On Sparse Grid Interpolation for American Option Pricing with Multiple Underlying Assets

ArXiv ID: 2309.08287 “View on arXiv”

Authors: Unknown

Abstract

In this work, we develop a novel efficient quadrature and sparse grid based polynomial interpolation method to price American options with multiple underlying assets. The approach is based on first formulating the pricing of American options using dynamic programming, and then employing static sparse grids to interpolate the continuation value function at each time step. To achieve high efficiency, we first transform the domain from $\mathbb{“R”}^d$ to $(-1,1)^d$ via a scaled tanh map, and then remove the boundary singularity of the resulting multivariate function over $(-1,1)^d$ by a bubble function and simultaneously, to significantly reduce the number of interpolation points. We rigorously establish that with a proper choice of the bubble function, the resulting function has bounded mixed derivatives up to a certain order, which provides theoretical underpinnings for the use of sparse grids. Numerical experiments for American arithmetic and geometric basket put options with the number of underlying assets up to 16 are presented to validate the effectiveness of the approach.

Keywords: American options, sparse grids, quadrature, multivariate interpolation, dynamic programming, Derivatives (Options)

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 7.5/10
• Quadrant: Holy Grail
• Why: The paper employs advanced mathematics including sparse grids, domain transformation, bubble functions, and rigorous theoretical bounds on mixed derivatives for high-dimensional interpolation. It presents extensive numerical experiments up to 16 assets with validation of accuracy and robustness, though it lacks real-market data or specific backtesting pipelines.

  flowchart TD
    A["Research Goal<br>Pricing American options<br>with multiple assets"] --> B["Key Methodology<br>Sparse Grid Interpolation<br>in d dimensions"]
    
    B --> C["Domain Transformation<br>ℝᵈ → (-1, 1)ᵈ via<br>scaled tanh map"]
    C --> D["Singularity Removal<br>Bubble function for<br>bounded mixed derivatives"]
    
    E["Data/Inputs<br>Arithmetic & Geometric<br>Basket Put Options"] --> F["Computational Process<br>Dynamic Programming<br>→ Sparse Grid Quadrature"]
    
    F --> G["Key Findings<br>Validated up to d=16 assets<br>High efficiency & accuracy"]
    
    D --> F