Performance of advanced stock price models when it becomes exotic: an empirical study

We calibrate several advanced stock price models to a time series of real market data of European options on the DAX. Via a Monte Carlo simulation, we price barrier down-and-out call options for all models and compare the modeled prices to given real market data of the barrier options. The Bates model reproduces barrier option prices very well. The BNS model overvalues and Lévy models with stochastic time-change and leverage undervalue the exotic options. The Heston model and a local volatility model undervalue the barrier option prices by about 5–6%. A heuristic analysis suggests that the different degree of fluctuation of the random paths of the models are responsible of producing different prices for the barrier options. Higher margins or additional risks like liquidity, calibration or model risk might economically explain why many advanced models undervalue barrier options.