Date of Submission

Spring 2019

Academic Programs and Concentrations

Mathematics

Project Advisor 1

Stefan Méndez-Diez

Abstract/Artist's Statement

The Nobel Prize-winning the Black-Scholes Model for stock option pricing has a simple formula to calculate the option price, but its simplicity comes with crude assumptions. The two major assumptions of the model are that the volatility is constant and that the stock return is normally distributed. Since 1973, and especially in the 1987 Financial Crisis, these assumptions have been proven to limit the accuracy and applicability of the model, although it is still widely used. This is because, in reality, observing a stock return distribution graph would show that there is an asymmetry or a leptokurtic shown in the stock return.

Therefore, we propose that by introducing the Heston Model, we can tackle these two problematic assumptions in the Black-Scholes Model. The Heston Model considers the leverage effect and the clustering effect, which allows the volatility itself to be random and also allows it to take the non-normally distributed stock return into account.

In our project, we aim to show whether the Heston model can actually improve the option pricing estimates by using the $S\&P$ 500 Index European Call Option to compare it to the Black-Scholes Model. We find that even though the results show that the Heston Model performs worse than the Black-Scholes Model when the option expiration date is soon to expire, the Heston Model significantly outperforms the Black-Scholes Model in almost all combinations of moneyness and maturity scenarios. There remains further work to improve the Heston Model.

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Open Access

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