Hedging performances of the Black-Scholes model in imperfect log-normal world

The study developed in the current master thesis concerns the Black-Scholes-Merton (BSM) method initially intended to give a fair price to options for which a geometric Brownian motion (GBM) drives the underlying asset. Indeed this model works remarkably well provided that the underpinned assumptions are respected with the most restrictive one stating that the distribution of the log-returns of the underlying asset has to be normal. However, empirical results show that this constraint of log-normality does not hold for real-life trials and consequently break the more key hypothesis of the aforementioned model. The primary purpose of the analysis developed in that work is to assess how wrong goes the BSM equation while the underlying asset is not a GBM. To that end, others models will be investigated, namely, the Merton jump diffusion (MJD) and Heston stochastic volatility (HSV) models. Each of it bears a new characteristic. The MJD brings jumps to the paths of the underlying processes whilst HSV comes with a volatility parameter that evolves with times unlike the deterministic one inherent to BSM. These specificities affect the log-returns distributions that are no more normal. Accordingly, once calibrated, those models will be used to simulate a world where the assumption of log-normality of the BSM model does not hold anymore. The study case explored is to measure how the BSM equation will react when its central assumption is broken. To do this, the performance of the delta-neutral portfolios of BSM, MJD and HSV processes will be benchmarked. As a subanalysis, another goal of that master thesis is to examine if, with a unique set of parameters, the HSV and MJD models can reproduce the volatility smiles constructed from the options prices provided by the market. A conclusion would be that those models are versatile enough to cover a broader range of market prices than the BSM model itself.