Five numerical methods for pricing American put options under Heston's stochastic volatility model are described and compared. The option prices are obtained as the solution of a two-dimensional parabolic partial differential inequality. A finite difference discretization on nonuniform grids leading to linear complementarity problems with M-matrices is proposed. The projected SOR, a projected multigrid method, an operator splitting method, a penalty method, and a componentwise splitting method are considered. The last one is a direct method while all other methods are iterative. The resulting systems of linear equations in the operator splitting method and in the penalty method are solved using a multigrid method. The projected multigrid method and the componentwise splitting method lead to a sequence of linear complementarity problems with one-dimensional differential operators that are solved using the Brennan and Schwartz algorithm. The numerical experiments compare the accuracy and speed of the considered methods. The accuracies of all methods appear to be similar. Thus, the additional approximations made in the operator splitting method, in the penalty method, and in the componentwise splitting method do not increase the error essentially. The componentwise splitting method is the fastest one. All multigrid-based methods have similar rapid grid independent convergence rates. They are about two or three times slower that the componentwise splitting method. On the coarsest grid the speed of the projected SOR is comparable with the multigrid methods while on finer grids it is several times slower.

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We consider the numerical pricing of American options under Heston's stochastic volatility model. The price is given by a linear complementarity problem with a two-dimensional parabolic partial differential operator. We propose operator splitting methods for performing time stepping after a finite difference space discretization. The idea is to decouple the treatment of the early exercise constraint and the solution of the system of linear equations into separate fractional time steps. With this approach an efficient numerical method can be chosen for solving the system of linear equations in the first fractional step before making a simple update to satisfy the early exercise constraint. Our analysis suggests that the Crank-Nicolson method and the operator splitting method based on it have the same asymptotic order of accuracy. The numerical experiments show that the operator splitting methods have comparable discretization errors. They also demonstrate the efficiency of the operator splitting methods when a multigrid method is used for solving the systems of linear equations. Mathematics Subject Classification (2000)35K85 · 65M06 · 65M55 · 65Y20 · 91B28

A parallel fast direct solver based on the Divide & Conquer method for linear systems with separable block tridiagonal matrices is considered. Such systems appear, for example, when discretizing the Poisson equation in a rectangular domain using the ve{point nite di erence scheme or the piecewise linear nite elements on a triangulated rectangular mesh. The Divide & Conquer method has the arithmetical complexity O(N log N), and it is closely related to the cyclic reduction, but instead of using the matrix polynomial factorization the so{called partial solution technique is employed. The method is presented and analyzed in a general base q framework and based on this analysis, the base four variant is chosen for parallel implementation using the MPI standard. The generalization of the method to the case of arbitrary block dimension is described. The numerical experiments show the sequential e ciency and numerical stability of the considered method compared to the well{known BLKTRI{implementation of the generalized cyclic reduction. The good scalability properties of the parallel Divide & Conquer method are demonstrated in a distributed memory Cray T3E computer.

Abstract. Partial integro-differential equation (PIDE) formulations are often preferable for pricing options under models with stochastic volatility and jumps, especially for American-style option contracts. We consider the pricing of options under such models, namely the Bates model and the so-called stochastic volatility with contemporaneous jumps (SVCJ) model. The nonlocality of the jump terms in these models leads to matrices with full matrix blocks. Standard discretization methods are not viable directly since they would require the inversion of such a matrix. Instead, we adopt a two-step implicit-explicit (IMEX) time discretization scheme, the IMEX-CNAB scheme, where the jump term is treated explicitly using the second-order Adams-Bashforth (AB) method, while the rest is treated implicitly using the Crank-Nicolson (CN) method. The resulting linear systems can then be solved directly by employing LU decomposition. Alternatively, the systems can be iterated under a scalable algebraic multigrid (AMG) method. For pricing American options, LU decomposition is employed with an operator splitting method for the early exercise constraint or, alternatively, a projected AMG method can be used to solve the resulting linear complementarity problems. We price European and American options in numerical experiments, which demonstrate the good efficiency of the proposed methods. It is well known that fitting empirically observed option prices into the Black-Scholes model typically implies a volatility distribution with a smile-like shape. This volatility smile becomes more pronounced near the maturity date. The usual modifications of the BlackScholes model to explain such implied volatility patterns include models with jumps and/or stochastic volatility. The underlying asset price can be completely modeled by an infinite activity model such as the Carr-Geman-Madan-Yor (CGMY) model [8]. Over long time intervals the component behaving like the geometric Brownian motion becomes the dominant part in the model. For options with long maturities, the stochastic volatility models, for example the Heston model [24], are often regarded as more appropriate. For options with short maturities, however, jumps become increasingly important as a purely geometric Brownian motion driven process would require extremely high levels of volatility to explain the pronounced volatility smile pattern. Well-known jump-diffusion models in the literature include the Merton [34]

Numerical methods are developed for pricing European and American options under Kou's jump-diffusion model which assumes the price of the underlying asset to behave like a geometrical Brownian motion with a drift and jumps whose size is log-double-exponentially distributed. The price of a European option is given by a partial integro-differential equation (PIDE) while American options lead to a linear complementarity problem (LCP) with the same operator. Spatial differential operators are discretized using finite differences on nonuniform grids and time stepping is performed using the implicit Rannacher scheme. For the evaluation of the integral term easy to implement recursion formulas are derived which have optimal computational cost. When pricing European options the resulting dense linear systems are solved using a stationary iteration. For American options two ways to solve the LCPs are described: an operator slitting method and a penalty method. Numerical experiments confirm that the developed methods are very efficient as fairly accurate option prices can be computed in a few milliseconds on a PC.

Giant monopole resonances and nuclear incompressibilities studied for the zero-range and separable pairing interactions Vesely, P.; Toivanen, J.; Carlsson, Gillis; Dobaczewski, J.; Michel, N.; Pastore, A. Citation for published version (APA): Vesely, P., Toivanen, J., Carlsson, G., Dobaczewski, J., Michel, N., & Pastore, A. (2012). Giant monopole resonances and nuclear incompressibilities studied for the zero-range and separable pairing interactions. Physical Review C (Nuclear Physics), 86(2), [024303]. https://doi.org/10.1103/PhysRevC.86.024303General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.• Users may download and print one copy of any publication from the public portal for the purpose of private study or research.• You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal Take down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.PHYSICAL REVIEW C 86, 024303 (2012) Giant monopole resonances and nuclear incompressibilities studied for the zero-range and separable pairing interactions Background:Following the 2007 precise measurements of monopole strengths in tin isotopes, there has been a continuous theoretical effort to obtain a precise description of the experimental results. Up to now, there is no satisfactory explanation of why the tin nuclei appear to be significantly softer than 208 Pb. Purpose: We determine the influence of finite-range and separable pairing interactions on monopole strength functions in semimagic nuclei. Methods: We employ self-consistently the quasiparticle random phase approximation on top of spherical HartreeFock-Bogoliubov solutions. We use the Arnoldi method to solve the linear-response problem with pairing. Results: We found that the difference between centroids of giant monopole resonances measured in lead and tin (about 1 MeV) always turns out to be overestimated by about 100%. We also found that the volume incompressibility, obtained by adjusting the liquid-drop expression to microscopic results, is significantly larger than the infinite-matter incompressibility. Conclusions:The zero-range and separable pairing forces cannot induce modifications of monopole strength functions in tin to match experimental data.