a. Build a simulation model in a spreadsheet to calculate the value of the option in today's dollars. Use RSPE to run three separate simulations to estimate the value of the call option and hence the price of the option in today's dollars. For the first simulation, run 100 trials of the simulation. For the second simulation, run 1,000 trials of the simulation. For the third simulation, run 10,000 trials of the simulation. For each simulation, record the price of the option in today's dollars.

b. Elise takes her calculations and recommended price to Michael. He is very impressed, but he chuckles and indicates that a simple, closed-form approach exists for calculating the value of an option: the Black-Scholes formula. Michael grabs an investment science book from the shelf above his desk and reveals the very powerful and very complicated Black-Scholes formula:

V = N [d1] P - N [d2] PV [K]

Where

N[x] = The Excel function NORMSDIST (x) where x = d1
Or x = d2
P = Current price of the stock
K = Exercise price
PV [K] = Present value of exercise price K / (1 + w)t
t = Number of weeks to exercise date
σ = Weekly volatility of stock
Use the Black-Scholes formula to calculate the value of the call option and hence the price of the option. Compare this value to the value obtained in part a.
c. In the specific case of Fellare stock, do you think that a random walk as described above completely describes the price movement of the stock? Why or why not?
Elise Sullivan moved to New York City in September to begin her first job as an analyst working in the Client Services Division of First-Bank, a large investment bank providing brokerage services to clients across the United States. The moment she arrived in the Big Apple after receiving her undergraduate business degree, she hit the ground running-or, more appropriately, working. She spent her first six weeks in training, where she met new First-Bank analysts like herself and learned the basics of First-Bank's approach to accounting, cash flow analysis, customer service, and federal regulations.
After completing training, Elise moved into her bullpen on the 40th floor of the Manhattan First-Bank building to begin work. Her first few assignments have allowed her to learn the ropes by placing her under the direction of senior staff members who delegate specific tasks to her.
Today, she has an opportunity to distinguish herself in her career, however. Her boss, Michael Steadman, has given her an assignment that is under her complete direction and control. A very eccentric, wealthy client and avid investor by the name of Emery Bowlander is interested in purchasing a European call option that provides him with the right to purchase shares of Fellare stock for $44.00 on the first of February-12 weeks from today. Fellare is an aerospace manufacturing company operating in France, and Mr. Bowlander has a strong feeling that the European Space Agency will award Fellare with a contract to build a portion of the International Space Station some time in January. In the event that the European Space Agency awards the contract to Fellare, Mr. Bowlander believes the stock will skyrocket, reflecting investor confidence in the capabilities and growth of the company. If Fellare does not win the contract, however, Mr. Bowlander believes the stock will continue its current slow downward trend. To guard against this latter outcome, Mr. Bowlander does not want to make an outright purchase of Fellare stock now.
Michael has asked Elise to price the option. He expects a figure before the stock market closes so that if Mr. Bowlander decides to purchase the option, the transaction can take place today.
Unfortunately, the investment science course Elise took to complete her undergraduate business degree did not cover options theory; it only covered valuation, risk, capital budgeting, and market efficiency. She remembers from her valuation studies that she should discount the value of the option on February 1 by the appropriate interest rate to obtain the value of the option today. Because she is discounting over a 12-week period, the formula she should use to discount the option is (Value of the Option / [1 + Weekly Interest Rate]12). As a starting point for her calculations, she decides to use an annual interest rate of 8 percent. But she now needs to decide how to calculate the value of the option on February 1.
Elise knows that on February 1, Mr. Bowlander will take one of two actions: either he will exercise the option and purchase shares of Fellare stock or he will not exercise the option. Mr. Bowlander will exercise the option if the price of Fellare stock on February 1 is above his exercise price of $44.00. In this case, he purchases Fellare stock for $44.00 and then immediately sells it for the market price on February 1. Under this scenario, the value of the option would be the difference between the stock price and the exercise price. Mr. Bowlander will not exercise the option if the price of Fellare stock is below his exercise price of $44.00. In this case, he does nothing, and the value of the option would be $0.
The value of the option is therefore determined by the value of Fellare stock on February 1. Elise knows that the value of the stock on February 1 is uncertain and is therefore represented by a probability distribution of values. Elise recalls from a management science course in college that she can use computer simulation to estimate the mean of this distribution of stock values.
Before she builds the simulation model, however, she needs to know the price movement of the stock. Elise recalls from a probability and statistics course that the price of a stock can be modeled as following a random walk and either growing or decaying according to a lognormal distribution. Therefore, according to this model, the stock price at the end of the next week is the stock price at the end of the current week multiplied by a growth factor. This growth factor is expressed as the number e raised to a power that is equal to a normally distributed random variable. In other words:
sn = eNSc
Where
sn = The stock price at the end of next week
sc = The stock price at the end of the current week
N = A random variable that has a normal distribution

Transcribed Image Text:

In[P/PV[K]] ¸ oVi di oVi 2 dz = d, - oVi