Levy processes and quantum mechanics : an investigation into the distribution of log returns

Le Roux, Christiaan Hugo (2021-03)

Thesis (MCom)--Stellenbosch University, 2021.

Thesis

ENGLISH SUMMARY : It is well known that log returns on stocks do not follow a normal distribution as is assumed under the Black-Scholes pricing formula. This study investigates alternatives to Brownian Motion which are better suited to capture the stylized facts of asset returns. Lévy processes and models based on Quantum Mechanical theory are described and t to daily log returns for various JSE Indices. Maximum likelihood estimation is used to estimate the parameters of the Lévy processes and the Cramer-von Mises goodness of t statistic is minimized to estimate the parameters of the Quantum Mechanical models. Q-Q plots and the Kolmogorov-Smirnov t statistic is presented to assess the fit of the various models. The results show that the Lévy processes, specically the Normal Inverse Gaussian process, are the best among the processes considered. The performance of the Quantum Mechanical models could be improved if more eigenstates are considered in the approximation, however the computational expense of these models makes them impractical.

AFRIKAANSE OPSOMMING : Dit is bekend dat log opbrengste op aandele nie 'n normale verdeling volg soos in die Black-Scholes prysingsformule aanvaar word nie. Hierdie studie ondersoek alternatiewe tot Brownse Beweging wat beter geskik is om die gestileerde feite van bate opbrengste vas te lê. Lévy prosesse en modelle gebaseer op die kwantummeganiese teorie word beskryf en aangepas tot daaglikse log opbrengste vir verskillende JSE-indekse. Maksimale aanneemlikheid beraming word gebruik om die parameters van die Lévy-prosesse te beraam, en die Cramer-von Mises passingstoets grootheid word geminimeer om die parameters van die kwantummeganiese modelle te beraam. Q-Q stippings en Kolmogorov-Smirnov passingsstatistieke word gebruik om die pasgehalte van die verskillende modelle te evalueer. Die resultate toon dat die Lévy prosesse, spesiek die Normaal Inverse Gaussiese proses, die beste presteer onder die prosesse wat oorweeg word. Die kwantummeganiese modelle kan beter presteer as meer eie state in die benadering gebruik word, maar die berekeningskoste van hierdie modelle maak dit onprakties.

Please refer to this item in SUNScholar by using the following persistent URL: http://hdl.handle.net/10019.1/109864
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