DATA 605 - Discussion 9, Ex. 11, p. 363

The price of one share of stock in the Pilsdorff Beer Company (see Exercise 8.2.12) is given by Yn on the nth day of the year. Finn observes that the differences Xn = Yn+1 − Yn appear to be independent random variables with a common distribution having mean µ = 0 and variance σ2 = 1/4. If Y1 = 100, estimate the probability that Y365 is:

The pnorm function can be utilized to solve this problem.

#Use X1 -X3 to represent the stock price delta divided by sqrt of days=(365-1).
X1 <- (100-100)/sqrt(364)
X2 <- (110-100)/sqrt(364)
X3 <- (120-100)/sqrt(364)

#The exercise provides the mean and sd:
mean <- 0
sd <- sqrt(1/4)

1. Greater than or equal to 100 => P(Y365−Y1≥0)

pnorm(X1, mean, sd, lower.tail=FALSE)

## [1] 0.5

2. Greater than or equal to 110 => P(Y365−Y1≥10)

pnorm(X2, mean, sd, lower.tail=FALSE)

## [1] 0.1472537

3. Greater than or equal to 120 => P(Y365−Y1≥20)

pnorm(X3, mean, sd, lower.tail=FALSE)

## [1] 0.01801584