A Sine Expansion on the Explicit Formula for Option-Implied Quantiles

Yash Deshpande -

Researcher

Víctor Domínguez Cámara

Faculty Advisor

Prof. Zhenyu Cui

Overview

This paper examines and expands upon the work of Cui, Liu, and Yao (2024) on the accuracy of the Cosine-Series expansion in approximating option-implied quantiles.

Key Findings

1. Cosine vs. Sine Expansion

  • The sine expansion performs similarly to the cosine expansion but is generally less accurate.
  • Under both the BSM and Heston models, the cosine series demonstrates superior accuracy.

2. Methodology

  • Fourier series expansions are used to approximate the quantile function.
  • Both analytical and model-free methods are explored to improve accuracy.
  • A stress test assesses performance under extreme parameter conditions.

3. Performance Under the BSM Model

  • Both cosine and sine expansions closely approximate theoretical quantiles.
  • The cosine expansion is more precise, reducing error by 5 orders of magnitude in some cases.
  • Performance deteriorates under extreme parameters such as high volatility and long maturity.

4. Performance Under the Heston Model

  • Since the Heston model lacks a closed-form solution, only the model-free method is applied.
  • Cosine and sine expansions yield similar results, but the cosine method is more stable in extreme conditions.

5. Stress Testing

  • Tests on time to maturity, risk-free rate, dividend yield, and volatility show robustness in real-world conditions.
  • Accuracy degrades with extreme parameter values, especially high volatility and long maturities.
  • The cosine expansion consistently outperforms the sine expansion.

Conclusion and Future Work

  • The cosine expansion is validated for option-implied quantiles, while the sine expansion requires improvements.
  • Future work involves applying these methods to real-world option market data.

Significance

This research enhances financial engineering by introducing alternative Fourier-based approximations for option pricing, crucial for risk management and derivatives pricing.