Question

A stock priced at $100 has a 70% probability of moving up and a 30% probability of moving down. If it moves up, it increases by a factor of 1.02. If it moves down, it decreases by a factor of 1/1.02. What is the probability that the stock will be $100 after two successive periods

Answer 1

The probability that after two period we will have the same value of $100 is:

`p =0.21+0.21 = 0.42`

Explanation:

We can analyze this situation by cases.

Stock moves up two times

For this case the final value would be:

$100 *1.02*1.02= $104.04

And the probability associated to this event is 0.7*0.7=0.49

Stock moves down two times

For this case the final value would be:

$100 *(1/1.02)*(1/1.02)= $96.117

And the probability associated to this event is 0.3*0.3=0.09

Stock moves up and then down

For this case the final value would be:

$100 *(1.02)*(1/1.02)= $100

And the probability associated to this event is 0.7*0.3=0.21

Stock moves down and then up

For this case the final value would be:

$100 *(1/1.02)*(1.02)= $100

And the probability associated to this event is 0.3*0.7=0.21

So then the probability that after two period we will have the same value of $100 is :

`p =0.21+0.21 = 0.42`