Variational inequalities and the pricing of American options

Jaillet, Patrick; Lamberton, Damien; Lapeyre, Bernard

doi:10.1007/bf00047211

Cited by 360 publications

(278 citation statements)

References 22 publications

(21 reference statements)

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“…For each time step, using the FFT to calculate the integral term in (3.10) costs O(N log N ) computations. On the other hand, the Brennan-Schwartz algorithm, which uses the LU decomposition to solve the sparse linear system in (3.10) (see [15] pp. 283), needs 2N computations for each time step.…”

Section: The Numerical Performance Of the Proposed Numerical Algorithmmentioning

confidence: 99%

Pricing American options for jump diffusions by iterating optimal stopping problems for diffusions

Bayraktar

Xing

2009

Math Meth Oper Res

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Abstract. We approximate the price of the American put for jump diffusions by a sequence of functions, which are computed iteratively. This sequence converges to the price function uniformly and exponentially fast. Each element of the approximating sequence solves an optimal stopping problem for geometric Brownian motion, and can be numerically computed using the classical finite difference methods. We prove the convergence of this numerical scheme and present examples to illustrate its performance.

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Section: The Numerical Performance Of the Proposed Numerical Algorithmmentioning

confidence: 99%

Pricing American options for jump diffusions by iterating optimal stopping problems for diffusions

Bayraktar

Xing

2009

Math Meth Oper Res

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“…See Harker and Pang [13] and Facchinei and Pang [8] for general references of variational inequality problems and applications. In particular, Jaillet et al [16] studied American option pricing problems in variational inequality form.…”

Section: Variational Inequality Formulationmentioning

confidence: 99%

“…Moreover, Dempster and Hutton [7] studied American option pricing problem using linear programming approach and Jaillet et al [16] presented variational inequality formulation of American option pricing problem. In this paper, we will construct an extremal problem equivalent to the variational inequality formulation and discuss the gradient projection method for the extremal problem.…”

Section: Introductionmentioning

confidence: 99%

Valuation of American options by the gradient projection method

Kwon

Friesz

2008

Applied Mathematics and Computation

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We study an equivalent optimization problem with an inequality constraint and boundary conditions, whose necessary condition for the optimality is the variational inequality presentation of American options. To solve the problem, we use the gradient projection method, with discretizations both in time and space. We tested the algorithm and compared with the projective successive over-relaxation method.

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“…In the special case of the classical American option for one asset there exist various analytic results and specialized numerical methods [13]. For d > 1, variational inequality (1.7) is usually solved by a finite element (or finite difference) discretization in space, and a backward Euler discretization in time.…”

Section: T ] : (Log S(t) T) ∈ C (14)mentioning

confidence: 99%

A posteriorierror analysis for parabolic variational inequalities

et al. 2007

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Abstract.Motivated by the pricing of American options for baskets we consider a parabolic variational inequality in a bounded polyhedral domain Ω ⊂ R d with a continuous piecewise smooth obstacle. We formulate a fully discrete method by using piecewise linear finite elements in space and the backward Euler method in time. We define an a posteriori error estimator and show that it gives an upper bound for the error in L 2 (0, T ; H 1 (Ω)). The error estimator is localized in the sense that the size of the elliptic residual is only relevant in the approximate non-contact region, and the approximability of the obstacle is only relevant in the approximate contact region. We also obtain lower bound results for the space error indicators in the non-contact region, and for the time error estimator. Numerical results for d = 1, 2 show that the error estimator decays with the same rate as the actual error when the space meshsize h and the time step τ tend to zero. Also, the error indicators capture the correct behavior of the errors in both the contact and the non-contact regions.Mathematics Subject Classification. 58E35, 65N15, 65N30.

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Variational inequalities and the pricing of American options

Cited by 360 publications

References 22 publications

Pricing American options for jump diffusions by iterating optimal stopping problems for diffusions

Pricing American options for jump diffusions by iterating optimal stopping problems for diffusions

Valuation of American options by the gradient projection method

A posteriorierror analysis for parabolic variational inequalities

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