Question

1. Suppose that an asset price is $100 and that its daily volatility is 1.6%.
   1. What would be a two-standard deviation move in the asset price in two days?
   2. You think that the asset price at the end of five days will be between $100 and 110. How confident should you be if you assume returns are normally distributed with mean 0?
   3. Again assuming normality (mean 0), what’s the probability that the stock price will be 90 or less at that time?

Answer 1

Asset Price = $100 | Daily Volatility = 1.6%

a) Z-score for a Standard normal distribution = 2 or 2 Standard Deviation

Time = 2 days

Change in Price = Asset Price * Number of standard deviation * Daily Volatility * (Time)^1/2

Change in Asset Price in two days = 100 * 2 * 1.6% * (2)^0.5

Change in Asset Price in two days = 3.2 * 1.41421

Change in Asset Price in two days = 4.5255 or $ 4.53

b) Time = 5 days | Price range = 100 to 110

For Price to be 100, change in Price from current Price = 100 - 100 = 0

Assuming a Standard Normal Distribution with Mean = 0 and Standard Deviation = 1.

Change in Price for Price of 100 = 0

On a standard normal distribution, Probability of 0 which is also the mean = 1 / 2 or 0.50

Therefore, Cumulative Probability for Price of 100 = 0.50

Below is screenshot showing the result of 0.50 along-with =NORM.DIST function use:

Now For Price of 110, Change in Price = 110 - 100 = 10

Change in Price = Asset Price * Number of Standard Deviation * Daily Volatility * (Time)^1/2

=> 10 = 100 * Number of SD * 1.6% * (5)^1/2

=> 10 = 1.6 * Number of SD * 2.23607

=> Number of SD = 10 /(1.6 * 2.23607)

Number of SD or Z-score = 2.79508 or 2.80

Now that we have the Z-score, we can use it in Excel's =NORM.DIST Function to find the Cumulative Probability of Price of 110.

Input: =NORM.DIST(x=2.79508,Mean=0,Standard Deviation=1,Cumulative=TRUE) [For Standard Normal Distribution, Mean = 0 and Standard Deviation = 1]

Below is the screenshot of how formula looks like and its result:

Hence, Cumulative Probability of 110 = 0.99741

Probability of Asset Price being between 100 and 110 = Cumulative Probability of 110 - Cumulative Probability of 100

Probability of Asset Price being between 100 and 110 = 0.99741 - 0.50

Probability of Asset Price being between 100 and 110 = 0.49741 or 49.74%

The Confidence level for Asset Price to be between 100 and 110 is 49.74%.

c) Asset Price expected = 90 | Time = 5 days

Change in Price = 90 - 100 = -10

Change in Price = Asset Price * Number of Standard Deviation * Daily Volatility * (Time)^1/2

=> -10 = 100 * Number of SD * 1.6% * (5)^1/2

=> -0.10 = Number of SD * 1.6% * 2.23607

=> Number of SD = -0.10 / (1.6% * 2.23607)

Number of SD for Price of 90 = - 2.79508

Using the Excel's NORM.DIST Function, we can calculate the Probability of stock price of 90.

Input: =NORM.DIST(x=(-2.79508),Mean=0,SD=1,Cumulative=TRUE) [Assuming Mean = 0 and SD = 1]

Below is the screenshot of NORM.DIST Function on excel along-with its result:

Hence, The Probability of Stock Price being 90 or lower in 5 days = 0.002594 or 0.2594% or 0.26%