# nodeSpacer: Non-Uniform Finite Difference Node Spacer In multiAssetOptions: Finite Difference Method for Multi-Asset Option Valuation

## 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 `leftBound` and `rightBound`. Increasing the parameter increases the node concentration around the strike. `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) ```

multiAssetOptions documentation built on May 29, 2017, 9:05 a.m.