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# VaR

#### [email protected]

##### New Member
At present the shares of Microsoft Corp. Trade at USD 50. The monthly risk neutral probability of the price increasing by USD 5 is 40% and the risk neutral probability of the price decreasing by USD 5 is 60%. You are required to find out the mean and Standard deviation of the price of Microsoft Corp. after 2 months if the change in price is independent for consecutive months?
a. Mean = 48, SD = 6.93
b. Mean = 48, SD = 6.12
c. Mean = 36, SD = 6.93
d. Mean = 36, SD = 6.12

#### Arka Bose

##### Active Member
Answer is (a) with Mean = 48, and Variance of 6.93
I dont know why you labelled it as VaR though.

For mean, just create the binomial tree and value the branches its going to be 60*0.4+50*0.6= 54 for the upper branch and 50*0.4+40*0.6=44
Now, we have to calculate this for another branch, and its 54*0.4+44*0.6=48
This is our mean, just like calculate the value of a call/put.
As for variance, its E(X^2)-E(X)^2 formula for one period comes to 2425-2401=24, thus the standard deviation is rt(24) =4.9
For this sum, since the period is 2, thus, by iid, sd= 4.9*rt(2) = 6.93

Or find out variance directly by the np(1-p) formula. (remember to adjust the percentage squared in this case carefully)

#### Dr. Jayanthi Sankaran

##### Well-Known Member
Hi @arkabose,

How did you get the value of E(X^2) - E(X)^2 = 2425 -2401 = 24. I have to brush up the binomial distribution.

Thanks

#### Dr. Jayanthi Sankaran

##### Well-Known Member
Hi @arkabose,

Please ignore my earlier message: Got it: E(X^2) = 55^2*0.4 + 45^2*0.6 = 2425, E(X) = (55*0.4 + 45*6) = 49. Therefore E(X)^2= 49^2 = 2401 and the rest follows.

Thanks

Subscriber

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @Deepak Chitnis I get the exact same result as @arkabose I like how he retrieves the 1-month variance of 24 and uses i.i.d. to retrieve the 2-month variance of 48 = 2*24. You can also recast this tree as: 0 jumps up to +5 with p = 40% or down to -5 with p = 60% . Then monthly drift is 40%*5 + 60%*-5 = -1, such that expected value after 2 months = 50 - 1*2 = 48; and one month variance = (40%*5^2 + 60%*-5^2) - (-1)^2 = 25 - 1 = 24 such that two month variance = 48 = 24*2. But i'm not sure that's easier. Thanks!

Thank you