Radial Basis Function Methods for Option Pricing in Exponential Lévy Models
Raymond Brummelhuis (Birkbeck, University of London, UK)
Joint work with Brad Baxter and Tat Lung (Ron) Chan

Wednesday June 4, 14:30-15:00 | session 5.1 | Computational Finance | room AB

The past decade has seen an increased interest in the use of Radial Basis Function (RBF) methodology for the numerical solution of the pricing equations of Finance. RBF interpolation seeks to interpolate an arbitrary function by a linear combination of translates of a given radially symmetric basis function or RBF, on a given set of interpolation points. This interpolation problem is uniquely solvable for a large class of basis functions which can be completely characterized. RBF interpolation also provides good approximations to the function which is being interpolated. The quality of the approximation depends on the RBF as well as on interpolation technique used (stationary or non-stationary) and may require basis functions which grow at infinity (and as such are the opposite end of the well-localized functions used in, for example, Finite Element methods). RBF interpolation can be used as the basis for a numerical scheme (a variant of the method of lines) to solve the parabolic Partial Integral-Differential Equations encountered in Finance. In the first part of this talk we report on results obtained when applying a stationary RBF scheme to pricing European and American options in exponential Lévy models of the CGMY - KoBoL class (Brummelhuis and Chan, Applied Math. Finance, to appear). We show how the scheme can deal with arbitrary singularities of the Lévy measure in 0, without introducing further approximations. In numerical experiments the scheme is found to be second order convergent in the log-price variable, including for infinite activity and unbounded variation Lévy processes. Since the scheme does not discretise the operator, it circumvents some of the complications encountered with Finite Difference (FD) methods, and leads to a better convergence rate in the unbounded variation case, thereby providing a viable alternative to FD. On the theoretical side, we rigorously determine the order of convergence of the scheme on a regular grid as the grid-size tends to 0. We show that in certain cases, the convergence is only apparent, saturating when the grid-size reaches a certain minimal level, but can nevertheless lead to a very good approximation. This is an illustration of the 'approximate approximation' phenomenon first noted by Mazj'a and Powell in the '90's.

General closed-form basket option pricing bounds
Ruggero Caldana (Università del Piemonte Orientale, Italy)
Joint work with Gianluca Fusai, Alessandro Gnoatto and Martino Grasselli

Wednesday June 4, 15:00-15:30 | session 5.1 | Computational Finance | room AB

This article presents lower and upper bounds for the basket option price, assuming very general dynamics for the n underlings. The only quantity we need to know explicitly is the joint characteristic function of the log-returns of the assets. All the bounds are general and do not require any additional assumption on the characteristic function specification. In particular, no affinity restriction on the characteristic function structure is made. Our procedure allows the computation for a very large class of stochastic dynamics like mean reverting and non-affine models. Moreover, the basket weights are not required to be positive. Our bounds involve the computation of a univariate Fourier inversion, hence they do not suffer from the curse of dimensionality. This makes our methodology particularly appealing for higher dimensional problems. We test the bounds on different models, including non Gaussian settings. Numerical examples are discussed and benchmarked against Monte Carlo simulations.

Simulation of Tempered Stable Lévy Bridges and its Applications to Option Pricing
Sojung Kim (KAIST, South-Korea)
Joint work with Kyoung-kuk Kim

Wednesday June 4, 15:30-16:00 | session 5.1 | Computational Finance | room AB

A Lévy bridge is a process conditioning a Lévy process on its endpoints, generalizing the notion of a Brownian bridge. Diffusion bridge and squared Bessel bridge are well-known. A gamma bridge (C. Ribero and N. Webber. Valuing Path Dependent Options in the Variance-Gamma Model by Monte Carlo with a Gamma bridge. Journal of Computational Finance, 7(2):81–100, 2004) and an inverse Gaussian bridge (C. Ribero and N. Webber. A Monte Carlo Method for the NIG Option Valuation Model using an Inverse Gaussian Bridge. Working paper, 2003) are examples of Lévy bridges, based on their closed forms of PDFs. However, since the PDF of a Lévy process is hardly known, only few approximation schemes have been developed, e.g., (P. Glasserman and K. Kim. Beta Approximations for Bridge Sampling. Proceedings of the 2008 Winter Simulation Conference, 2008).
In this work, we consider a tempered stable Lévy subordinator, which is an exponentially tempered version of a stable process, with a stable index less than one. The purpose of this paper is to investigate the bridge distribution of the process, which we call a tempered stable Lévy bridge, and also provide an efficient simulation method. The tempered stable Lévy bridge contains gamma and inverse Gaussian bridge as its special case, and this is in fact the same as the bridge of a stable process.
An approximate conditional PDF, that is, the PDF of the tempered stable Lévy bridge, is derived using the double saddle-point approximation. This not only provides the exact PDFs of gamma and inverse Gaussian bridges, but also shows high accuracy in approximation over a wide range of parameters. Then an acceptance-rejection algorithm is proposed using the PDFs of the known bridges as its proposal densities, depending on a stable index of the tempered stable process.
The proposed sampling method is applied to the pricing of path-dependent options under subordinated Brownian motion models. We illustrate accuracy and bias on numerical examples for Asian, look-back, and barrier options, compared to the existing sampling method. Moreover, the difference-gamma bridge sampling method in the paper (A. N. Avramidis and P. L’Ecuyer. Efficient Monte Carlo and Quasi-Monte Carlo Option Pricing Under the Variance Gamma Model. Management Science, 52(12):1930–1944, 2006) is extended to two sided tempered stable processes, e.g., finite variation CGMY processes.