jiefeiy/glsm-american

Gradient-enhanced sparse Hermite polynomial expansions for pricing and hedging high-dimensional American options

glsm-american

This repository includes numerical examples for the paper Gradient-enhanced sparse Hermite polynomial expansions for pricing and hedging high-dimensional American options by Jiefei Yang and Guanglian Li, 2024.

Quick start

To see how G-LSM method works in a 2-d max-call example, git clone this repo, and run ./quick_start_glsm.m.

Reproducibility of the numerical examples

Parameters used in examples are listed as follows.

Example	Parameters
1. Geometric basket put	$K = 100, T = 0.25, r = 0.03, \delta_i = 0, \sigma_i = 0.2,\rho_{ij} = 0.5, N = 50$
2. Geometric basket call	$K = 100, T = 2, r = 0, \delta_i = 0.02, \sigma_i = 0.25, \rho_{ij} = 0.75, N=50$
3. Max-call with $d$ symmetric assets	$K = 100, T = 3, r = 0.05, \delta_i = 0.1, \sigma_i = 0.2, \rho_{ij} = 0, N=9$
4. Max-call with $d$ asymmetric assets	If $d\le 5$, $\sigma_i = 0.08 + 0.32\times(i-1)/(d-1)$; if $d>5$, $0.1 + i/(2d)$
5. Put option under Heston model	$K=10, T=0.25, r=0.1, v_0 = 0.0625, \rho = 0.1, \kappa = 5, \theta = 0.16, \nu = 0.9, N=50$

• ./ex1_geobaskput/ includes tests for example 1.
• ./ex2_geobaskcall/ includes tests for example 2.
• ./ex3_maxcall_sym/ includes tests for example 3.
• ./ex4_maxcall_asym/ includes tests for example 4.
• ./ex5_heston/ includes tests for example 5.

Other methods

• Least squares Monte Carlo (LSM): ./ex1_geobaskput/lsm_geobaskput.m tests with LSM. Longstaff, F. and Schwartz, E. (2001). Valuing American options by simulation: a simple least-squares approach.
• Cosine method (COS) under Heston model: ./ex5_heston/cos_heston.m tests with COS. Fang, F. and Oosterlee, C. W. (2011). A Fourier-based valuation method for Bermudan and barrier options under Heston’s model.

Citation

@article{yang2024gradient,
  title={Gradient-enhanced sparse Hermite polynomial expansions for pricing and hedging high-dimensional American options},
  author={Yang, Jiefei and Li, Guanglian},
  journal={arXiv preprint arXiv:2405.02570},
  year={2024}
}