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# Garch Option Pricing and Volatility Scaling for Leveraged Exchange-Traded Funds
- URL: https://fsc.stevens.edu/garch-option-pricing-and-volatility-scaling-for-leveraged-exchange-traded-funds/
- Published: 2019-10-24T15:27:21.000Z
- Updated: 2026-05-16T22:40:37.000Z
- Author: hfsladmin
- Tags: FE 800 Projects, 2016 Fall, Option Pricing, Research, #Import 2024-05-30 10:26, #Import 2025-11-08 22:14

### **Abstract**

The main purpose of this project is to develop a new pricing method for options on [Leveraged Exchange-Traded Funds](https://en.wikipedia.org/wiki/Exchange-traded%5Ffund?ref=fsc.stevens.edu#Leveraged%5FETFs). It compares the traditional **Heston-Nandi GARCH* ( Heston and Nandi, 2000) with the new Beta-Leveraged method. The traditional HN-GARCH model is calibrated using log-returns on the LETF, and the Beta-Leveraged model is calibrated using log-returns on the ETF. The parameters are estimated using the maximum likelihood estimator. The data set that we use are DDM ( X2 Leverage on DIA ETF) and DOG ( -X1 Leverage on DIA ETF). For DDM, the new model performs better than the traditional HN-GARCH method. For DOG, the two models are performing closely to each other, which is reasonable because the absolute scaling factor is 1.

### **Researchers:**

****Research Group (2016 Fall)*:**

Amit Kumar Singh, Master in Financial Engineering, Graduated in May 2017  
Mehrab Kooner, Master in Financial Engineering, Graduated in May 2017  

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

### **Research Topics:**

Heston-Nandi GARCH model, Beta-Leveraged Model, option pricing, volatility smile, LETF, ETF

### **Main Results:**

For DDM, the option prices predicted by our model are within the bid and ask spread of the market option quotes. Where as, the traditional HN-GARCH model overprices the option. Also, the HN-GARCH model generates relatively flat volatility smiles, but our model generates smiles that follows closely to the shape of the underlying ETFs option volatility smile.

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

For DOG, the two models are producing similar option prices. This is because that the absolute scaling factor is 1.

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

### **Conclusions**

This project evaluates the performance of Heston and Nandi's closed-form Garch model versus our Beta-Leveraged HN-GARCH model to evaluate which model prices options and scaled volatility discrepancies better between ETFs and LETFs. We find that our model outperforms or tie the traditional HN-GARCH for pricing options in all benchmarks. Also, our Beta-Leveraged model was able to generate a volatility smile that mi micks the shape of the volatility smile of the underlying ETF. Furthermore, we believe that our model could be used on other models that contain a daily return innovation process.