/multi-asset-American-option

price American arithmetic and geometric put options with d assets

Primary LanguageMATLABGNU General Public License v3.0GPL-3.0

multi-asset-American-option

This repository includes the algorithms, figures, tables of the paper 'On Sparse Grid Interpolation for American Option Pricing with Multiple Underlying Assets' by Jiefei Yang and Guanglian Li. The code is published to ensure the reproducibility of the numerical examples.

  • The algorithm can be used to price American arithmetic and geometric put options with d assets.
  • The following two Matlab Toolboxes are needed.
    • "Parrallel Computing Toolbox",
    • "Statistics and Machine Learning Toolbox".
  • Before running examples, you will need to add necessary files to the MATLAB path. Just execute
init.m

Main algorithms

The folder ./algorithms contains implementation of Algorithm 1 of the paper with various quadrature methods including

  1. Genz-Keister sparse grid quadrature,
  2. Gauss-Hermite sparse grid quadrature,
  3. normal Leja sparse grid quadrature,
  4. QMC with scramble Sobol sequence,
  5. preintegration strategy and QMC with scramble Sobol sequence.

Quick start

To see a 2-d example with the above algorithms, run ./examples/examples_diff_quad.m. The following is expected to display in the command window:

The results of pricing values are 
1. The value using Genz-Keister sparse grid quadrature is 3.188
2. The value using Gauss-Hermite sparse grid quadrature is 3.1825
3. The value using normal Leja sparse grid quadrature is 3.1833
4. The value using QMC with scramble Sobol sequence is 3.1818
5. The value using preintegration strategy and QMC with scramble Sobol sequence is 3.1877

(The results of 4,5 might be slightly different due to the randomness.)

Reproducibility of the numerical examples

To get the results in Section 5 (Numerical experiments) of the paper, run executable scripts in ./numerical_tests.

  • For Section 5.1, get into the directory ./numerical_tests/section5_1/ first, then

    • run t1_arithm_bask_put.m to obtain values in Table 1 of the paper;
    • run t2_geo_bask_put.m to obtain values in Table 2 of the paper;
    • run t3_exact_geo_bask_put.m to obtain the reference Bermudan price in Table 2 of the paper;
    • run tab1_results_arithBask.m and tab2_results_geoBask.m to compute relative errors of Table 1 and 2, respectively.

  • For Section 5.2, get into the directory ./numerical_tests/section5_2/ first, then

    • run convergence_plot.m to obtain Figure 4 of the paper.

  • For Section 5.3, get into the directory ./numerical_tests/section5_3/ first, then

    • run t1_fig_5a_onestep_error.m to obtain Figure 5(a);
    • run t2_fig_5b_total_error.m to obtain Figure 5(b);
    • run t3_fig_6_convergence_rqmc.m to obtain Figure 6.

  • For Section 5.4, get into the directory ./numerical_tests/section5_4/ first, then

    • run t1_different_beta.m to obtain Figure 7(a);
    • run t2_different_L.m to obtain Figure 7(b);
    • run t3_different_K.m to obtain Figure 7(c).

Reproducibility of Figure 1,2,3

First, get into the directory ./figures123/, then

  • run fig1_b_and_c.m to obtain Figure 1(b)(c).
  • run fig2_transform_gridpoints.m to obtain Figure 2(a)(b).
  • run fig3_compare_num_points.m to obtain Figure 3.
  • To obtain Figure 1(a), the toolbox "Algorithm 847: Spinterp: piecewise multilinear hierarchical sparse grid interpolation in MATLAB" (https://dl.acm.org/doi/abs/10.1145/1114268.1114275) is required. After installation, run fig1_a_maxnorm_no_boundary.m.

References

The implementation of sparse grids is based on the Sparse Grids Matlab Kit, a MATLAB toolbox for high-dimensional quadrature and interpolation, see https://sites.google.com/view/sparse-grids-kit.