for
non-regular problems. We propose to use the inherent smoothing property of the
density of the underlying in the above models to mollify the payoff function by
means of an exact conditional expectation. The resulting conditional
expectation is unbiased and yields a smooth integrand, which is amenable to the
efficient use of adaptive sparse grid cubature. Numerical examples indicate
that the high-order method may perform orders of magnitude faster compared to
Monte Carlo or Quasi Monte Carlo in dimensions up to 35
Smoothing the payoff for efficient computation of basket option prices
We consider the problem of pricing basket options in a multivariate Black Scholes or Variance Gamma model. From a numerical point of view, pricing such options corresponds to moderate and high dimensional numerical integration problems with non-smooth integrands. Due to this lack of regularity, higher order numerical integration techniques may not be directly available, requiring the use of methods like Monte Carlo specifically designed to work for non-regular problems. We propose to use the inherent smoothing property of the density of the underlying in the above models to mollify the payoff function by means of an exact conditional expectation. The resulting conditional expectation is unbiased and yields a smooth integrand, which is amenable to the efficient use of adaptive sparse grid cubature. Numerical examples indicate that the high-order method may perform orders of magnitude faster compared to Monte Carlo or Quasi Monte Carlo in dimensions up to 25
Transporting Higher-Order Quadrature Rules: Quasi-Monte Carlo Points and Sparse Grids for Mixture Distributions
Integration against, and hence sampling from, high-dimensional probability
distributions is of essential importance in many application areas and has been
an active research area for decades. One approach that has drawn increasing
attention in recent years has been the generation of samples from a target
distribution using transport maps: if
is the pushforward
of an easily-sampled probability distribution under
the transport map , then the application of to
-distributed samples yields
-distributed samples. This paper proposes the
application of transport maps not just to random samples, but also to
quasi-Monte Carlo points, higher-order nets, and sparse grids in order for the
transformed samples to inherit the original convergence rates that are often
better than , being the number of samples/quadrature nodes. Our
main result is the derivation of an explicit transport map for the case that
is a mixture of simple distributions, e.g.\ a
Gaussian mixture, in which case application of the transport map requires
the solution of an \emph{explicit} ODE with \emph{closed-form} right-hand side.
Mixture distributions are of particular applicability and interest since many
methods proceed by first approximating by a mixture
and then sampling from that mixture (often using importance reweighting).
Hence, this paper allows for the sampling step to provide a better convergence
rate than for all such methods.Comment: 24 page
Efficient hierarchical approximation of high-dimensional option pricing problems
A major challenge in computational finance is the pricing of options that depend on a large number of risk factors. Prominent examples are basket or index options where dozens or even hundreds of stocks constitute the underlying asset and determine the dimensionality of the corresponding degenerate parabolic equation. The objective of this article is to show how an efficient discretisation can be achieved by hierarchical approximation as well as asymptotic expansions of the underlying continuous problem. The relation to a number of state-of-the-art methods is highlighted
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