PRICING UNDER PRESSURE
Elise Sullivan moved to New York City in September to begin her first job as an analyst working in the Client Services Division of FirstBank, a large investment bank providing brokerage services to clients across the United States. The moment she arrived in the Big Apple after graduating with an undergraduate degree in industrial engineering that included a concentration in finance, she hit the ground running-or more appropriately-working. She spent her first six weeks in training, where she met new FirstBank analysts like herself and learned the basics of FirstBank's approach to accounting, cash flow analysis, customer service, and federal regulations. After completing training, Elise moved into her bullpen on the fortieth floor of the Manhattan FirstBank 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 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.
(a) Elise knows that on February 1, Mr. Bowlander will take one of two actions: either he will exercise the option to 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 an operations research course in college that she can use 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:
To begin her analysis, Elise looks in the newspaper to find that the Fellare stock price for the current week is $42.00. She decides to use this price to begin her 12-week analysis. Thus, the price of the stock at the end of the first week is this current price multiplied by the growth factor. She next estimates the mean and standard deviation of the normally distributed random variable used in the calculation of the growth factor. This random variable determines the degree of change (volatility) of the stock, so Elise decides to use the current annual interest rate and the historical annual volatility of the stock as a basis for estimating the mean and standard deviation
The current annual interest rate is r = 8 percent, and the historical annual volatility of the aerospace stock is 30 percent. But Elise remembers that she is calculating the weekly change in stock-not the annual change. She therefore needs to calculate the weekly interest rate and weekly historical stock volatility to obtain estimates for the mean and standard deviation of the weekly growth factor. To obtain the weekly interest rate w, Elise must make the following calculation:
The historical weekly stock volatility equals the historical annual volatility divided by the square root of 52. She calculates the mean of the normally distributed random variable by subtracting one half of the square of the weekly stock volatility from the weekly interest rate w. In other words:
The standard deviation of the normally distributed random variable is simply equal to the weekly stock volatility. Elise is now ready to build her simulation model.
(1) Describe the components of the system, including how they are assumed to interrelate.
(2) Define the state of the system.
(3) Describe a method for randomly generating the simulated events that occur over time.
(4) Describe a method for changing the state of the system when an event occurs.
(5) Define a procedure for advancing the time on the simulation clock.
(6) Build the simulation model to calculate the value of the option in today's dollars.
(b) 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 iterations of the simulation. For the second simulation, run 500 iterations of the simulation. For the third simulation, run 1,000 iterations of the simulation. For each simulation, record the price of the option in today's dollars
(c) 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:
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 (b).
(d) 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?