An Analytic COS Method for Compound Option Valuation

2026-07-28

AI Summary

This study addresses the computational inefficiency of multi-stage numerical integration in pricing compound options by proposing an analytical Fourier cosine (COS) method. By deriving closed-form expressions for the cosine coefficients at each compounding stage, the approach eliminates the need for numerical integration at intermediate exercise dates and is applicable to a broad class of stochastic processes with known characteristic functions, including jump-diffusion models. This work presents the first analytical computation of cosine coefficients for multi-stage compound options, accommodating generalized payoff functions and nested decision structures. Numerical experiments demonstrate that the method achieves high accuracy while significantly improving computational efficiency, and its flexibility and robustness are further validated through successful application to multi-stage real option valuation.

Abstract

We develop an analytic Fourier cosine (COS) method for the valuation of compound options. By deriving closed-form expressions for the cosine coefficients at all compound stages, the proposed method eliminates the need for numerical quadrature in intermediate exercise stages while retaining the convergence properties of the underlying COS approximation. The formulation extends to multi-stage compound structures and a broader class of payoffs, and remains applicable to a wide class of stochastic models characterized by known characteristic functions, including jump-diffusion dynamics. Numerical experiments demonstrate improved computational efficiency compared with quadrature-based implementations while maintaining high accuracy. Applications to staged real-option problems further illustrate the flexibility of the method in handling nested decision structures under different uncertainty dynamics.

Problem

• compound option
• valuation
• multi-stage
• stochastic models
• jump-diffusion

Innovation

• analytic COS method
• compound option valuation
• closed-form cosine coefficients
• jump-diffusion models
• multi-stage real options