Question

A stock sells today for $130. The price of the stock in a year is expected to be $140. The annual volatility of the stock is 30%.
a. Calculate the probability that in six years the stock will sell for more than $150.
b. Calculate the probability that in six years the stock will sell for less than $115.
c. Calculate the probability that in six years the stock will sell for a price between $120 and $160.
d. You are 85% confident the stock price in six years will be between what two values?
e. There is an 80% probability that in 6 years the stock price will exceed ___________

Answer 1

a) We assume Black-Scholes accurately models stock price movements. We also assume we know the future volatility "sigma" of the stock's price action. Assume the stock price today is "P". Assume the price-to-be-touched is "S" (the "strike price").

The probability "X" that the stock will touch or exceed the strike price S, within T days, can be found thus:

Z = ln(S/P) / (sigma * sqrt(T/365))
X = CNDF(Z)

ln() = natural logarithm = log to the base e
Z = Zscore = size of price move from P to S, in standard deviations
CNDF() = Cumulative Normal Distribution Function

In this case S is $150, Current Price is $130; T = 6 x 365 and Sigma is the Volatility of the stock price

Therefore Z = ln (150/130)/(0.3*Sqrt(6).

We determine the value of z from the above as 0.194736

Now we have to use the cumulative Normal distribution function to determine the Cumulative probability that the Stock prive shall exceed the strike price of $150.

We can do this very easily in excel using the Normsdist function. In a particular cell we key in the formula

= Normdist(z) and we obtain the cumulative probability that the stock price shall touch or exceed the strike price which in this case if $150.

Using this formula in excel we find the value is 0.5772.

Therefore the answer is that there is 57.72 % probability that the stock price shall sell for a price more than $150.

b) We again us the same formula as earlier and this time we take the strike price as $115.

Therefore Z =ln(115/130)/Sigma* Sqrt (6)

Calculating thus the value of Z = -0.16684

Using the Normsdist function the value we obtain is 0.4337 or 43.37%

This is probability that the stock price shall be equal to or exceed $ 115.

Therefore the probability that the stockprice shall be below $ 115 is 1 - 0.4337 i.e 0.5663. or 56.63%

c) We repeat the process in this case use the strike prices as $ 120 and $ 160

The Z values are calculated as 0.282561 and -0.10892

The corresponding probabilities are calculated as 61.18% and 45.66%

Subtracting the probabilities we find that there is 15.52 % probability that the stock price remains between $ 120 and $ 160

Hence the answer is 15.52%