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Binomial Trees

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Problem 1: Binomial Trees and Real Options

Suppose that the spot price, 6-month forward price, 12-months forward price and 18-months forward price for wheat are 250, 260, 270.4 and 281.216 cents per bushel, respectively.[1] A farmer has a project that involves an immediate expenditure of $20,000 and a further expenditure of $90,000 in six months. The farmer collects two wheat harvests per year, and if this project is implemented, it will increase wheat that is harvested and sold by 20,000 bushels in one year and in 18 months (as compared to his typical harvest). Assume that there is no uncertainty about the harvests (all uncertainty comes from future wheat prices). The continuously compounded risk-free rate is 5% p.a.

a) If the project has no flexibility (a “yes or no” decision), what is the value of the project? Notice that you do not have to use the binomial tree to answer this question. b) Suppose that the wheat price follows a binomial tree process with parameters u=1.1 and d=0.95 (determine y from forward prices and remember to take this variable into account when computing risk-neutral probabilities). Suppose that the farmer can abandon the project in six months and avoid paying the $90,000 cost at that time (in this case, the increase in harvest will be 5,000 bushels per year instead of 20,000). What is the value of this abandonment option? Is this abandonment option a put or a call? What is its strike? Explain. c) Now, assume that there is no option to abandon. Instead, suppose that the project can be delayed for 6 months: the investments can be made in six months and in one year, and the increases in harvests will be observed in 18 months and 2 years. What is the value of this option to delay? d) If the option to abandon and the option to delay are both available, what is the value of the project? Is the value of the portfolio of these two options equal to the sum of their individual values? Explain why.

Problem 2:

Suppose that a company has a project whose value of distributed lognormally with volatility (. The current value of the project is Vt. The project has a constant continuous payout rate (. Suppose the company has an opportunity to increase the size of the project by 20% at cost K1. There is a one-time chance to make the expansion decision T years from now.

a) How much does this type of flexibility add to the company value? Explain.

Suppose the company has a choice between increasing the size of the project by 20% at cost K1 and increasing it by 50% at cost K2 (2K2>5K1). How much does this type of flexibility add to the company value? Explain. (Hint: this is equivalent to having a chooser option with two underlying calls. Show that this chooser option is equivalent to the portfolio consisting of 0.2 units of a European call option written on one unit of the project with strike 5K1 and 0.3 units of a European call option written on one unit of the project with strike [pic]. Then, the prices of these options can be estimated using the BS formula.)

Problem 3:

Suppose that a company is considering replacing each of its existing six widget machines with new machines. The new machines cost $1,000 each and have a five-year life. The management has estimated the expected per-machine cash flows; these flows are defined as the incremental cash flow of replacing a single old machine by a new machine and include the sale of the old machine. The management does not know the exact realization of these annual cash flows, but knows their means. The expected cash flows for each new machine over their lives are: $220, $300, $400, $200 and $150 (these cash flows are uncertain from the firm’s managers’ viewpoint just because the managers have no information about the performance of these machines and there are no other sources of uncertainty).
The financial analyst working on the replacement project has estimated a cost of capital for the project of 12% (the discount rate). a) What is the NPV of this replacement project? b) Suppose that supplier of the new machines gives the following option to the manager in charge of the widget line: replace one machine at cost $2,000 now and get an opportunity to replace the other five machines next year at cost $1000. Assuming that the Black-Scholes formula provides a good approximation, and (=20% (the volatility of the replacement project’s return), rf=6%, what is the value of the project? What is the intuition behind this increase in the project’s value? If you need to make any extra assumptions, state them explicitly.

[1] The forward price of a commodity is given by [pic], where y is the difference between the storage cost and the convenience yield. The variable y corresponds to a (negative of a) dividend yield ( in the case of a dividend-paying stock; however, it is typically positive for commodities. If y is not equal to zero, the expected growth rate of a commodity price under the risk-neutral probability measure is equal to [pic]:
Alternatively, [pic] is the growth rate of the “cum-convenience-yield” commodity price [pic].…...

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