Joint modelling of S&P 500 options and VIX derivatives

Specialeforsvar ved Emilie Majland

Titel: Joint modelling of S&P 500 options and VIX derivatives

Abstract: In this thesis we study joint modeling of S&P 500 options and VIX derivatives using the well-known Heston model, the pure diffusion 3/2 model and a 3/2 model with jumps in the stock price. We use option quotes for S&P 500 and the VIX index for one specific date across maturity and strike levels to calibrate the three models, which to the extent of our knowledge has not been previously done in the literature. We find that all three models produce good fits to S&P 500 option quotes when the calibration is done separately to this market. In the joint calibration the 3/2 models fit significantly better than the Heston model. The effect of adding jumps in the stock price in the 3/2 model is limited for all calibrations, in alignment with the well-known fact that including jumps in the stock price mainly has an influence on options with short maturity. All the investigated models produce a higher level of implied volatility for VIX options when the time to maturity is short compared to long. This reflects the market. The 3/2 models with and without jumps in the stock price are also able to produce upward sloping implied volatility smiles for VIX options consistent with what is seen in the market. In contrast the Heston model fails to capture this phenomenon. We explain the differences in the slopes, by showing that the density of VIX log returns in the 3/2 models is skewed to the right, while the density in the Heston model is skewed to the left. 

Vejleder:  David G. Skovmand
Censor:   Thomas Kokholm, Aarhus Universitet