Space–time adaptive finite difference method for European multi-asset options

Lötstedt, Per; Persson, Jonas; Sydow, Lina von; Tysk, Johan

doi:10.1016/j.camwa.2006.09.014

Cited by 60 publications

(45 citation statements)

References 23 publications

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“…Nevertheless, in our numerical experiments the accuracies of all studied methods were similar. One way to improve the accuracy of discretizations is to make it adaptive like in [52].…”

Section: Discussionmentioning

confidence: 99%

Efficient numerical methods for pricing American options under stochastic volatility

Ikonen

Toivanen

2007

Numerical Methods Partial

143

132

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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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“…Nevertheless, in our numerical experiments the accuracies of all studied methods were similar. One way to improve the accuracy of discretizations is to make it adaptive like in [52].…”

Section: Discussionmentioning

confidence: 99%

Efficient numerical methods for pricing American options under stochastic volatility

Ikonen

Toivanen

2007

Numerical Methods Partial

143

132

View full text Add to dashboard Cite

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“…Hence, we first solve forũ n+1 using (17) and (19) and then update for u n+1 using (18) and (20), respectively.…”

Section: Operator Splitting Methodsmentioning

confidence: 99%

“…As the grid used for error estimation can be fairly coarse the computional cost of the error estimation is small while the obtained fine grid is nearly optimal. An alternative, more elaborate approach would be to use a goal oriented adjoint equation based method described in [19]. For more discussion on adaptive discretizations see the book [1].…”

Section: Introductionmentioning

confidence: 99%

Adaptive finite differences and IMEX time-stepping to price options under Bates model

Sydow

Toivanen

Zhang

2015

International Journal of Computer Mathematics

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In this paper we consider numerical pricing of European and American options under the Bates model, a model which gives rise to a partial-integro differential equation. This equation is discretized in space using adaptive finite differences while an IMEX scheme is employed in time. The sparse linear systems of equations in each time-step are solved using an LU-decomposition and an operator splitting technique is employed for the linear complementarity problems arising for American options. The integral part of the equation is treated explicitly in time which means that we have to perform matrixvector multiplications each time-step with a matrix with dense blocks. These multiplications are accomplished through fast Fourier transforms. The great performance of the method is demonstrated through numerical experiments.

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“…The results showed that accurate discretizations are essential for pricing American options under Heston's model accurately and fast. One possibility to further improve the accuracy of discretizations is to use adaptive grids; see [1], [27], for example.…”

Section: Newtonmentioning

confidence: 99%

Lagrange Multiplier Approach with Optimized Finite Difference Stencils for Pricing American Options under Stochastic Volatility

Toivanen¹

2009

SIAM J. Sci. Comput.

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Abstract. The deterministic numerical valuation of American options under Heston's stochastic volatility model is considered. The prices are given by a linear complementarity problem with a twodimensional parabolic partial differential operator. A new truncation of the domain is described for small asset values while for large asset values and variance a standard truncation is used. The finite difference discretization is constructed by numerically solving quadratic optimization problem aiming to minimize the truncation error at each grid point. A Lagrange approach is used to treat the linear complementarity problems. Numerical examples demonstrate the accuracy and effectiveness of the proposed approach.Key words. American option pricing, stochastic volatility model, linear complementarity problem, finite difference method, quadratic programming, multigrid method, Lagrange method, penalty method AMS subject classifications. 35K85, 65M06, 65M55, 65Y20, 91B281. Introduction. The seminal papers [2] and [28] by Black & Scholes, and Merton, respectively, laid the foundations of the modern theory of pricing financial options. Since then vast body of scientific work has been devoted to the development of methods for pricing options. Particularly American options are challenging to evaluate due to their early exercise possibility and various approaches to approximate their price have been proposed. The paper [4] by Brennan and Schwartz was one of first ones to formulate a linear complementarity problem (LCP) and then to solve it using a finite difference discretization. Empirical evidence has shown that the assumption on log-normal behavior of the value of the underlying asset is not realistic for many asset categories including shares of companies. To alleviate this, many generalizations have been introduced for the log-normal model. We consider stochastic volatility models which assume that the volatility of the value of the asset follows a stochastic process; see [11] and references therein. Particularly, our model problem for American options is based on Heston's model [14], but the techniques considered in this paper can be generalized also for other stochastic volatility models like the Hull-White model [17] and the Stein-Stein model [33].Based on Heston's stochastic volatility model, a LCP with a parabolic partial differential operator can derived for the price of American options with the value of underlying asset and its variance being the spatial variables. Near the axes, the first-order derivatives dominate the second-order derivatives in the operator. It also includes the second-order cross derivative. Due to these properties, it is not easy to construct an accurate and stable discretization. In financial literature, most often finite differences are used for the discretization while sometimes also finite elements are used; see [1], [34], [38], for example. It is desirable to use discretizations leading to matrices with the M-matrix property [39]; see also [41]. This property guarantees the stability of the ...

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Space–time adaptive finite difference method for European multi-asset options

Cited by 60 publications

References 23 publications

Efficient numerical methods for pricing American options under stochastic volatility

Efficient numerical methods for pricing American options under stochastic volatility

Adaptive finite differences and IMEX time-stepping to price options under Bates model

Lagrange Multiplier Approach with Optimized Finite Difference Stencils for Pricing American Options under Stochastic Volatility

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