A European call option gives a person the right to buy a particular stock at a given price (the strike price) on a specific date in the future (the expiration date). This type of call option is typically sold at the NPV of the expected value of the option on its expiration date. Suppose you own a call option with a strike price of $54. If the stock is worth $59 on the expiration date, you would exercise your option and buy the stock, making a $5 profit. On the other hand, if the stock is worth $47 on the expiration date, you would not exercise your option and make $0 profit. Researchers have suggested the following model for simulating the movement of stock prices:

Pk+1 = Pk(1 + mt + zα√1)

Where

Pk = price of the stock at time period k

m = v + 0.5 α2

v = The stock's expected annual growth rate

α = The standard deviation on the stock's annual growth rate

t = time period interval (expressed in years)

z = a random observation from a normal distribution with mean 0 and standard deviation of 1.

Suppose a stock has an initial price (P0) of $80, an expected annual growth rate (v) of 15%, and a standard deviation (α) of 25%.

a. Create a spreadsheet model to simulate this stock's price behavior for the next 13 weeks (note t = 1/52 because the time period is weekly).

b. Suppose you are interested in purchasing a call option with a strike price of $75 and an expiration date at week 13. On average, how much profit would you earn with this option?

c. Assume a risk-free discount rate is 6%. How much should you be willing to pay for this option today?

d. If you purchase the option, what is the probability that you will make a profit?