Implied volatility modelling and non-linear machine learning estimation of payoff replication performances in the Black-Scholes framework.
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- This thesis provides a thorough work on the modelling of the implied volatility of the Black- Scholes model and the estimation of the hedging error through non-linear machine learning techniques. The implied volatility will be computed and some adjustments to moneyness will be made as depicted in the work of MacBeth and Merville followed by estimating a simultaneous implied volatility and implied risk-free rate as in the work of Bianconi and al. Afterwards, based on the constant volatility hypothesis of the Black-Scholes model, the replicating portfolio will be built and the hedging error at the option’s maturity will be computed. The thesis wishes to know if there is a difference in using the MacBeth and Merville adjustment or the Bianconi and al. adjustment. Furthermore, a non-linear machine learning algorithm will be trained to predict hedging errors. In this part, the thesis wishes to know if it can precisely predict hedging errors as well as the hedging type. The results that emerge from this thesis are that there is a difference in the hedging performance using one implied volatility adjustment versus another one where the sole moneyness adjustment as MacBeth and Merville tends to have a higher hedging error compared to the adjustments of Bianconi and al. The results regarding the prediction of the machine learning algorithm are that it can predict whether the replicating portfolio under or over hedges with an accuracy of 84.28% and predicting the exact value of the hedging error leads to an MSE of 26.83 and a RMSE of 5.18. The implications coming forward in this thesis are that there is no perfect method to estimate the implied volatility as it depends on the goal of the investor and that hedging error prediction is possible to a certain extent.