Violations In The Returns On European Options Under The Black Scholes Model

This paper describes the Itô processes for the continuously compounded returns on European call and put stock options under the one-dimensional diffusion assumption and the Black Scholes pricing model. It uses the Itô processes to motivate discrete time approximations for the returns on calls and puts. Theses models are used in a simulation study to compute the probability of an option return violation as defined by Bakshi et al (2000). Two specific cases are described in some detail. The main findings of the study are that, at both daily and intraday intervals, option returns are not perfectly correlated with underlying returns. Call (put) returns may move in the opposite (same) directions as that of the underlying return and call and put returns may move together. Even at high frequencies, such as the 30-minute sampling interval, some violation occurrence rates are not low, in particular for short-term and out of the money options. It may therefore be argued that the effect of time decay in short intervals is not always negligible and that the sign of the change in the price of an option may not be correctly predicted by the sign of the price change in the underlying stock. The findings confirm that violations are likely to present difficulties when using options for either hedging or speculating and that there is a need for further development of parametric models of option returns.