Efficient Gradient-Enhanced Sparse Hermite Polynomial Expansions for Pricing and Hedging High-Dimensional American Options

Core Concepts

The authors propose an efficient and easy-to-implement gradient-enhanced least squares Monte Carlo method (G-LSM) for computing prices and Greeks (derivatives) of high-dimensional American options. It employs sparse Hermite polynomial expansions as a surrogate model for the continuation value function and exploits the fast evaluation of gradients.

Abstract

The key methodological innovations of the proposed G-LSM method include:

Using sparse Hermite polynomial space with a hyperbolic cross index set as the ansatz space for approximating the continuation value functions (CVFs). This allows for efficient computation of gradients with nearly no extra cost.

Incorporating the gradient information for computing the expansion coefficients by solving a linear least squares problem, which differs from the projection-based approach in the standard least squares Monte Carlo (LSM) method.

The authors analyze the convergence of G-LSM using BSDE techniques, stochastic and Malliavin calculus, and establish an error bound in terms of time step size, statistical error of the Monte Carlo approximation, and the best approximation error in weighted Sobolev space.
Numerical experiments show that G-LSM outperforms the state-of-the-art LSM method in terms of accuracy for prices, Greeks, and optimal exercise strategies, with nearly identical computational cost. It can also deliver comparable results with recent neural network-based methods up to 100 dimensions.

Stats

The authors provide the following key figures and metrics:

The complexity of G-LSM is O(NMNb), where N is the number of time steps, M is the number of sample paths, and Nb is the number of basis functions. This is nearly identical to the complexity of the standard LSM method.
The authors analyze the one-step error propagation and provide an error bound for the proposed G-LSM method.

Quotes

"The key methodological innovations includes using sparse Hermite polynomial space with a hyperbolic cross index set as the ansatz space for approximating the continuation value functions (CVFs), and incorporating the gradient information for computing the expansion coefficients."
"Numerical experiments show that the accuracy of G-LSM is competitive with DNN-based methods for dimensions up to d = 100."

Key Insights Distilled From

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

by Jiefei Yang,... at arxiv.org 05-07-2024

https://arxiv.org/pdf/2405.02570.pdf

Deeper Inquiries

How can the proposed G-LSM method be extended to handle more complex option payoff structures or underlying asset dynamics

The proposed G-LSM method can be extended to handle more complex option payoff structures or underlying asset dynamics by adapting the sparse Hermite polynomial expansions to accommodate the increased complexity. For more complex option payoff structures, the basis functions in the Hermite polynomial expansion can be adjusted to capture the non-linearities and intricacies of the payoff function. This may involve using higher-order polynomials or introducing additional basis functions to better approximate the payoff function.
Similarly, for more complex underlying asset dynamics, the Hermite polynomial expansion can be modified to include terms that account for the specific characteristics of the assets, such as jumps, stochastic volatility, or correlation structures. By incorporating these additional features into the expansion, the G-LSM method can provide more accurate pricing and hedging for options with complex underlying dynamics.

What are the potential limitations or drawbacks of using sparse Hermite polynomial expansions compared to other approximation techniques, such as deep neural networks

While sparse Hermite polynomial expansions offer advantages such as computational efficiency and ease of implementation, there are potential limitations and drawbacks compared to other approximation techniques like deep neural networks.
One limitation is the restricted flexibility of sparse Hermite polynomials in capturing highly non-linear and complex relationships between variables. Deep neural networks, on the other hand, have the ability to learn intricate patterns and dependencies in the data, making them more suitable for modeling complex option pricing problems with non-linear payoffs and high-dimensional input spaces.
Additionally, sparse Hermite polynomial expansions may struggle with high-dimensional problems that require a large number of basis functions to achieve accurate approximation. This can lead to increased computational costs and memory requirements, especially in scenarios with a large number of underlying assets or complex payoff structures.
Furthermore, sparse Hermite polynomial expansions may not generalize well to unseen data or adapt easily to changing market conditions compared to deep neural networks, which have the capacity to learn from new data and adjust their models accordingly.

Can the gradient-enhanced approach be applied to other high-dimensional stochastic optimization problems beyond American option pricing

The gradient-enhanced approach used in the G-LSM method can be applied to other high-dimensional stochastic optimization problems beyond American option pricing. By leveraging gradient information to enhance the accuracy of approximation methods, this approach can be beneficial in various optimization problems in finance, engineering, machine learning, and other fields.
For example, the gradient-enhanced technique can be utilized in portfolio optimization, risk management, asset allocation, and derivative pricing for a wide range of financial instruments. By incorporating gradient information into the optimization process, practitioners can improve the efficiency and accuracy of their models, leading to better decision-making and risk mitigation strategies.
In engineering, the gradient-enhanced approach can be applied to optimization problems in design, control systems, signal processing, and image recognition. By utilizing gradients to guide the optimization process, engineers can achieve faster convergence, better solutions, and improved performance in complex systems.
Overall, the gradient-enhanced approach has broad applicability in high-dimensional stochastic optimization problems, offering a powerful tool for enhancing accuracy, efficiency, and robustness in a variety of domains.