Why is it that the lognormal model of security prices implies weak form efficiency?

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Well, weak form efficiency essentially tells us that the history of an asset price cannot be used to predict the future of the asset price. The future price is random.

Under the continuous log-normal model we have mutual independence of returns. So the return you should get between now and tomorrow on a stock (i.e. S(t+1)/S(t)) will be independent of the return that you got between when you bought the stock and today (i.e. S(t)/S(0)) This comes from the fact that the geometric Brownian Motion shares the Markovian property of the Brownian Motion.