With the book’s approach, the distribution for ST is replaced with E[ST] and Var[ST], that is in terms of moments only. While the shape of the distribution for ST can be reflected in the choice of optimistic, pessimistic and most likely values, this knowledge of the shape is lost when using two moments only. A third moment of skewness is necessary to incorporate knowledge of the distribution shape.
A distribution with a tail to the right has a positive skewness, with a tail to the left a negative skewness and a symmetrical distribution a zero skewness.
And so if a particular distribution is wanted for ST, for example a lognormal distribution, this cannot be obliged. A flow on from this is that if ST can only take positive values, then this can only be guaranteed through manual intervention, or where stock prices have expected value and variance magnitudes such that the probability of the stock price being negative is very small.
How might a lognormal shape for ST be incorporated into the approach given?
Also consider: A lognormal distribution for the present worth of a stock price might be used. This would require separately from this (for a call option) the (deterministic) present worth of the exercise price to be subtracted. Is this workable? A lognormal distribution could not be fitted to ST − K collectively, because this could take negative values.