Saddlepoint approximations to tail expectations under non-Gaussian base distributions: option pricing applications

Area of Science:

• Probability and Statistics
• Quantitative Finance

Background:

• Saddlepoint approximations are used for tail expectation analysis via Laplace integrals.
• Current methods typically rely on a Gaussian base distribution.

Purpose of the Study:

• To generalize saddlepoint approximation formulas for tail expectations to arbitrary base distributions.
• To investigate criteria for selecting optimal base distributions.
• To assess the accuracy improvements in financial applications.

Main Methods:

• Generalization of saddlepoint approximation formulas.
• Analysis of cumulant generating functions.
• Numerical comparison of approximation accuracy with various base distributions.
• Application to pricing European options under the Heston model.

Main Results:

• Developed generalized saddlepoint approximation formulas applicable to non-Gaussian base distributions.
• Identified criteria for selecting base distributions that enhance approximation accuracy.
• Demonstrated improved accuracy of generalized formulas in pricing European options on continuous integrated variance.

Conclusions:

• The choice of base distribution is critical for the accuracy of saddlepoint approximations.
• Generalized saddlepoint approximations offer enhanced precision, particularly in complex financial models.