# Using a meshless kernel-based method to solve the Black–Scholes variational inequality of American options

## Abstract

Under the Black–Scholes model, the value of an American option solves a free boundary problem which is equivalent to a variational inequality problem. Using positive definite kernels, we discretize the variational inequality problem in spatial direction and derive a sequence of linear complementarity problems (LCPs) in a finite-dimensional euclidean space. We use special kind of kernels to impose homogeneous boundary conditions and to obtain LCPs with positive definite coefficient matrices to guarantee the existence and uniqueness of the solution. The LCPs are then successfully solved iteratively by the projected SOR algorithm.

- Authors:

- Shahid Beheshti University, Department of Mathematics (Iran, Islamic Republic of)

- Publication Date:

- OSTI Identifier:
- 22769380

- Resource Type:
- Journal Article

- Journal Name:
- Computational and Applied Mathematics

- Additional Journal Information:
- Journal Volume: 37; Journal Issue: 1; Other Information: Copyright (c) 2018 SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0101-8205

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICAL METHODS AND COMPUTING; ALGORITHMS; BOUNDARY CONDITIONS; EUCLIDEAN SPACE; ITERATIVE METHODS; KERNELS; MATHEMATICAL SOLUTIONS; MATRICES; VARIATIONAL METHODS

### Citation Formats

```
Moradipour, M., and Yousefi, S. A., E-mail: s-yousefi@sbu.ac.ir.
```*Using a meshless kernel-based method to solve the Black–Scholes variational inequality of American options*. United States: N. p., 2018.
Web. doi:10.1007/S40314-016-0351-7.

```
Moradipour, M., & Yousefi, S. A., E-mail: s-yousefi@sbu.ac.ir.
```*Using a meshless kernel-based method to solve the Black–Scholes variational inequality of American options*. United States. doi:10.1007/S40314-016-0351-7.

```
Moradipour, M., and Yousefi, S. A., E-mail: s-yousefi@sbu.ac.ir. Thu .
"Using a meshless kernel-based method to solve the Black–Scholes variational inequality of American options". United States. doi:10.1007/S40314-016-0351-7.
```

```
@article{osti_22769380,
```

title = {Using a meshless kernel-based method to solve the Black–Scholes variational inequality of American options},

author = {Moradipour, M. and Yousefi, S. A., E-mail: s-yousefi@sbu.ac.ir},

abstractNote = {Under the Black–Scholes model, the value of an American option solves a free boundary problem which is equivalent to a variational inequality problem. Using positive definite kernels, we discretize the variational inequality problem in spatial direction and derive a sequence of linear complementarity problems (LCPs) in a finite-dimensional euclidean space. We use special kind of kernels to impose homogeneous boundary conditions and to obtain LCPs with positive definite coefficient matrices to guarantee the existence and uniqueness of the solution. The LCPs are then successfully solved iteratively by the projected SOR algorithm.},

doi = {10.1007/S40314-016-0351-7},

journal = {Computational and Applied Mathematics},

issn = {0101-8205},

number = 1,

volume = 37,

place = {United States},

year = {2018},

month = {3}

}

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