# Non-Uniform Finite Difference Node Spacer

### Description

`nodeSpacer`

implements the spatial discretization scheme from Hout and Foulon (2010) with arbitrary left and right bounds. The function additionally includes logic for mesh shifting.

### Usage

1 | ```
nodeSpacer(K, leftBound, rightBound, nodes, density, kShift)
``` |

### Arguments

`K` |
scalar; option strike. |

`leftBound` |
scalar; near spatial boundary of the underlying domain. |

`rightBound` |
scalar; far spatial boundary of the underlying domain. |

`nodes` |
integer; number of nodes used in the spatial discretization. |

`density` |
scalar; impacts the concentration of nodes around the option strike. At 0, nodes are uniformly distributed between the |

`kShift` |
case; if 0, no mesh shifting, if 1, adjusts the node spacing such that the strike falls exactly between two nodes, if 2, adjusts the node spacing such that the strike falls exactly on a node. See Tavella and Randall (2000). |

### Details

Mesh shifting is accomplished by multiplying the vector of gridpoints by a scalar. For multi-asset options, this `nodeSpacer`

is called iteratively to discretize each underlying's spatial domain.

### Value

`nodeSpacer`

returns a vector of gridpoints used in spatial discretization in the finite difference method. The `nodes`

input determines the length of the output vector.

### Author(s)

Michael Eichenberger and Carlo Rosa

### References

Hout, K. J., Foulon, S., 2010. ADI finite difference schemes for option pricing in the Heston model with correlation. *International Journal of Numerical Analysis and Modeling*, **7** (2), 303–320.
http://www.math.ualberta.ca/ijnam/Volume-7-2010/No-2-10/2010-02-06.pdf

Pooley, D. M., Vetzal, K. R., Forsyth, P. A., 2002. Convergence remedies for non-smooth payoffs in option pricing. http://www.cs.uwaterloo.ca/~paforsyt/report.pdf

Tavella, D., Randall, C., 2000. Pricing Financial Instruments: The Finite Difference Method. John Wiley & Sons, Inc., New York.

### Examples

1 2 3 | ```
# sample mesh spacing
plot(nodeSpacer(100, 0, 500, 26, 5, 1), rep(0, times=26), main='Non-Uniform Mesh Spacing',
xlab='Underlying Price (Strike = 100)', ylab='', yaxt='n', type='p', cex=0.5, pch=16)
``` |

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