A simplified model for the movement of the price of a stock supposes that on eac

Question

A simplified model for the movement of the price of a stock supposes that on each day the stock's price either moves up one unit with probability p or it moves down one unit with probability 1-p. The changes on different days are assumed to be independent (a) What is the probability that after two days the stock will be at its original price? (b) What is the probability that after three days the stock's price will have increased by one unit? (c) Given that after three days the stock's price has increased by one unit, what is the probability that it went up on the first day?

Explanation / Answer

let u and d are events of stock going up and going down.

a)P(after two days stock on its original position)=P(ud)+P(du)=p*(1-p)+(1-p)*p=2*p(1-p)

b)

P(P(after 3 days stock increased by one unit)=P(udu)+P(uud)+P(duu)

=p*(1-p)*p+p*p*(1-p)+(1-p)*p*p=3p2*(1-p)

c)

P(went up first day|increased by one unit)

=P(went up on first day and increased by one unit)/P(increased by one unit)

=(P(udu+uud)/P(udu+duu+uud)=(p*(1-p)*p+p*p*(1-p))/(3p2*(1-p)) =2/3

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