Question 1 (15pts)
A stock is priced at $100. It pays a continuous dividend of 2% per year. The riskless rate is 3% per year continuously compounded. The volatility of the stock is 23% per year.
a) [5pts]Construct a 2 period binomial lattice over a six month time horizon showing the stock prices. Identify the values for u, d and the risk neutral probability, p as well as the discount factor for each period.
b) [5pts] Compute the price of an American call option with strike $90 that expires in six months. Show the option prices at each node of the lattice and indicate the nodes where the option should be exercised.
( c ) [5pts] Compute the price of an European call option on the stock with strike 90 for the above problem. Establish the initial replicating portfolio for the European call. Show your calculations and demonstrate that this portfolio does indeed replicate the payout of the European call option over the first period.
Question 2 (10 points)
[5pts] (a) Suppose that observations on a stock price (in dollars) at the end of each of 6 consecutive weeks are as follows:
20.0; 21.2; 23.6; 21.9; 20.5; 21.7
Estimate the annualized volatility of logarithmic returns that is used in the Black Scholes formula. Show your calculations.
[5pts] (b) The plot of the implied volatility for one year options against the stock price was flat, but now has taken the shape of a smile. That is it is U shaped. Provide some reasons for the change in shape. Be very clear and precise.
[10pts] Question 3
The Webber company is an international conglomerate with a real estate division that owns the right to erect an office building on a parcel of land in downtown Cleveland over the next year. The building would cost 55 million dollars to construct. Due to low demand for office space such a building would be worth 53.5 million dollars today. If demand increases, the building would be worth 60 million dollars a year from now. However, if demand decreases the same building would be worth 49.2 million dollars. The firm can borrow and lend at the risk free annual effective rate (i.e simple rate) of 4.5 percent. A local competitor in the real estate business has recently offered 1.3 million for the right to build an office building on the land. Should the company accept this offer. Provide details explaining your results.
Question 4 [10 pts]
A European call option and put option on a stock both have a strike price of 20 dollars and an expiration date of 3 months. Both sell for 3 dollars. The risk free rate for all maturities is 10% per year continuously compounded and remains unchanged over time. The current stock price is 19 dollars. A one dollar dividend is expected in one month. Identify the arbitrage opportunity open to a trader and provide the exact details of generating the quantity. Be very specific showing the exact trading strategy.
Question 5 (15pts)
A project has two phases each of one year duration. You pay 15 dollars today. The first phase has a 60% chance of success. If successful, the firm can pay 30 dollars for the second phase. The second phase has a 20% chance of being successful. In this case, the firm can pay 100 dollars. Viewed from date 0, the expected revenue in year 2 of a successful project is estimated to be 500 dollars.
The volatility of the logarithmic returns on the revenues is sigma = 0.693147. The riskless rate is zero. The risk of the project is fully diversifiable.
(a) Establish the lattice of present values for a successful project.
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