This paper price leveraged exchange traded funds(LETF) options using the Heston-Nandi GARCH(1,1) model. We derive the price dynamics of LETFs from that of ETFs in order to have them structurally match.The options valuation is carried out through Fast Fourier Transform(FFT) via Carr-Madan Inversion. Discretization, the Simpson Rule, Cublic Spline Interpolation are all utilized as part of our methodology. We compare our results to market prices. Conclusions about accuracy of model will be drawn according to fit with market prices

Research Topics:

leveraged exchange traded funds(LETF), Heston-Nandi GARCH, options valuation, Fast Fourier Transform(FFT)


Research Group (2016 Fall)

Caroline DeLuca, Master in Financial Engineering, Graduated in May 2017
Kristen Goncalves, Master in Financial Engineering, Graduated in May 2017
Alexandre Roumi, Master in Financial Engineering, Graduated in May 2017


Dr. Zhenyu Cui

Main Results:

Upon calculating the HN-Parameters using the daily returns data and the maximum likelihood we found that the initial estimate used had a large effect on the resulting calibrated parameters. The accuracy of the options prices were in turn affected by the fitting of the model. We obtained our initial estimates based off of the parameters used in the Christoffersen article. When using these parameters the options prices obtained were very inaccurate. We then discovered that changing the initial parameters even by a small amount, as small as a hundred thousandth for certain parameters the resulting parameters were very different. We also determined that a, b, and gamma played a bigger role in the maximum likelihood estimation that λ, and ω did.

After the initial parameters were determined we moved on the determining how the time period used to calibrate the parameters affected the accuracy of the European Call Options prices. Three different time periods were used for calibration and two different times to expiration were used to compare options prices. The three time periods the calibration was carried out from were July 1, 2007-July 1, 2012, July 01, 2006-July 01, 2015, and March 23, 2015-March 23, 2017. The initial time period that was tested was from March 23, 2015-March 23, 2017. This was selected because it was two years before the date the the options data was taken from. The 2007-2012 date range was selected to represent the time during the crash of 2008. Lastly, the 20062015 range was selected because 2006 was when the data started and it was a ”long” 9 year period. The 2006-2015 range included the stock market crash and overlapped with the 2015-2017 period for a couple of months. The three month and six month time to expiration were used because they were the times with the most strike prices to compare to.


This project contribute to the continuing research on LETFs. This project provide pricing dynamics which structurally match ETFs and LETFs which allows one to carry out valuation using the existing characteristic function. This project successfully apply Fast Fourier Inversion to option valuation for the Heston-Nandi GARCH(1,1) model. When carried out, we successfully price options which generally fall within or close to the bid-ask spread. This project were able to keep our error small; however, the three month options had a higher error due to a few strikes having a bid-ask spread of zero. Given this we were still able to successfully price many options in the money. Furthermore, for six month options we were able to successfully price multiple options both in the money and out of the money. Our estimation and further methodology helped us reach these fair and successful results. Moving forward we would recommend preforming more out of sample analysis to better test the predictive performance of our model and possibly carry out calibration using implied volatility.