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# Options Valuation and Calibration for Leveraged Exchange-Traded Funds
- URL: https://fsc.stevens.edu/options-valuation-and-calibration-for-leveraged-exchange-traded-funds/
- Published: 2019-10-24T16:48:21.000Z
- Updated: 2026-05-16T22:40:42.000Z
- Author: hfsladmin
- Tags: FE 800 Projects, 2016 Spring, Research, #Import 2024-05-30 10:26, #Import 2025-11-08 22:14

### **Abstract**

The main purpose of this project is to develop parameter-consistent calibration method for [leveraged exchange-traded funds](https://en.wikipedia.org/wiki/Exchange-traded%5Ffund?ref=fsc.stevens.edu#Leveraged%5FETFs). First we model the LETF movement by [Heston stochastic volatility model](https://en.wikipedia.org/wiki/Heston%5Fmodel?ref=fsc.stevens.edu) and derive the characteristic function, then we price the option by fast fourier transform based on the characteristic function in Heston model. Finally, we calibrate heston model using both ETF and LETF option prices in weighted least square fit.

![](https://155.246.104.53/projects/sites/default/files/pictures/800_LETF_consist.png)

### **Researchers:**

****Research Group (2016 Spring)*:**  
 Yang Lu, Master in Financial Engineering, Graduated in May 2016  
 Tiancheng Chen, Master in Financial Engineering, Graduated in May 2016  
 Zihan Yi, Master in Financial Engineering, Graduated in May 2016  

****Advisor*:**  
[Dr. Zhenyu Cui](https://web.stevens.edu/hfslwiki/index.php?title=Zhenyu%5FCui&ref=fsc.stevens.edu)

### **Research Topics:**

Heston Stochastic model, Consistent Calibration, Individual Calibration

### **Main Results:**

The data sources that we conduct the calibration on are ETF from SPY, LETF from SPXL (bull, 3x) and SSO (bull, 2x).

![](https://155.246.104.53/projects/sites/default/files/pictures/800_LETF_calibra.png)

The model robustness is also performing well:

![](https://155.246.104.53/projects/sites/default/files/pictures/800_LETF_robust.png)

### **Conclusions**

While the individual calibration does not match the theoretical relationship, the consistent calibration gives out good estimation on theoretical relationship between ETF and LETF parameters. We have achieved in finding a good model for pricing the LETFs in this project. This new calibration method can be developed to estimate future option price. Our testing result proves the reasonability of this forecasting model.