En empirisk analyse af Heston og Heston-Nandi modellen som udvidelse til Black Scholes

Casper Ehlers

Student thesis: Master thesis


This thesis examines the validity of the assumptions in the original1973 Black Scholes framework and subsequently builds upon them through analysesof both the Heston and Heston-Nandi model. The original framework assumed that the returns of an underlying asset were lognormally distributed with constant volatility. As is easily displayed in this thesis’ first section, this is clearly not the case when looking at empirical returns of the S&P 500 index.The empirical returns distribution exhibits, amongst other things, fatter tails along with a higher level of kurtosis,than what should be expected from a theoretical lognormal distribution. In addition, I found that options traded on the S&P 500 exhibits the well-known volatility smile. To address the issues found when examining the implicit volatility of traded options, I introduce the 1993 Heston model. By analyzing and examining its added parameters, I find that it is useful in correcting the observed volatility smile. Furthermore, to add to the findings of the empirical returns found in the first section, I also apply the Heston-Nandi model and combine it with a thorough analyses of the GARCH component. The GARCH component proves useful in forecasting volatility upwards of 30 trading days into the future. At last I perform an empirical comparison of the 3 models ability to price options traded in the market. In order to perform this comparison, I initially calibrate the parameters in the Heston and Heston-Nandi model through application of the Nelder-Mead algorithm. Due to the GARCH models ability to forecast volatility realistically upwards of 30 days, the Heston-Nandi model outperforms the two other models, when comparing option pricing performance in the short term. Moving to the medium and long-term horizon, the Heston model dominates and becomes the prevailing option pricing model. In order to solidify my conclusions,I round with analysisof the 3 models’ ability to delta hedge options over a short, medium and long-term period into the future. Again, both the Heston and Heston-Nandi model seem to outperform the classic Black Scholes model. My conclusion is that both models serve as an improvement to the original Black Scholes model.

Educations, (Graduate Programme) Final Thesis
Publication date2018
Number of pages102