Sparse wavelet methods for option pricing under stochastic volatility

Hilber, Norbert; Matache, Ana-Maria; Schwab, Christoph

doi:10.21314/jcf.2005.131

Cited by 58 publications

(41 citation statements)

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“…We mention only mathematical finance (pricing of derivative contracts on baskets of n assets under stochastic volatility models with d m − 1 ≥ 0 'hidden' volatility drivers for the mth risky asset with the case d m = 1 corresponding to deterministic volatility; see, e.g., [HMS05]), multiscale problems (elliptic homogenization problems with n separated length scales); see, e.g., [HS05], stochastic PDEs (the computation of n-point correlation functions for random solutions); see, e.g., [vPS06].…”

Section: Pde's On (High Dimensional) Product Domainsmentioning

confidence: 99%

“…The diffusion operator (2.1) is (part of) the infinitesimal generator of geometric Brownian Motion which appears in mathematical finance. Other generators A appear in connection with other Markovian processes; they may be degenerate (see, e.g., [HMS05] and, for the Riesz basis property in this case, [BSS04]), but always fit into our abstract tensor framework.…”

Section: Pde's On (High Dimensional) Product Domainsmentioning

confidence: 99%

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Adaptive wavelet algorithms for elliptic PDE's on product domains

Schwab

Stevenson

2008

Math. Comp.

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Abstract. With standard isotropic approximation by (piecewise) polynomials of fixed order in a domain D ⊂ R d , the convergence rate in terms of the number N of degrees of freedom is inversely proportional to the space dimension d. This so-called curse of dimensionality can be circumvented by applying sparse tensor product approximation, when certain high order mixed derivatives of the approximated function happen to be bounded in L 2 . It was shown by Nitsche (2006) that this regularity constraint can be dramatically reduced by considering best N -term approximation from tensor product wavelet bases. When the function is the solution of some well-posed operator equation, dimension independent approximation rates can be practically realized in linear complexity by adaptive wavelet algorithms, assuming that the infinite stiffness matrix of the operator with respect to such a basis is highly compressible. Applying piecewise smooth wavelets, we verify this compressibility for general, non-separable elliptic PDEs in tensor domains. Applications of the general theory developed include adaptive Galerkin discretizations of multiple scale homogenization problems and of anisotropic equations which are robust, i.e., independent of the scale parameters, resp. of the size of the anisotropy.

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Section: Pde's On (High Dimensional) Product Domainsmentioning

confidence: 99%

Section: Pde's On (High Dimensional) Product Domainsmentioning

confidence: 99%

Adaptive wavelet algorithms for elliptic PDE's on product domains

Schwab

Stevenson

2008

Math. Comp.

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“…One has to find suitable boundary conditions on the artificial boundaries. This is discussed qualitatively in [1], and more quantitatively by A.M. Matache and Christopher Schwab in [15]. Then one can use the finite element method described above for the problem in the truncated domain.…”

Section: Concerning the Termmentioning

confidence: 99%

“…Of course, the error comes from both the artificial boundary conditions and the discrete method. The error due to artificial boundary conditions is analyzed in [15].…”

Section: Concerning the Termmentioning

confidence: 99%

A partial differential equation connected to option pricing with stochastic volatility: Regularity results and discretization

Achdou

Franchi²,

Tchou³

2004

Math. Comp.

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Abstract. This paper completes a previous work on a Black and Scholes equation with stochastic volatility. This is a degenerate parabolic equation, which gives the price of a European option as a function of the time, of the price of the underlying asset, and of the volatility, when the volatility is a function of a mean reverting Orstein-Uhlenbeck process, possibly correlated with the underlying asset. The analysis involves weighted Sobolev spaces. We give a characterization of the domain of the operator, which permits us to use results from the theory of semigroups. We then study a related model elliptic problem and propose a finite element method with a regular mesh with respect to the intrinsic metric associated with the degenerate operator. For the error estimate, we need to prove an approximation result.

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“…Further, in the paper (Hilbert et al, 2004), the authors introduce the pricing problem in terms of parabolic partial differential equations. They show how one can construct optimal diagonal preconditioners based on wavelet norm equivalences.…”

Section: Introductionmentioning

confidence: 99%

On the convergence of the wavelet-Galerkin method for nonlinear filtering

Nowak¹,

PasłAwska-PołUdniak²,

Twardowska³

2010

International Journal of Applied Mathematics and Computer Science

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The aim of the paper is to examine the wavelet-Galerkin method for the solution of filtering equations. We use a wavelet biorthogonal basis with compact support for approximations of the solution. Then we compute the Zakai equation for our filtering problem and consider the implicit Euler scheme in time and the Galerkin scheme in space for the solution of the Zakai equation. We give theorems on convergence and its rate. The method is numerically much more efficient than the classical Galerkin method.

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Sparse wavelet methods for option pricing under stochastic volatility

Cited by 58 publications

References 0 publications

Adaptive wavelet algorithms for elliptic PDE's on product domains

Adaptive wavelet algorithms for elliptic PDE's on product domains

A partial differential equation connected to option pricing with stochastic volatility: Regularity results and discretization

On the convergence of the wavelet-Galerkin method for nonlinear filtering

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