Black-Scholes-Merton Model
3 important questions on Black-Scholes-Merton Model
What about the lognormal property of stock prices?
Stock prices have a lognormal distribution, as the value is between zero and infinity. This implies that the distribution is skewed, mean, median and mode are all different. E(St) = S0 * e^mu*t. VAR(St) = S0^2 * e^2mu*t * (e^var/vol*t - 1).
What about the distribution of the rate of return?
Lognormal property provides information on the probability distribution of the continuously compounded rate of return earned on a stock between times 0 and T.
The distribution of continuously compounded rate: mean = mu - variance/2 and vol = Variance/T.
The expected return/mean is not equal to mu due to the difference between geometric and arithmetic means. Luckily, the value of the options when expressed in terms of the value of the underlying stock does not depend on mu (the expected return on the stock).
The distribution of continuously compounded rate: mean = mu - variance/2 and vol = Variance/T.
The expected return/mean is not equal to mu due to the difference between geometric and arithmetic means. Luckily, the value of the options when expressed in terms of the value of the underlying stock does not depend on mu (the expected return on the stock).
What about risk-neutral pricing?
The equation does not involve any variables that are affected by the risk preferences of investors (all independent). Good, because this means we can discount at the risk-free rate (as no premium is required).
Moving to the real world, expected growth and discount rate change but these two effects offset each other.
Moving to the real world, expected growth and discount rate change but these two effects offset each other.
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