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JackSwan
02-08-2010, 04:16 AM
(Random Walk) A stock has the value of 100 on day 0. Each day, the value changes randomly either
one unit upwards with probability 2/3 or two units downwards with probability 1/3. The daily changes are
independent.
a) Compute the probability that, after 50 days, the value > 110.
b) Compute the probability that, after 200 days, the value > 110.
c) Compute the probability that, after 200 days, the value < 90.

I know I should apply central limit theorem to this, but how do I do it?
I've calculated:
Expected value = 0
and
Variance = 2

Can anyone please help out?

BGM
02-08-2010, 05:34 AM
Let Z ~ Bernoulli(2/3).
Then we can let 3Zi - 2 be the daily movement
Also note that ∑(3Zi - 2) will be the aggregate n days movement, ∑Zi ~ Binomial(n, 2/3)
and thus we can use Central Limit Theorem do find the approximate probabilities
as Zi are i.i.d. random variables
E[Zi] = 2/3, Var[Zi] = (2/3)(1 - 2/3) = 2/9
E[∑(3Zi - 2)] = n(3*2/3 - 2) = 0
Var[∑(3Zi - 2) = n(9*2/9) = 2n

a)
Pr{100 + ∑(3Zi - 2) > 110} = Pr{∑(3Zi - 2) > 10}
= Pr{[∑(3Zi - 2) - 0]/√(2*50) > (10 - 0)/√(2*50)}
≈ Pr{Z > 1}, where Z ~ N(0, 1)

Part b & c should follow the same idea

JackSwan
02-08-2010, 06:30 AM
Thank you!