## Abstract

The famous Black Scholes model assumes that the underlying asset follows a geometric Brownian motion with constant drift and volatility. This assumption implies that returns are lognormally dis-tributed and that the volatility is constant, although it is very easy to show that this is not the case for empirical return series. In this thesis we examine the effect of changing the Black Scholes as-sumption in such a way that we address the two implications. First, we implement a Normal Inverse Gaussian (NIG) option pricing model that addresses the implication of the marginal distribution of the logreturns. Second, we implement S. Heston’s stochastic volatility model that addresses the im-plication of constant volatility, thereby also changing the marginal distribution of the logreturns. To ensure that we use liquid options in our calibration, we examine Volume and Open Interest and find a suitable slice of strikes. We calibrate the parameters of our three models daily to liquid S&P 500 plain vanilla options under a risk neutral probability measure Q. We use several ‘goodness of fit’ criteria to ensure that our comparison of the three models does not depend on a single criterion. We see that the Heston model fits observed option prices reasonably well but that the NIG model dis-plays systematical errors, which suggest that it is not a very good model. In order to decide whether one model outperforms the other models, we use the calibrated parameters to perform three differ-ent types of delta hedges. The main result of the paper is that the Heston model outperforms the two other models, especially if we aggregate the P&L of the individual options into a portfolio. Al-though the Heston model is superior to the Black Scholes model, it should be noted that the Black Scholes model is much easier to implement and that the Black Scholes model performs quite well and outperforms the NIG model. The NIG model turns out to be the worst model on all accounts.

Educations | MSc in Mathematics , (Graduate Programme) Final Thesis |
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Language | Danish |

Publication date | 2009 |

Number of pages | 151 |