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 Garman-Kohlhagen model, which is an extension of the Black-Scholes Merton model, for valuing foreign currency options.
Highlights
Black and Scholes (1973) as we know, obtained exact formulas for valuing call and put options on non-dividend paying stocks, by assuming that stock prices follow a lognormal process
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 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%
Summary
Black and Scholes (1973) as we know, obtained exact formulas for valuing call and put options on non-dividend paying stocks, by assuming that stock prices follow a lognormal process. N(X) is the cumulative probability distribution function for a standard normal variable, and σ is the standard deviation of the rate of return on the stock
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