The Black-Scholes Merton Model<br/>—Implications for the Option Delta and the Probability of Exercise

Abstract

This paper analyzes the implications of the Black-Scholes-Merton model of option pricing, for the deltas of call and put options and their respective probabilities of exercise at expiration. It derives a threshold value of the stock price and shows that in certain cases the options will have a delta in excess of 0.50, and will also have more than a 50% probability of exercise, while other options will have a delta that is lower than 0.50 and a probability of exercise that is lower than 50%. Similar results are obtained for the Gar man-Kohlhagen model, which is an extension of the Black-Scholes Merton model, for valuing foreign currency options.