[Paper Review] A fast method for pricing American options under the variance gamma model

Weilong Fu, Ali Hirsa|arXiv (Cornell University)|Mar 18, 2019

Stochastic processes and financial applicationsEconomics, Econometrics and Finance22 references3 citations

TL;DR

This paper proposes a fast, accurate method for pricing American options under the variance gamma (VG) model by combining quadratic approximation with kernel regression to correct approximation errors. The approach pre-calculates correction terms using machine learning, achieving speed comparable to analytical approximations while maintaining high accuracy, outperforming finite difference and Monte Carlo methods in both speed and error metrics.

ABSTRACT

We investigate methods for pricing American options under the variance gamma model. The variance gamma process is a pure jump process which is constructed by replacing the calendar time by the gamma time in a Brownian motion with drift, which makes it a time-changed Brownian motion. In general, the finite difference method and the simulation method can be used for pricing under this model, but their speed is not satisfactory. So there is a need for fast but accurate approximation methods. In the case of Black-Merton-Scholes model, there are fast approximation methods, but they cannot be utilized for the variance gamma model. We develop a new fast method inspired by the quadratic approximation method, while reducing the error by making use of a machine learning technique on pre-calculated quantities. We compare the performance of our proposed method with those of the existing methods and show that this method is efficient and accurate for practical use.

Motivation & Objective

• Address the lack of fast, accurate approximation methods for American options under pure-jump Lévy processes like the variance gamma (VG) model.
• Overcome the limitations of existing methods—such as finite difference methods (slow) and Monte Carlo simulations (slow and noisy)—for practical option pricing.
• Develop a hybrid approach that leverages the speed of quadratic approximation while reducing its error through data-driven correction using kernel regression.
• Create a practical, scalable method that can be extended to other pure-jump models like CGMY and NIG.
• Achieve a balance between computational speed and pricing accuracy suitable for real-time risk management and market making.

Proposed method

• Adapt the Ju-Zhong quadratic approximation framework to the VG model, using a simplified PIDE-based approximation to avoid time-stepping like finite difference methods.
• Introduce a correction term to the approximated option price to reduce error from the initial quadratic simplification.
• Use nonparametric kernel regression to estimate the correction term as a function of model parameters (r, q, T, σ, ν, θ), trained on pre-calculated option prices.
• Train the kernel regression on a Cartesian product of parameter grids (6D input, N+1 response variables), with N=6 for the stock price grid.
• Optimize kernel bandwidth via repeated cross-validation (5 runs) and average results for robustness.
• Implement the final pricing routine as a lookup and regression-based function, enabling fast evaluation once the correction model is pre-trained.

Experimental results

Research questions

• RQ1 Can a fast approximation method for American options under the VG model be developed that is significantly faster than finite difference or Monte Carlo while maintaining high accuracy?
• RQ2 How effective is the combination of quadratic approximation with machine learning-based correction in reducing pricing error for pure-jump models?
• RQ3 To what extent does the performance of the proposed method degrade as time to maturity increases, particularly beyond T=0.5?
• RQ4 Can the method be generalized to other pure-jump Lévy models such as CGMY and NIG with minimal modification?
• RQ5 Is it feasible to pre-compute correction terms using a limited number of training points such that the resulting model remains accurate and efficient for real-time use?

Key findings

• The proposed method achieves a favorable balance between speed and accuracy, outperforming both finite difference (FDcoarse) and Monte Carlo methods in both error and computational time.
• The method is significantly faster than FDfine (the high-accuracy reference) and MC simulation, with speed comparable to analytical approximations.
• Even when the finite difference method is accelerated (FDcoarse), it still exhibits slightly higher error than the proposed method for T ≤ 0.5.
• For T = 1.0, the method’s performance degrades slightly due to the difficulty of approximating the true American option premium surface with an exponential form over longer maturities.
• The kernel regression-based correction reduces the error of the initial quadratic approximation to within or below the bid-ask spread in most cases, making it practically useful.
• The method is robust and scalable, with the pre-computation phase being the only expensive step, enabling fast online pricing.