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# A Sine Expansion on the Explicit Formula for Option-Implied Quantiles
- URL: https://fsc.stevens.edu/a-sine-expansion-on-the-explicit-formula-for-option-implied-quantiles/
- Published: 2025-03-23T03:32:48.000Z
- Updated: 2026-05-16T22:35:29.000Z
- Author: Yash Deshpande
- Tags: FE 800 Projects, 2024 Fall, #Import 2025-11-08 22:14

## **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.