Option pricing models: A comparison between models with constant and stochastic volatilities as well as discontinuity jumps

The purpose of this thesis is to compare option pricing models. We have investigated the constant volatility models Black-Scholes-Merton (BSM) and Merton’s Jump Diffusion (MJD) as well as the stochastic volatility models Heston and Bates. The data used were option prices from Microsoft, Advanced Micro Devices Inc, Walt Disney Company, and the S&P 500 index. The data was then divided into training and testing sets, where the training data was used for parameter calibration for each model, and the testing data was used for testing the model prices against prices observed on the market. Calibration of the parameters for each model were carried out using the nonlinear least-squares method. By using the calibrated parameters the price was calculated using the method of Carr and Madan. Generally it was found that the stochastic volatility models, Heston and Bates, replicated the market option prices better than both the constant volatility models, MJD and BSM for most data sets. The mean average relative percentage error for Heston and Bates was found to be 2.26% and 2.17%, respectively. Merton and BSM had a mean average relative percentage error of 6.90% and 5.45%, respectively. We therefore suggest that a stochastic volatility model is to be preferred over a constant volatility model for pricing options.